From cube-lovers-errors@mc.lcs.mit.edu  Thu Sep 17 17:18:28 1998
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Date: Mon, 14 Sep 1998 14:14:22 -0400 (Eastern Daylight Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re: Weak Local Maxima, 6f from Start
In-Reply-To: <14Sep1998.111621.Hoey@AIC.NRL.Navy.Mil>
To: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9809141422.E@GN209A.PSTCC.CC.TN.US>

> It turns out that 6f is indeed the shortest.  There are two such positions
> unique to symmetry which are 6f from Start, the Pons and one other.  The
> other one is quite pretty:
>
> L2 R2 D2 U2 B' F  (6f*)
>

I didn't notice it originally, but this position is in the slice
subgroup, and is only one slice move from Pons.  Half turns such
as L2 can be written equally well as either LL or as L'L',
so we can write (L2 R2) as (L'R)(L'R) and (D2 U2) as (D'U)(D'U).
Thus, the weak local maximum 6f from Start can be written as five
slices, one slice short of Pons.

L'R  L'R  D'U  D'U  B'F

All we would have to do to get the Pons would be to add one more
B'F slice.

----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Thu Sep 17 20:27:13 1998
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From: "Chris and Kori Pelley" <ck1@home.com>
To: "Cube Lovers" <cube-lovers@ai.mit.edu>
Subject: DOGIC solution
Date: Mon, 14 Sep 1998 18:01:56 -0400
Message-Id: <000a01bde02b$49b5aae0$da460318@CC623255-A.srst1.fl.home.com>

Using Noel Dillabough's PUZZLER program for MS Windows, I was able to verify
the basic moves needed to solve the new DOGIC puzzle.

SPOILER WARNING!  If you wish to solve the puzzle yourself, read no further.


What seems fairly obvious is that the DOGIC is essentially a superset of
Impossi-Ball, which is basically the corners of MegaMinx.  On DOGIC,
however, these corners have been flattened and could more properly be called
"centers."  Using classic 3x3x3 techniques, you can position and orient
these pieces using the following moves:

Center 3-cycle:

(R' U L U') (R U L' U')

Center orientation (pair):

(R' D R) (F D F') U'
(F D' F') (R' D' R) U

Note that these faces refer to the large pentagons and must be "translated"
to fit the dodecahedral nature of DOGIC.  R, U, and L form a horseshoe, and
F intersects all three.  The D face is not really D, in fact it touches the
U face at one point.

The remaining triangular pieces turn out to be fairly trivial, and any two
can be swapped with the simple sequence:

R u R' u'

In this case, R is a large pentagon and u is any intersecting smaller
pentagon.

A general strategy would be to manually place the top "centers" followed by
their adjacent centers (if you have solved ImpossiBall this should not be
difficult).  Then apply the first two moves above to complete the remaining
centers.  Finally, place all the smaller triangles with the third move.

Despite having more permutations than most magic puzzles, DOGIC seems to be
fairly easy to solve.

Chris Pelley
ck1@home.com
http://www.chrisandkori.com

From cube-lovers-errors@mc.lcs.mit.edu  Fri Sep 18 14:29:06 1998
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Date: Tue, 15 Sep 1998 12:21:58 -0400 (EDT)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199809151621.MAA19286@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Two-face and three-face subgroups

I've been playing with the two-face subgroup [%] of the 3-Cube and got
to wondering - how much work has been done on the two-face and
three-face subgroups?  Certainly the two-face subgroup "feels" like a
much smaller object than even the 2-Cube (though perhaps more tedious
for human solution), perhaps about the size of the Pyraminx.

[%] Okay, strictly speaking there are two different two-face subgroups,
    but one of them is not even the least bit interesting.

And what about the three-face subgroups?  Certainly the three- and
four-face subgroups are smaller than the whole Cube group, though ISTR
that the five-face (sub)group is actually the whole thing.  But how
much smaller, and how difficult of human solution?  I'd expect one of
the three-face groups (the one involving two opposite faces - call it
the L-F-R one) to be more tedious but no more difficult than the
two-face group, whereas the other one (involving one face from each
pair of opposite faces - U-F-R, say) should have more interest.

In particular, the two-face subgroup is smaller than the set of all
position that leave unchanged the cubies that the two-face subgroup
never touches.  (To put it another way, I'm saying that the subgroup
generated by {R,F} is smaller than the set of positions of the full
group that leaves unmoved the 11 cubies that don't touch either of
those two faces - 7 if you don't count face cubies.)  I can see a
factor of 128 smaller, since it's not possible to flip edge cubies in
the two-face group, but I haven't thought about the corners, so it may
be even smaller than that.

What about the three-face subgroups?  The L-F-R subgroup is also
smaller, if for no other reason than an inability to flip edge cubies,
like the two-face group.  But is the U-F-R subgroup the same as the
subset of the full group that leaves untouched the 7 (4 if you don't
count face centers) cubies in the DBL corner?

What about human solvability?  I've taught myself to solve the two-face
group, and with the tools I developed (largely powers, reorientations,
and inverses of F' R' F R) I feel confident I can handle the L-F-R
three-face group or even the L-F-R-B four-face group.  Can anyone
comment on how humanly difficult the U-F-R group, or for that matter
the U-F-R-L four-face group, is?

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

From cube-lovers-errors@mc.lcs.mit.edu  Tue Sep 22 16:10:10 1998
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Date: Fri, 18 Sep 1998 15:52:33 -0400 (EDT)
From: Nicholas Bodley <nbodley@tiac.net>
To: Hana Bizek <hbizek@ameritech.net>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Rubik's cube kingdom
In-Reply-To: <35F350B1.626F@ameritech.net>
Message-Id: <Pine.SUN.3.95.980918154427.13428O-100000@sunspot.tiac.net>

My apologies for a delayed reply.

Hana's essay was rather philosophical, and contained some uncommon
points of view; it was appropriate, in my opinion.

One aspect of the Cube (and related puzzles) that seemed to be ignored
is the remarkable ingenuity of their internal mechanisms. I maintain
that the mechanism of the original (i.e., 3^3) Rubik's Cube is one of
the most ingenious ever invented.

I recall being very fatigued, riding the West Side IRT subway in NYC
about 2 AM, perhaps, and catching sight of someone manipulating what
must have been one of the very first Cubes, probably from Hungary*. I
was fairly sure I wasn't hallucinating, but was very troubled that what
I'd seen simply appeared impossible. I've been a somewhat-casual student
of mechanisms all my life.

*This was probably several weeks, or more, before the Scientific
American article, and the later explosion of its popularity.

My regards to all,

|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Tue Sep 22 18:09:16 1998
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Date: Thu, 17 Sep 1998 00:03:47 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: More on Calculating Weak Local Maxima
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Message-Id: <Pine.PMDF.3.95.980916234641.7009A-100000@PSTCC6.PSTCC.CC.TN.US>

In the process of adding the code to my God's Algorithm program to
calculate weak local maxima in the face turn metric, I realized that the
algorithm I posted previously to do so was incomplete in one subtle but
very important respect.  This message will provide the missing piece to
the algorithm.

I have posted much of this before, but my program is in general jumping
ahead by more than one move at a time.  For example, suppose we can store
all the positions up to five moves from Start.  Then, we can determine all
the positions which are eight move s from Start by calculating all the
products xy where x is a position of length five and y is a position of
length three.

Obviously, just because the length of x is five and the length of y is
three does not mean that the length of xy is eight.  In fact, the length
of xy could be anywhere from two through eight.  To determine the true
length of xy, we compare xy to the stored positions of length two, three,
four, and five.  In addition, we compare xy to the calculated positions of
length six and seven, which are calculated in the same manner as is xy.
If xy fails to match all such shorter positions, then its length is indeed
eight.

Next we focus on the quarter turn metric.  For some fixed q in Q, the set
of twelve quarter turns, what is the length of xyq if the length of xy is
eight with the length of x equal to five and the length of y equal to
three?  First of all, it must be either seven or nine.  Second of all, the
length of yq must be either two or four.  If the length of yq is two, then
we know that the length of xyq must be seven.  But if the length of yq is
four, then we are still not sure.  The reason is that there might be some
u not equal to x of length five and some v not equal to y of length three
such that xy=uv, but where the length of vq is two.  If so, then the
length of xyq is the same as the length of uvq which is seven.

The basic idea is that if z=xy where the length of x is five and the
length of y is three, then there may be many, many x and y pairs of length
five and three respectively whose product yields z.  The length of zq is
nine only if for every such y the length of yq is four.  Even if all but
one yq is of length four, it only takes one yq of length two to spoil the
pudding, as it were.

The mechanism which I have posted previously to capture this concept is
the Ends-with function E(z).  E(z) is defined to the be set of all moves
with which a minimal maneuver for z can end.  So in the case at hand,
since the length of z is eight, the length of zq is nine only if E(z) does
not contain q'.  E(z) can be calculated in the case at hand as the union
of E(y) taken over all the y values of length three which can be composed
with an x of length five to create z.  Therefore, to say that E(z) does
not contain q' is the same thing as saying that none of the E(y) contain
q'.

So far, so good and there is nothing new here which I haven't posted
before.  But let's consider the exact same issue in the face turn metric.
If the length of x is five and the length of y is three, then the length
of xy can be in the range of two through eight as before.  And as before,
if we compare xy with all positions of length two through seven without
finding a match, then the length of xy is indeed eight.

But this time we need to consider xyf, where f is some fixed face turn in
the set Q+H of twelve quarter turns and six half turns.  What is the
length of xyf?  For starters, it is either seven or eight or nine.  Also,
the length of yf is two or three or four.

If the length of yf is two, then the length of xyf is guaranteed to be
seven.

If the length of yf is three, then the length of xyf is guaranteed to be
no more than eight.  But the length nevertheless might be seven, because
as in the quarter turn case, there may be some u of length five and some v
of length three such that uv=xy, but such that the length of vf is only
two.  If so, the length of xyf is the same as the length of uvf which is
guaranteed to be seven.

The definition of Ends-with is the same in the face turn case as in the
quarter turn case, namely E(z) is the set of all face turns with which a
minimal maneuver for z can end.  If z=xy then E(z) can be calculated as
the union of E(y) over all the y value s of length three which can be
combined with an x value of length five to form z.  To say that the length
of zf is at least eight is the same thing is saying that E(z) does not
contain f' which is the same thing as saying that none of the E(y) contain
f'.

Next, let's suppose that indeed E(z) does not contain f'.  We are still
left with the issue of whether the length of z is eight or nine, having
eliminated seven as a possibility.  The test is still the length of all
the yf, with a length of two having been eliminated as a possibility.  If
all of the yf are of length 4, then xyf is of length nine.  But if even so
many as one of the yf are of length three, then xyf is of length 8.

The mechanism I have posted before to capture this concept is the
Ends-with2 function.  E2(z) is a little tricky to describe.  Informally,
we might say that E2(z) is the set of all f in Q+H with which z can end
without changing it's length.  It is probably better to say that E2(z) is
the set of all f in Q+H such that the length of zf' is the same as the
length of z.  The technique which I have posted before (and which I must
now correct) to calculate E2(z) is to form the union of E2(y) over all y
values of length three which can be combined with an x value of length
five to form z.

If the length of zf is eight or nine, then this mechanism is fine.  But if
the length of zf turns out to be seven, there is a problem.  That is,
there may be one y where the length of yf is two and where E(y) contains
f, and there may be another y where the length of yf is is three and where
E2(y) contains f.  In such a case, both E(z) and E2(z) would contain f.
Hence, we must always calculate E(z) prior to calculating E2(z), and we
must omit from E2(z) any f values which are already contained in E(z).

With this correction, everything works.  A local maximum is a position z
for which |E(z)|+|E2(z)|=18, a strong local maximum is a local maximum z
for which |E(z)|=18 and |E2(z)|=0, and a weak local maximum is a local
maximum z for which |E(z)| < 18 and |E 2(z)| > 0.  All my examples have
been specific to y values of length 3 for clarity of exposition, but the
calculation of E(z) and E2(z) is totally general, and is the union of E(y)
and E2(y), respectively, over all y values which can be used to form a z
of the form z=xy, and with any values which are in E(z) omitted from
E2(z).

Finally, my programs also calculate a Starts-with and a Starts-with2
function, which are defined analogously.  The same correction must be made
to the Starts-with2 function as was made for the Ends-with2 function.
Equivalently, we can define S(z)=E'(z') and S2(z)=E2'(z'), where E' and
E2' are the set of all inverses of the elements of E and E2, respectively.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
Pellissippi State                            (423) 539-7198
10915 Hardin Valley Road                     (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990

From cube-lovers-errors@mc.lcs.mit.edu  Tue Sep 22 19:05:00 1998
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Date: Sat, 19 Sep 1998 09:13:58 -0400 (Eastern Daylight Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re: Two-face and three-face subgroups
In-Reply-To: <199809151621.MAA19286@Twig.Rodents.Montreal.QC.CA>
To: der Mouse <mouse@rodents.montreal.qc.ca>
Cc: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9809190958.A@GN209A.PSTCC.CC.TN.US>

On Tue, 15 Sep 1998 12:21:58 -0400 (EDT) der Mouse
<mouse@rodents.montreal.qc.ca> wrote:

> I've been playing with the two-face subgroup [%] of the 3-Cube and got
> to wondering - how much work has been done on the two-face and
> three-face subgroups?  Certainly the two-face subgroup "feels" like a
> much smaller object than even the 2-Cube (though perhaps more tedious
> for human solution), perhaps about the size of the Pyraminx.
>

The <U,R> subgroup has been explored fairly thoroughly.  For
example, look in the archives 8/31/1994 for a summary of the
first complete God's Algorithm search of this particular
subgroup.  There are a number of articles in the archives
thereafter.  <U,R> has been searched in both the quarter turn
metric and the face turn metric, and local maxima have been
investigated as a part of the search.

<U,R> has a very small branching factor and a corresponding
large diameter of 25 in the quarter turn metric, at least I
think it's a large diameter for such a small group. Until Mike
Reid recently showed that the diameter of G in the quarter turn
metric was at least 26, the diameter of <U,R> was the largest
known for the 3x3x3 cube or any of its subgroups.

Frey and Singmaster's book discusses both two face and three
face subgroups, among other things giving their sizes.

To my knowledge, no God's Algorithm searches have been performed
for the three face subgroups.

We have |<U,R>|=73483200, so <U,R> is slightly smaller than the
corners group at 88179840.  The 2x2x2 is 24 times smaller than
the corners group, at 3674160.  However, I am of the school
of thought that tends not to equate the size of the group (or
search space, for problems that are not actually groups) with
the difficulty of the problem.

----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Tue Sep 22 19:50:07 1998
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Date: Mon, 21 Sep 1998 16:34:29 -0400 (Eastern Daylight Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Local Maxima which Fix the Corners, 12q from Start
To: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9809211629.A@GN209A.PSTCC.CC.TN.US>

I am making a run to calculate God's Algorithm out to 12 moves
from Start in the quarter turn metric.  It has been running
several weeks, and will probably run several more.

I have made some changes to my program to make it easier to
extract the positions for local maxima, and to checkpoint the
local maxima data.  As a part of the checkpointing, I can
actually see the local maxima as they are generated without
having to wait for the program to end.

It is becoming apparent that there are a *lot* of local maxima
12q from Start.  It is already known that there are only four
(unique to symmetry) which are 10q from Start (the shortest ones
in the quarter turn metric), and that there are none 11q from
Start. So I am a little surprised that I am seeing so many.

I have looked at quite a few of them, and most of them are not
all that interesting.  But the ones which fix the corners are
all quite pretty.  Because the positions are being produced in
lexicographic order, and because I am sorting by corners first,
edges second, the positions which fix the corners are the first
ones to appear.  There are eight of them as follows.

1.  F2 L2 F2 B2 R2 B2
2.  F  B' U2 D2 F' B  R2 L2
3.  F  B  R2 F' B' U  D  L2 U' D'
4.  D' F  B' R  F  R' F' B  U  F'  U' D
5.  F  B  R2 L2 F  B  U2 D2
6.  R  L' F2 B2 R  L' F2 B2
7.  F2 B2 U2 D2 R2 L2
8.  R  L' U  D' F  B' R2 L2 U  D'

#1 is a 2-H pattern (only four edge cubies are moved).

#2 is a 4-H.

#3 moves four edge cubies, leaving eight of the nine
facelets the same color on four faces, and a solid color on the
other two faces.

#4 moves three edge cubies, leaving eight of the nine facelets
the same color on all six faces.

#5 has 2 H's, 2 checkerboards, and 2 solid faces  -- with the
respective H's, checkerboards, and solid faces opposing each
other.

#6 has 4 H's and 2 checkerboards, with the 2 checkerboards
opposing each other.

#7 is the Pons Asinorum, and is included only for completeness
because we already knew that the Pons was a local maximum of
length 12q.

#8 has all six faces being sort of a "three colored
checkerboard".

Some of these positions may have appeared on Cube-Lovers in some
other context, but the only one I recognize for sure is the Pons.
In some ways, #4 is the most interesting to me, because it a
simple 3-cycle on the edges, and who would have thought that
such a position would turn out to be a local maximum?  #1 and #3
both consist of two 2-cycles on the edges, and are about as
striking to me as is #4.

----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Wed Sep 23 12:37:03 1998
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Date: Tue, 22 Sep 1998 13:47:17 -0700 (PDT)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: Moves to this pattern
To: Cube-Lovers@ai.mit.edu

Hi I just joined the list!

I was wondering if someone could help me with a
pattern that has been bugging me for a while.
I have been able to solve the cube to this pattern so
i know it is a valid one.
I used one of those on-line solvers entered the
pattern and then reversed the sequence the solver
gave me and it indeed gets the pattern i want but
somehow it seemed too many moves for me for such a
simple pattern.
the pattern I am talking about is: the 4 cubelets
that make the vertex formed by FDR are exchanged with
the vertex from BDL

Can any one give me the set of moves to get to this
pattern from a solved cube? Does this pattern have a
name?

[Here is a] set of moves (they work but I am
sure there is a shorter way):

D2 F B- L2 F- B D- F B- L F- B D2 F B- L2 F- B D F B-
L- F- B F B- L- F- B D2 F B- L- F- B D2 R- D- R D- R-
D2 R D2 L- D- L B D- B- L- D L2 D L- D- F- D- F D B-
D- B D R D R- D F- B D B- F R D- R- T B- T- B- D- B-

This long set of moves reminds me of something else:
There are many Rubik cube annimations that you move
the faces with either the keypad or the mouse. Does
anyone know of one that follows sets of orders you
write?

It would be neat to try this long patterns in one of
those simulators.

Thanks
===
Jorge E Jaramillo

From cube-lovers-errors@mc.lcs.mit.edu  Wed Sep 23 19:54:38 1998
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Date: Tue, 22 Sep 1998 14:32:21 -0700 (PDT)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: Is it only mine?
To: cube <Cube-Lovers@ai.mit.edu>

In order to solve the cube faster I developed a
method that would turn a lot the middle faces. I
don't know how you call them in this list I am
talking about the face between Top and Bottom, the
face between Right and Left and even the face between
Front and Back.
Well after twisting it a few times my cube came
undone, fell apart and I thought "Damn made in Taiwan
cubes" (although the ones I can buy here do not say
where are they made). I was going through my second
cube in a short while (I just regained interest in
the cube a short while ago after say 15 years) and it
fell apart again. I took off the plastic color of the
center cubelet that came apart and found a screw and
a spring that keeps the screw tight. I re screwed it
but did not have any glue at hand so I kept on
playing without the color of the center cubelet I was
doing one of these center face moves and saw how the
screw was turning counterclockwise in other words the
way that makes the cube fall apart. Needless to say I
had to re design my method.
This long story is to ask if all the cubes are built
this way or only the ordinary ones I can buy here?
Thanks.


===
Jorge E Jaramillo

From cube-lovers-errors@mc.lcs.mit.edu  Fri Sep 25 18:05:43 1998
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Message-Id: <360964EA.D07A79F8@t-online.de>
Date: Wed, 23 Sep 1998 23:15:22 +0200
Reply-To: Rainer.adS.BERA_GmbH@t-online.de
Organization: BERA Softwaretechnik GmbH
To: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Moves to this pattern
References: <19980922204717.20206.rocketmail@web1.rocketmail.com>
From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring)

Jorge E. Jaramillo wrote:

> the pattern I am talking about is: the 4 cubelets
> that make the vertex formed by FDR are exchanged with
> the vertex from BDL.

R2 U  R2 U2 B2 D  L2 U2 L2 B2 D' B2 U2

Rainer adS

PS
If you have a WINTEL system you should download Herbert Kociemba's cube
program.

[ Moderator's note: Jerry Bryan provides another 13f process,
R2 U' B2 R2 U2 R2 U' B2 U2 R2 U' B2 U2, which hasn't been proven optimal.
Steve LoBasso has a longer one. ]

From cube-lovers-errors@mc.lcs.mit.edu  Sat Sep 26 00:05:04 1998
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Date: Fri, 25 Sep 1998 08:36:22 -0400 (Eastern Daylight Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Summary of Local Maxima, Face Turn Metric
To: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9809250822.A@GN209A.PSTCC.CC.TN.US>

I have posted maneuvers for a number of specific strong and weak 
local maximal positions (the strong local maxima at 9f and 10f, 
and the weak local maxima at 6f) , but I haven't really posted a 
summary of the numbers. Here are the numbers I have so far.  In 
order to complete the table through 10f from Start for weak 
local maxima, I would have to repeat a rather long run.  As 
might be expected, it appears that the number of weak local 
maxima will greatly exceed the number of strong local maxima.

As is the usual case, patterns are M-conjugacy classes (symmetry 
classes), and represent the number of positions which are unique 
up to symmetry.



Distance   Strong    Strong     Weak      Weak
  from     Lclmax    Lclmax     Lclmax    Lclmax
  Start   Patterns   Positions  Patterns  Positions

   0f          0        0          0          0
   1f          0        0          0          0
   2f          0        0          0          0
   3f          0        0          0          0
   4f          0        0          0          0
   5f          0        0          0          0
   6f          0        0          2          7
   7f          0        0          1          6
   8f          0        0         37        739
   9f          2       32        327      13014
  10f          6      107          

----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Tue Oct  6 15:05:42 1998
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Date: Sun, 27 Sep 1998 20:35:33 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Corners Only, Ignoring Twist
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Reply-To: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Message-Id: <Pine.PMDF.3.95.980927193742.73560A-100000@PSTCC6.PSTCC.CC.TN.US>

I have been playing around with the idea of trying to calculate God's
Algorithm all the way to the bitter end for the group which results from
ignoring all twists of the corners and flips of the edges.  It's a pretty
big group.  The order is |G|/(3^7)/(2^11), which is about 9.7*10^12, call
it about 10^13 to make it a round number.  (Another way to calculate it is
8!12!/2.)

This is probably right at the bare edge, maybe even slightly past the bare
edge, of the size of problem I can handle right now, which makes it a
worthy endeavor.  Also, it would provide a lower limit on the diameter of
G (although the limit might not be any better than the ones we already
have), which again makes it a worthy endeavor.

Such a result might be suitable as the estimator function required by IDA*
searches.  The distance from Start in the no-twist, no-flip group would
certainly be a lower bound for every position where twist and flip *are*
considered.  My only hesitation about suggesting this group as a suitable
IDA* estimator function is that there are obvious pathological cases such
as the superflip where this function would be a very poor estimator.

In developing a no-twist, no-flip version of the program, I decided to try
it out on the corners only case.  Here are the results.


Distance
 from           Patterns     Positions
Start


  0q                   1               1
  1q                   1              12
  2q                   5             114
  3q                  24             876
  4q                 119            4931
  5q                 301           12972
  6q                 364           15066
  7q                 166            6300
  8q                   3              48



Distance
 from           Patterns       Positions
Start

 0f                    1               1
 1f                    2              18
 2f                    9             243
 3f                   68            2646
 4f                  302           12516
 5f                  418           17624
 6f                  170            7080
 7f                   14             192

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
Pellissippi State                            (423) 539-7198
10915 Hardin Valley Road                     (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990

From cube-lovers-errors@mc.lcs.mit.edu  Tue Oct  6 16:52:26 1998
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Date: Fri, 2 Oct 1998 23:18:37 +0200 (METDST)
From: Martin Moller Pedersen <tusk@daimi.aau.dk>
Message-Id: <199810022118.XAA19053@stargazer.daimi.aau.dk>
To: cube-lovers@ai.mit.edu
Subject: cubes at spielmessen in Essen

There will soon be a big gathering for games in Germany - Essen a so-called 
Spielmessen.

I am attending this gathering for the first time in three years so I am
looking for companies who will came to the spielmessen and who sells cubes.

The places is big so it would be nice for me to have same names to look for.

and hopefully I will have a real 4x4x4 and 5x5x5 cube to play with in a few weeks :-)

/Martin

From cube-lovers-errors@mc.lcs.mit.edu  Tue Oct  6 18:38:32 1998
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Message-Id: <v04003a01b23d5b4d06f8@[198.252.208.218]>
Date: Sun, 4 Oct 1998 11:12:27 -0600
To: cube-lovers@ai.mit.edu
From: Steve LoBasso <slobasso@dtint.com>
Subject: Cube Solver for Macintosh

I just wrote a Macintosh port to Dik T. Winter's cube solving code.
I put it on my web page at the link below. I haven't had a chance to make
an info page for it so the link below is just the application.

It will run on both 68k and PPC Native.
Be warned the 68k version runs fairly slow and the initialization phase
takes quite a while.

Cube Solver

By Steve LoBasso
  slobasso@dtint.com

Written using algorithm code by
Dik T. Winter based on algorithm described by Herbert Kociemba.

http://members.tripod.com/~slobasso/downloads/Cube_Solver.hqx

--
Steve LoBasso                          mailto:slobasso@dtint.com
Digital Technology International    or mailto:slobasso@hotmail.com
500 West 1200 South, Orem, UT, 84058   http://members.tripod.com/~slobasso
(801)226-6142  ext.265                 FAX (801)221-9254

From cube-lovers-errors@mc.lcs.mit.edu  Tue Oct  6 20:00:34 1998
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From: "Chris and Kori Pelley" <ck1@home.com>
To: <cube-lovers@ai.mit.edu>
Subject: That's Incredible!
Date: Sun, 4 Oct 1998 22:37:28 -0400
Message-Id: <002701bdf009$180cb5e0$da460318@CC623255-A.srst1.fl.home.com>

I recently obtained (courtesy of Peter Beck) the Rubik's Cube-a-Thon video
from the TV show "THAT'S INCREDIBLE" and digitized it in RealVideo format.
The file is rather large (18.1 megabytes) but it's worth a download if
you're into cubic nostalgia.  Eleven and a half minutes long, it features
Minh Thai, Jeff Varasano, Kris Wunderlich, and others that may be on this
list.

Here's the URL:

http://www.chrisandkori.com/incredible.htm

It requires the RealPlayer 5.0 or later to view it.  Note that you must
download the file, then view it.  I do not have a streaming video server.
If anyone would like to host the file on a streaming server, please contact
me.

Chris Pelley
ck1@home.com

From cube-lovers-errors@mc.lcs.mit.edu  Thu Oct  8 19:04:05 1998
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Date: Thu, 8 Oct 1998 15:11:54 +0100
From: David Singmaster       <david.singmaster@sbu.ac.uk>
To: cube-lovers@ai.mit.edu
Message-Id: <009CD656.72A81016.35@ice.sbu.ac.uk>
Subject: Davenport's pattern

	The pattern given by Jacob Davenport is what I called a cube in a cube
in a cube.  I discovered this in 1979 or 1980 and was very pleased with it.
Indeed, I used the cube in a cube as the logo of the late and much lamented
David Singmaster Ltd. in 1980-1982 (approx. dates since I'm not where my
records are).  The pattern is in my Notes.  There
are various ways to generate the pattern, but the one that I can remember uses
what Roger Penrose called the Y-commutator, which has the form  FR'F'R.  The
reason this is the Y-commutator is that it affect the three edges adjacent to a
corner and the corner and its three adjacent corners.  I.e. the affected pieces
form a Y, while the pieces affected by the ordinary commutator  FRF'R'  form a
Z.  Combining three Y-commutators as follows:  FR'F'R  RU'R'U  UF'U'F  gives a
process that twists the corner and the three adjacent edges as a unit and
twists an adjacent corner the opposite direction.  NOTE - I'm doing this from
memory and I have a suspicion that the middle group may need to be inverted??
	By moving the odd corner to the right place adjacent to the opposite
corner and applying the inverse of the above, one gets the same sort of pattern
at the opposite corner and the odd corner has been restored.  Now one 3-cycles
the centers, as is easily done by a commutator of slice moves, and one has the
cube in a cube.  Now one can twist the two opposite corners to get the cube in
a cube in a cube, though I find this not as visually dramatic as the cube in a
cube.
	Someone - Mike Reid ? - sent me a minimal method for one of these
patterns, but it's not very memorable.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Thu Oct  8 19:47:11 1998
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Date: Thu, 8 Oct 1998 16:45:31 +0100
From: David Singmaster       <david.singmaster@sbu.ac.uk>
To: cube-lovers@ai.mit.edu
Message-Id: <009CD663.86DEF2D6.16@ice.sbu.ac.uk>
Subject: Nicholas Bodley's message of 22 Sep 1998.

	Nicholas Bodley's message reminds me of when I wrote about the Cube in
1978 or early 1979, I think in the Observer, which seems to have been the first
article outside Hungary.  I mentioned that the mechanical problem seemed even
harder than the mathematical problem and this led to about six submissions of
mechanisms from readers.  All but one of these were clearly impossible, but the
last was Rubik's mechanism with slight differences - e.g. he had the undersides
of the centers rounded.  The submitter of this was a UK patent agent with
obvious mechanical aptitude.
	However, one of my students, who had bought a cube from me, told me
that a friend rang her up and asked if she had seen the hoax article about a
cube that moved in all directions.  The friend had just proven that such an
object was impossible.  My student had to disabuse her.
	When the cubes first came out of Hungary, we didn't know what the
mechanism was and they were too precious to fiddle with.  Roger Penrose said he
had one face center piece come undone and he carefully wrapped thread around
the exposed part of the screw and worked the screw into place and pulled on the
thread to screw the screw  back into the central piece.  Sometime in late 1978,
a friend had trouble with his cube and took a screwdriver to it and discovered
the cover plates and the screw heads inside!
	Enough for now.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Fri Oct  9 18:40:38 1998
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Message-Id: <361E8D3F.6597@ameritech.net>
Date: Fri, 09 Oct 1998 17:25:03 -0500
From: Hana Bizek <hbizek@ameritech.net>
Reply-To: hbizek@ameritech.net
To: cube-lovers@ai.mit.edu
Subject: comments on "davenport's pattern"

David Singmaster claims to have discovered this pattern in 1979 or
1980, so it should be credited to him. In my own book I present a
number of patterns, but I would never dare to claim authorship to any
of them. Singmaster's comments prompted me to look at my own books. In
CUBE GAMES (Taylor and Rylands} this pattern appears on the top of page
37.I have a strong suspicion this pattern could be a combination of the
two cyclicity-three patterns on page 36 therein. One may use this
pattern as a corner in a 3-color design.

Design-construction is a step beyond pattern-construction. My question
has not yet been satisfactorily answered. Has anyone seen construction
of 3-dimensional "sculpture-like" designs? People referred me to
Davenport's creations, but my own designs are quite different. 

Hana

From cube-lovers-errors@mc.lcs.mit.edu  Fri Oct 30 13:56:08 1998
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Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Oct 15 11:02:13 1998
Message-Id: <19981015145704.8314.rocketmail@attach1.rocketmail.com>
Date: Thu, 15 Oct 1998 07:57:04 -0700 (PDT)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: Moves to this other pattern
To: cube <Cube-Lovers@ai.mit.edu>

Please if this is not one of the purposes of this
list someone let me know I don't meant to be rude.

Could someone please tell me the moves to get from a
solved cube to the following pattern:
The top and bottom faces keep their colors.
The 4 columns in the middle of every side face stay
with their color.
The left column on the front face moves to the right
and the right column moves to the left.
The left column on the back face moves to the right
and the right column moves to the left.

Thanks

===
Jorge E Jaramillo

[ Moderator's note: We have a lot of requests for processes for various
  and have got a lot of optimal processes.  Maybe the hard part is
  figuring out how to look them up.  This is one of the "4-" patterns,
  and probably appears among the quasi-continuous partial isoglyphs. --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Fri Oct 30 14:17:00 1998
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Date: Mon, 26 Oct 1998 23:58:27 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: 12q From Start
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Message-Id: <Pine.PMDF.3.95.981026235420.29075A-100000@PSTCC6.PSTCC.CC.TN.US>


 |x|     Patterns     Lcl    Positions    Lcl    Branching
                      Max                 Max      Factor


  0q             1     0              1     0
  1q             1     0             12     0    12
  2q             5     0            114     0     9.5
  3q            25     0           1068     0     9.368
  4q           219     0          10011     0     9.374
  5q          1978     0          93840     0     9.374
  6q         18395     0         878880     0     9.366
  7q        171529     0        8221632     0     9.355
  8q       1601725     0       76843595     0     9.347
  9q      14956266     0      717789576     0     9.341
 10q     139629194     4     6701836858    42     9.337
 11q    1303138445     0    62549615248     0     9.333
 12q   12157779067   103   583570100997  2913     9.330

The last time a new level was calculated for the quarter turn metric was 4
February 1995.

The cumulative number of positions now identified is 653625391832, or
about 6.5*10^11.  This is well past the "geometric halfway point"  of
sqrt(|G|), which is about 6.5*10^9.  However, it is known that the
diameter of G is at least 26q, strongly indicating that there is a bit of
a tail to the distribution of positions by length.

Of the 103 local maxima of length 12q, 70 of them also have their inverse
as local maxima.  For the other 33, the inverse is not a local maximum.
For one of them, the inverse has 11 moves which go closer to Start.  For
seven of them, the inverse has 10 moves which go closer to Start.  For
eleven of them, the inverse has 8 moves which go closer to Start.  For six
of them, the inverse has 6 moves which go closer to Start.  For two of
them, the inverse has 4 moves which go closer to Start.  And for six of
them, the inverse has only 2 moves which go closer to Start.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov  2 09:40:50 1998
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Message-Id: <363B1799.3534@hrz1.hrz.tu-darmstadt.de>
Date: Sat, 31 Oct 1998 14:58:49 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: cube-lovers@ai.mit.edu
Subject: Unauthorized Use of RUBIK'S CUBE and CUBE Design Marks?

About 2 weeks ago I received the following message and it seems to me
that it might be interesting for you too:

> Subject:
>         Use of RUBIK'S mark
>   Date:
>         Sun, 18 Oct 1998 21:05:31 EDT
>   From:
>         CK4IPLAW@aol.com
>     To:
>         kociemba@hrz1.hrz.tu-darmstadt.de
> 
> 
> CLEARY, KOMEN & LEWIS, LLP
> 600 Pennsylvania, Avenue, S.E.
> Suite 200
> Washington, D.C. 20003-4316
> Telephone: 202 675-4700
> Telecopier: 202 675-4716
> E-Mail: ck4iplaw@aol.com
> 
> 
> October 18, 1998
> 
> Via Electronic Mail
> 
> Herbert Kociemba
> kociemba@hrz1.hrz.th-darmstadt.de
> 
> Re:     Unauthorized Use of RUBIK'S CUBE and CUBE Design Marks
> 
> Dear Mr. Kociemba:
> 
> This firm is intellectual property counsel to Seven Towns Limited ("Seven
> Towns"), the manufacturer and worldwide distributor of the RUBIK'S CUBE three-
> dimensional puzzle and provider of an electronic version of the puzzle via its
> official web site, which is located at http://rubiks.com.
> 
> The RUBIK'S CUBE mark is famous throughout the world.  The distinctive
> overall appearance of the RUBIK'S CUBE puzzle also is a famous trademark owned
> by Seven Towns.  These marks are registered or are the subject of pending
> trademark applications in most of the major countries of the world.
> 
> It has come to our attention that your web site features a program under the
> name of Rubik's Cube Explorer.   I must advise that your unauthorized use of
> the RUBIK'S CUBE mark owned by Seven Towns constitutes trademark infringement.
> Specifically, the use of this mark in designating the origin of your program
> confuses the public into believing mistakenly that it derives from, is
> associated with, or is endorsed or sponsored by the owner of this commercial
> symbol (i.e., Seven Towns).  Moreover, apart from causing consumer confusion,
> your use of the well-known mark dilutes its distinctive value in violation of
> the federal and state anti-dilution laws.
> 
> Seven Towns appreciates your interest in the RUBIK'S CUBE puzzle, and it
> certainly does not wish to inhibit legitimate discussion of the puzzle on the
> Internet or in any other medium.  However, it also must be vigilant in
> maintaining the value and integrity of its intellectual property.  It cannot
> afford to lose control over its commercial reputation, or damage to its
> substantial goodwill, by permitting another party to use its trademarks or
> trade dress in a manner that causes source confusion or otherwise dilutes
> their selling power.  Thus, Seven Towns requests that you remove from your web
> site the electronic version of the RUBIK'S CUBE manipulative puzzle, and that
> you discontinue any further use of the term RUBIK'S CUBE or any similar
> designation in the prominent, source-indicating manner of a trademark.
> 
> I hope that you are understanding of our client's position, and I thank you
> in advance on behalf of Seven Towns for your prompt attention to this matter.
>  
>                                                         Sincerely yours,
> 
>                                                         //sjm//
> 
>                                                         Scott J. Major



Indeed the headline of my homepage at

http://home.t-online.de/home/kociemba/cube.htm

was "Rubik's Cube Explorer 1.5". So I removed the word "Rubik's" and
added some note at the bottom of the page (...blah blah is not derived
from, is not associated with blah blah...).

I definitely will not remove the program from my homepage. This seems
ridiculous. I know, that other cube fans received similar mail because
they have some similar statements on their homepages now.

What do you think about that?


Herbert Kociemba

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov  2 14:41:00 1998
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Date: Sat, 31 Oct 1998 23:06:34 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: All the Local Maxima at 12q
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Message-Id: <Pine.PMDF.3.95.981031222133.81424A-100000@PSTCC6.PSTCC.CC.TN.US>

I had originally decided not to post all the 12q local maxima because
there are so many and because not all of them look all that interesting. 
But I have been looking at them a bit more, and I think it's worth the
effort.  Some of them will be very familiar and some of them not.  I would
highlight the following ones. 

#9  is a partial isoglyph  --  four short T's (or four short U's).
#57 is the well-known four spot.
#68 is one of the more striking of several pseudo two spots.
#80 is worthy of some study.  It's the only one where the maximality
    of the inverse is 11  --  almost but not quite a local maximum.
#97 along with #98 and #99 are very striking pseudo six spots.  #97
    and #99 are almost the same pattern, and it took me a while to
    see how they differ.

In the table below, the right most column gives the maximality of the
inverse of the pattern, where the maximality is the number of moves which
go closer to Start.  The maximality of the inverse gives an indication of
how close the inverse comes to being a local maximum, with 12 indicating a
local maximum.  (The maximality of the pattern itself is not given, since
it is 12 by definition.)  The table gives the 103 local maxima of length
12q which are unique up to symmetry.

   1.  F  F  L  L  F  F  B  B  R  R  B  B    12
   2.  F  B' U  U  D  D  F' B  R  R  L  L    12
   3.  F  B  R  R  F' B' U  D  L  L  U' D'   12
   4.  U' R' L  B  L  B' R  L' D  L' U  D'   12
   5.  F  B  R  R  L  L  F  B  U  U  D  D    12
   6.  R  L' F  F  B  B  R  L' F  F  B  B    12
   7.  F  F  B  B  U  U  D  D  R  R  L  L    12
   8.  R  L' U' D  F' B  R  R  L  L  U' D    12
   9.  F  B  U  U  D  D  F' B  R  R  L  L    12
  10.  U  B' D  F  D' B' D  F' D' B  B  U'   12
  11.  L  B  B  R  D  D  R' L  B  B  L  L    12
  12.  U  R' U  L' U' R' U  L  U' R  R  U'   12
  13.  F  F  U' D  R  R  U  D  D  B  B  D'   12
  14.  F  B  D  D  F  B' L  L  U  U  F  F    12
  15.  U  D' F  U  D  R' L' U  D  F  U  D'   12
  16.  U  U  R  L  F' B' U  D' R  R  L  L    10
  17.  R  R  U  D  F' B  U' D  F  B' U' D     8
  18.  U  U  D  D  F  B' R  R  L  L  B  B    12
  19.  R' L  F' B  U  U  F  F  B  B  U' D'   12
  20.  U  D' F  F  U' D  R  L  F  B' U' D    12
  21.  F  B  R' L  D  D  F  B' R' L  U' D    12
  22.  L  L  U  D' B  B  R  L  U' D  B  B    12
  23.  F  L  L  F' D  D  F  B' L  L  B  B    12
  24.  F' R  R  F' U  U  F' B  L  L  B  B    12
  25.  F  B  U  U  L  L  B  B  L  L  U  U     8
  26.  U' D' R  L' F  F  B  B  L  L  U' D     4
  27.  U  D  R  L' F  F  B  B  L  L  U  D'    4
  28.  F  B' U  U  B  B  L  L  B  B  U  U    12
  29.  F' B  U  D  L  L  B  B  R  R  U  D    12
  30.  F' B  U  D  R  R  F  F  L  L  U  D    12
  31.  F  B' U  U  F  F  R  R  F  F  U  U    12
  32.  F  B' U  U  L  L  B  B  L  L  U  U    12
  33.  F  B' U  U  R  R  F  F  R  R  U  U    12
  34.  L  L  U  D  L  L  F' B  R  L  U  U    10
  35.  F' R  R  F  U  U  F  B' R  R  B  B    12
  36.  L  L  F' B  D  D  R  R  F  F  B  B    12
  37.  L  L  F' B  D  D  F  F  B  B  L  L    12
  38.  D  D  F' B  U  D' F  F  B  B  U  D'   10
  39.  F  F  R' L  F  B' U' D  F  F  L  L    12
  40.  U  R' L  F  U  D' R' L  F  F  B  B     8
  41.  U  D  R  L  F  B' U  D' R' L  F  F    12
  42.  U  D  F  F  U' D  R  L  F  B' U' D    12
  43.  U  D  B  B  U' D  R' L' F  B' U' D    12
  44.  F  F  R  L' F  B' U  D' F  F  L  L    12
  45.  U  D  F' B  R  L  U' D  B  B  U' D    12
  46.  U  D  F' B  R' L' U' D  F  F  U' D    12
  47.  F  B' D  R' L' U  U  R  L  D  B  B     8
  48.  B' U' D  R' L  B' U' D  R  R  L  L     6
  49.  B' U  D' R  L' F  U  D' R  R  L  L     6
  50.  U  D  F' B' U' D  L  L  F  B' U' D'    8
  51.  U' F' U' D  F  B' U' D  R  R  L  L     2
  52.  U' F  F  B  R' L  U' D  F  F  B  B     8
  53.  D  R  R  L  F' B  U' D  R  R  L  L     2
  54.  R  U' F' B  R  L' U  D' F  F  U  D'    2
  55.  L  F  F  B  R  L' F  B' R  R  L  L     2
  56.  L  F  F  B  U  D' R' L  F  F  B  B     8
  57.  F  B' U  U  D  D  F  B' R  R  L  L    12
  58.  F  B' R  L' U' D  F' B' R  L' U' D'   12
  59.  F  B  R  L  U  D' F  B  R' L' U  D'   12
  60.  F  B' U  U  D  D  F' B  U  U  D  D    12
  61.  R  L' U' D  F' B  L  L  U' D  F  F    12
  62.  R  L  B  B  U  D' R  R  F  B  U  D'   12
  63.  F  B' U  D' R  L' F  F  B  B  U  D'   12
  64.  F  B  R  R  L  L  U  D' R' L' U  D'   12
  65.  F  B' U  D' R  L' B  B  U  D' L  L    12
  66.  F  F  U' D  R  R  U  D  F  F  U  U    12
  67.  F  L  L  B  U  U  F' B  L  L  F  F    12
  68.  B  R  L  F  F  R' L' B  U  U  L  L    12
  69.  U' F  B' L' U  D' R  L' F  F  B  B     2
  70.  F  B  R' L' U  D' F' B' R' L' U' D'   12
  71.  F  B  R  L  U  D' F  B  R' L' U' D'   12
  72.  U' R  L' U' D  F  B' R  R  L  L  U'   12
  73.  F' B  R' L  U' D  B  R' L  U' D  D    12
  74.  U' R' L  F  B' U' D  F  B  B  U' D     6
  75.  U  R' L  F' B  U  D' B  F  F  U  D'    6
  76.  F  B  U  D  R' L  F  F  R' L  U' D    12
  77.  R' L' F' B  R  R  L  L  U  D  B  B    12
  78.  R  L  F  B' R  R  L  L  U' D' F  F    12
  79.  F  F  R  R  F' B' U' D' L  L  U  D     8
  80.  U  U  R' L  F' B' U' D  R  R  L  L    11
  81.  U  U  D  R  L' B  U  D' R  R  L  L     6
  82.  U  D  F' B  U' D  F  B' U' D  R  R    12
  83.  F  B' U  D  R  L' B  B  R  L' U' D    12
  84.  U  D' F  F  U  D' R' L' F  B' U' D    12
  85.  B  R  L' D' F  B' U  D' R  R  L  L     2
  86.  F' R' L  U' F  B' R' L  U  D' B  B    12
  87.  F' R' L  U' F' B  R' L  U' D  B  B    12
  88.  R' L  F  B' U  D  R  R  L  L  D  D    12
  89.  F  B  R  L' F' B  R  L' U  D' F  F    12
  90.  U' D' R  L' F  F  U  D' R  R  L  L    12
  91.  F  F  R' L  F' B' U' D' L  L  B  B    10
  92.  F  U  D  B  B  U' D' F  R  L  U  D'   10
  93.  R  L  U  D' F  B' R' L' F  B' U' D     8
  94.  F  B' U' D  L  L  F  B' U' D' F  F    10
  95.  F  B' U  U  D  D  F  B' R' L' U  D'   10
  96.  R' L  L  F' B  U  R  L' F  F  B  B     6
  97.  F  B' U' D  R  L  F' B  U  U  B  B    12
  98.  F  B  U' D  R' L  F' B  D  D  B  B    12
  99.  F  B' U' D  R  L  F  B' D  D  F  F    12
 100.  U' D  F  B' U  D' F  F  B  B  U  D'   12
 101.  U  R  L' F' U  D' R  L' F  F  B  B     8
 102.  U  D' R  L' F' B  U  D' F  F  B  B    12
 103.  F' B' L  L  U' D  F  B  U' D  R  R     8

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
Pellissippi State                            (423) 539-7198
10915 Hardin Valley Road                     (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov  2 18:00:09 1998
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Message-Id: <3.0.32.19981102164235.0094e100@mail.spc.nl>
Date: Mon, 02 Nov 1998 16:42:36 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: Re: Unauthorized Use of RUBIK'S CUBE and CUBE Design Marks?

At 14:58 31-10-1998 +0100, you wrote:
>About 2 weeks ago I received the following message and it seems to me
>that it might be interesting for you too:

[Message deleted for brevity]
>
>Indeed the headline of my homepage at
>
>http://home.t-online.de/home/kociemba/cube.htm
>
>was "Rubik's Cube Explorer 1.5". So I removed the word "Rubik's" and
>added some note at the bottom of the page (...blah blah is not derived
>from, is not associated with blah blah...).
>
>I definitely will not remove the program from my homepage. This seems
>ridiculous. I know, that other cube fans received similar mail because
>they have some similar statements on their homepages now.
>
>What do you think about that?
>

I had the same 'problem' a couple of months ago.

A handheld computer users group I was active in, once published a
program that could be described as 'well, it looks a bit like Tetris
((R), if those lawyers are reading this, as well :-), but it's a long
way off the mark'.

The program was published in our magazine, and was also placed on a
web-page.

A couple of months ago, I received a letter from a Belgian lawyer firm,
addressed to the user's group. This user's group, by the way, died about
5 years (!) ago.

They told us to 'cease and decist' (a couple of things), including publishing
this article on 'our web-page'. Well, since this was someone else's web-
page, there was nothing we could do about it.

We told them (in friendly terms) that the user's group no longer existed,
that we were not affiliated to the user having the article on his web-page,
and that we nover sold the program.

We haven't heard from them (not even a letter stating that they received our
reply!) since.

It's kinda sad: The program was for an HP28S. I would have _loved_ to see
those lawyers type the program in (a couple of kilobytes, via the keyboard!),
only to find out that it was 'almost, but not quite, entirely unlike Tetris'
(Douglas, if you're reading this, quotes are ok, no?).

Just tell them you will take down the name, and they will probably be
off your back.

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Tue Nov  3 06:37:22 1998
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Date: Mon, 2 Nov 1998 15:32:29 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199811022032.PAA24281@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Unauthorized Use of RUBIK'S CUBE and CUBE Design Marks?
Cc: kociemba@hrz1.hrz.tu-darmstadt.de

> > Re:     Unauthorized Use of RUBIK'S CUBE and CUBE Design Marks
> >
> > Dear Mr. Kociemba:

[ ... ]

> What do you think about that?

i think it's not good business strategy to attack the people who
are the biggest fans of their product!  to claim ownership of
"the distinctive overall appearance of the RUBIK'S CUBE puzzle"
is ludicrous!

the changes you've described to your web page seem quite
reasonable and appropriate (given the circumstances), without
compromising too much.  i hope you have no further trouble with
seven towns.  but if you do, please let me know about it.

to make this situation even more ridiculous, i just checked out
their "official" website, which features a java cube
(http://rubiks.com/VRCUBE.html).  their applet is stolen
from karl ho"rnell!  (http://www.tdb.uu.se/~karl/java/rubik.html)

mike

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov  9 17:39:21 1998
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Date: Tue, 3 Nov 1998 19:29:38 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Query for Corners-First Method Rubik Solution
To: cube-lovers@ai.mit.edu


Hi,

Does anyone know of any websites that describe the Corners-first
method of the solving the rubik's cube?  I know of many layer-first
methods such as Jiri Fridrich's (for which I have spent many hours
learning), but I really haven't seen a comprehensive explanation of
the corners-first method.  I'm really curious to understand how anyone
can solve the cube under 30secs by solving the corners first.

-Han-

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov  9 18:35:15 1998
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Date: Wed, 04 Nov 1998 16:41:26 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Local Maxima Whose Inverses are not  Local Maxima
To: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9811041626.D@GN209A.PSTCC.CC.TN.US>

On 30 June 1997 I reported that if you could find a local
maximum whose inverse was not a local maximum, then you could
also find a longer local maximum.  For example, suppose x is a
local maximum in the quarter turn metric and x' is not.  Then,
there exists q in Q such that |x'q| = |x'| + 1 = |x| + 1.  But
we know that q'x is a local maximum and we also know that
|q'x| = |x| + 1 because |q'x| is the same as |x'q|.

Because we now have at 12q a good number of local maxima  whose
inverses are not local maxima as specimens, I have begun to
wonder if the same process might be able to be repeated several
times to yield progressively longer local maxima.  For example,
if x is a local maximum and (q1)x is a local maximum, might also
(q2)(q1)x be a local maximum and also (q3)(q2)(q1)x etc.  It
seems to me that good candidates to investigate in this regard
might be those local maxima at 12q whose inverses have a very
small maximality.  For example, if x is a local maximum where
the maximality of x' is 2 (and there are several such cases),
then we know that there are 10 local maxima of the form qx.

I am not sure if I have time to investigate this question
further, but I certainly would love to hear from anyone who has
the time and the computing resources to do so.

----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us



From cube-lovers-errors@mc.lcs.mit.edu  Tue Nov 10 06:10:31 1998
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Date: Mon, 09 Nov 1998 15:37:18 -0800
From: Derek Bosch <bosch@sgi.com>
To: Han Wen <hansker@yahoo.com>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Query for Corners-First Method Rubik Solution
References: <19981104032938.14284.rocketmail@send102.yahoomail.com>

Here's the method I use to solve Rubik's Cube in 30 seconds or less.

My notation is as follows:

  F  = turn front face clockwise 90 degrees.
  F' = turn front face counter-clockwise 90 degrees.
  F" = turn front face 180 degrees.

  U = turn top face clockwise 90 degrees.
  R = turn right face   "      "    "
  L = turn left face    "      "    "
  B = turn back face    "      "    "
  D = turn bottom face  "      "    "

  ^  = move middle slice 90 degrees up.
  v  = move middle slice 90 degrees down.

OK.  Now the order I do things is corners, edges on two OPPOSITE sides (Right
and Left), followed by the middle slice edges.

(1) Corners:
  I solve my corners a bit wierdly, but I find it is really fast.  I position
any four corners of the same color on a side.  I don't care what colors are
on the adjoining faces right now, as I fix them later.

(1a)
  Once I have the 4 corners of the same color, I turn the cube so that those
colors are on the down face.  Now there are a few combinations that can occur
on the top face:

  All corners on top face same color:   Goto (1b)

  Three corners need to rotate clockwise (position like below o=no rotate)
                                         (+ = needs clockwise rotation)
					 (- = needs counter-clkwise rot.)
       + +     Move: R'U"RUR'UR    and goto (1b)
       + o

  Three corners need counter-clockwise rotate:

       - o     Move: RU"R'U'RU'R'  and goto (1b)
       - -
  One corner needs clockwise rotate, One needs counter-clockwise rotate:

  3 cases:
       + -     Move: RU"RU"RUR"    and goto (1b)
       o o

       - +     Move: RUR'U'F'U'F   and goto (1b) 
       o o

       o -     Move: R'URUBU'B'    and goto (1b)
       + o

   Two corners need clockwise rotate, Two need counter clockwise:

   2 cases:
       + -     Move R"U"RU"R"      and goto (1b)
       - +

       - +     Move RUR"F'R"UR'    and goto (1b)
       - +

(1b)
   Now, you should have two opposite sides, with the corners of those
   two sides the proper color.

   We have to correct the 4 remaining sides to get corners in the right
   place, before we can move onto edges.  To do this, count the number
   of sides that have the upper pair of corners the same color.  Also
   counter the number of sides that have the lower pair of corners the
   same color.

   All four sides (upper and lower) corner pairs match.  Goto (2)

   No sides' corner pairs match.   Do Move R"F"R".  Goto (2)
    
   One Bottom corner pair matches.
      Move that corner pair to the Down-Left position.
      Move R"UR"U'R"UR"U'R.  Goto (2)
   One Top corner pair matches.  Turn Cube over, and do previous moves.

   One Top and one Bottom pair matches.
      Move both corner pairs to the front face.
      Move R"UR"U"F"UF".  Goto (2)

   All Bottom pairs match.
      Move R"UR"U"F"UF"U"L"UL".  Goto (2)
   All Top pairs match.  Turn Cube over, and do previous move.

   All Bottom pairs match.  One Top pair match.
      Move Top Pair to Left-Upper position.
      Move R"UR"U'R"F"U'F"UF".  Goto(2)
   All Top pairs match.  One Bottom pair match.  Turn Cube over, and do prev.

(2)  Solving two Opposite Sides.
   Now, all the corners should be solved.  You should move the center of
   each cube to its respective corners, to get an X on each side (at least
   on two opposite sides).  From now on orient the cube so that the two 
   opposite sides are right and left.

(2a)
   Solve three edges on the left face with the following moves.   

   U'^U -  moves the edge piece in the Front-Down position to the Up-Left
             position.
   UvU' -  moves the edge piece in the Back-Down position to the Up-Left
             position.

   This is easier done with a cube in your hand, and try and see how this
works.  This will mess up the centers and edges in the middle slice, as well
as the Up-Right edge.  Don't worry about this.  As long as you keep this 
orientation, and rotate the left face to get ready for a new edge to be moved
you can solve three out of four of the edges on the left face.

(2b)
  Solve four edges on the right face:
  First, rotate the left face, so that the unsolved edge is in the Up-Left
     position.

  Then, using the following moves, solve all four edges (similarly to step 2a).

   U^U' -  moves the edge piece in the Front-Down position to the Up-Right
             position.
   U'vU -  moves the edge piece in the Back-Down position to the Up-Right
             position.

(2c)
  Solve remaining edge on left face:
  2 cases (other than already solved):
    edge in place, needs to be flipped:  Use U'vUvUvU'

    otherwise, move edge to Down-Front position, using v or ^.
       if the front color (of the DF edge) is the same as the left face color,
					 Use U'vU"vU'
       else Use vU^U"^U

(3)
  Solve middle slice edges.
  First use ^ or v to position middle slice centers in proper faces.

(3a) Position edges:
  3 cases (other than all in place):
     only three edges out of place:
        position cube such that DF needs to go to UB and UB needs to go to UF.
        Use ^U"vU".
     all four edges need to move:
        if UF needs to go to DB, Use ^F"B"vF"B".
     otherwise, position so that UF needs to go to UB, Use U'^^U'^^.

(3b) Flip required edges:
   3 cases (other than no flips needed).
     all four need flipping, use FR'F'^U^U^U^UFRF'

     two edges need flipping, both on same face.  Turn cube so that these
       edges are the UF and UB edges.  use ^U^U^U"vUvUvU"
     otherwise, turn cube so UB and DF need flipping, use F"^U^U^U"vUvUvU"F"

That should do it.  I apologize for the roughness of this solution.  I think
diagrams would help it a lot.  If you have any criticism or ideas that could
help this solution become more readable, let me know.

Note, this solution is very close to Jeff Verasano (sp) and
Minh Thai's methods...

D
-- 
Derek Bosch        "A little nonsense now and then
(650) 933-2115      is relished by the wisest men"... W.Wonka
bosch@sgi.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Nov 10 07:33:39 1998
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Date: Mon, 9 Nov 1998 19:15:58 -0500 (EST)
From: Alchemist Matt <monroem@email.unc.edu>
Reply-To: Alchemist Matt <monroem@email.unc.edu>
To: Han Wen <hansker@yahoo.com>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Query for Corners-First Method Rubik Solution
In-Reply-To: <19981104032938.14284.rocketmail@send102.yahoomail.com>
Message-Id: <Pine.GSO.3.95L.981109190803.17496B-100000@sunny.isis.unc.edu>


        My page at http://www.unc.edu/~monroem/rubik.html describes a
method that is sort of "corners first".  Although, in my first step I say
to solve the first layer before going on, one could effectively simply
place only the four corners in the top layer, then move on to the four
corners in the bottom layer (specified in steps 2 and 3), then begin
filling in the gaps on the top and bottom layers (steps 4 and 5), and
lastly finish the middle layer.

        In fact, a chemistry professor at my current school, Holden Thorp,
competed in one of the Rubik's cube playoff contests that was aired on the
TV show That's Incredible.  Someone posted the video of it about a month
ago (and mentioned it in this discussion list), and he saw it here at my
school after I downloaded it.  He then looked at my page and mentioned
that the winner of the contest actually used the solution shown on my page
(probably modified slightly).  I can only solve a well-scrambled cube in 2
to 3 minutes using the solution, but I'm sure someone quite adept, nimble,
and fast could push it to under one minute.  (Please note this isn't "my"
solution; I simply learned it from a book many years ago.  Further, I have
never been in a cube solving competition).

                Matt

-----------------------------------------------------------------------
Matthew Monroe				Monroem@UNC.Edu
Analytical Chemistry      		http://www.unc.edu/~monroem/
UNC - Chapel Hill, NC                   This tagline is umop apisdn

From cube-lovers-errors@mc.lcs.mit.edu  Thu Nov 12 14:18:47 1998
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Date: Wed, 11 Nov 1998 09:03:41 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: RE: Query for Corners-First Method Rubik Solution
To: Noel Dillabough <Noel.Dillabough@ineacorp.com>
Cc: cube-lovers@ai.mit.edu

Hi,

Thanks for the link to your Puzzler program.

You're not going to believe this, but you can still purchase the
Professor's Cube (5x5x5) and the Megaminx! Since it's difficult... no,
impossible to find anyone that sell these puzzles, I think it's worth
mentioning. You can get them from Meffert's site:
http://ue.net/mefferts-puzzles/

Your Puzzler program is a tremendously useful tool to develop moves.
I've got 11/12 sides of the Megaminx solved.  But for the last side, I
need to figure out corner/edge twisting/permuting moves.  You're
Puzzler program's great for that.  I'm surprised how many of my
Rubik's cube moves can be applied with minor modifications to the
Megaminx.

-Han-

---Noel Dillabough <Noel.Dillabough@INEACORP.COM> wrote:

> I actually solve all the cubes this way (or at least centers ->
> corners -> edges for larger cubes) I just find it more logical and
> easier to memorize than other methods.

> You can check out my solution at
> http://www.mud.ca/puzzler/puzzler.html.  Its in the puzzler help
> file under "solving the cube".  I will be adding other solutions
> soon that are clearer, let me know if you would like them I could
> mail them to you.

> -Noel.

From cube-lovers-errors@mc.lcs.mit.edu  Thu Nov 12 15:00:46 1998
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Date: Thu, 12 Nov 1998 15:01:40 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: cube-lovers@ai.mit.edu
Message-Id: <009CF1D5.D1348312.50@ice.sbu.ac.uk>
Subject: Use of the name Rubik's Cube

        The lawyers are being obsessively zealous as the name is
certainly well on its way to becoming a common noun.  It was included
in the Oxford English Dictionary in the mid-1980s.  Other examples are
Kleenex and Aspirin, which were both originally tradenames and their
owners fought to retain them but eventually lost.  Xerox is fighting a
rear-guard action on its name.  If you don't want to get involved in
legal hassle, I suggest that you use the name  Magic Cube  which was
the original name and is such a common term that they can't claim it
is a trademark.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

[ Moderator's note: I am still dropping messages that consist mainly of
  generic comments on intellectual property issues.  There a great variety
  of individualistic and contentious debate on these topics that you may
  follow in dedicated fora such as the Usenet group misc.int-property.  I
  am not yet persnickety enough to elide the third and fourth sentences
  from the above, but they are on the edge.

  I will also note that the term "Magic Cube" is also used to refer to a
  cubical array of natural numbers whose orthogonal and diagonal rows sum
  to the same number, as a generalization of "Magic Square", so it is
  advisable to include context such as "The geometrical puzzle originally
  known as the Hungarian Magic Cube." ]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Nov 18 12:52:47 1998
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Date: Thu, 12 Nov 1998 17:05:37 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Han Wen <hansker@yahoo.com>
Cc: Noel Dillabough <Noel.Dillabough@ineacorp.com>, cube-lovers@ai.mit.edu
Subject: Solutions (Was: RE: Query for Corners-First Method Rubik Solution)
In-Reply-To: <19981111170341.14375.rocketmail@send105.yahoomail.com>
Message-Id: <Pine.SUN.3.95.981112170248.21046A-100000@sunspot.tiac.net>

On Wed, 11 Nov 1998, Han Wen wrote:

{snips}

}mentioning. You can get them from Meffert's site:
}http://ue.net/mefferts-puzzles/

It might be of interest to mention that Meffert has solutions to many
puzzles at his Web site. The Contributors section gives generous credit
to a number of experts; they wrote the solutions.

Best,

|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Wed Nov 18 13:27:34 1998
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Message-Id: <3.0.32.19981113091853.00948790@mail.spc.nl>
Date: Fri, 13 Nov 1998 09:18:55 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: MegaMinx/5^3 (Was: RE: Query for Corners-First Method Rubik Solution)

At 09:03 11-11-1998 -0800, you wrote:
>Hi,
>
>Thanks for the link to your Puzzler program.
>
>You're not going to believe this, but you can still purchase the
>Professor's Cube (5x5x5) and the Megaminx! Since it's difficult... no,
>impossible to find anyone that sell these puzzles, I think it's worth
>mentioning. You can get them from Meffert's site:
>http://ue.net/mefferts-puzzles/

Also, a store in The Netherlands sells these! Last time I was there
(last saturday), they had;
- a couple of 3^3's
- a couple of 5^3's
- skewb

I can't recall if they had a Megaminx at that time.

The store is based in Eindhoven. If anybody wants some, I can buy
them and send them out.

5^3 costs F. 50 (about $25). 3^3 costs F. 10 (about $5).

Before anybody gets this wrong:
- I do not work for them, I'm a happy customer
- I don't get paid to do this
- I make no money out of this

You can call them at: +31-40-2461376

Business hours are 0900 to 1800. The Netherlands is at CET
(differs +6 hours with NY, +9 with CA)

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Wed Nov 18 17:23:08 1998
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Message-Id: <19981114070437.9789.rocketmail@attach1.rocketmail.com>
Date: Fri, 13 Nov 1998 23:04:37 -0800 (PST)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: RE: Moves to this other pattern
To: David Singmaster <david.singmaster@sbu.ac.uk>, <Cube-Lovers@ai.mit.edu>

Maybe (although I don't think so since some people
already answered what I was asking) I made a mistake
when describing the position I wanted to accomplish.
What I wanted was:
      L B R
      L B R
      L B R
F F F T T T F F F D D D
L L L T T T R R R D D D
B B B T T T B B B D D D
      L F R
      L F R
      L F R

And until now the best solution is:
F B L F2 T D- L2 B- T- D- F2 R- T- D L- R D

===
Jorge E Jaramillo

From cube-lovers-errors@mc.lcs.mit.edu  Wed Nov 18 18:05:42 1998
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Message-Id: <365115AC.22369936@erco.com>
Date: Tue, 17 Nov 1998 07:20:28 +0100
From: "michael ehrt" <m.ehrt@erco.com>
Reply-To: m.ehrt@erco.org
To: Cube Lovers Mail <Cube-Lovers@ai.mit.edu>
Subject: Getting 2x2x2 cubes

If anyone is interested in getting 2x2x2 cubes, during my holiday in the
UK two weeks ago I found a shop in Sheffield which has them in stock.
It's called "The Puzzle Shop" and in situated in Meadowhall shopping
centre. The cubes are GBP 5 each, and they have a few other things like
keyring 3x3x3s etc.

Michael

From cube-lovers-errors@mc.lcs.mit.edu  Thu Nov 19 13:41:00 1998
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Message-Id: <19981117105414.18936.rocketmail@attach1.rocketmail.com>
Date: Tue, 17 Nov 1998 02:54:14 -0800 (PST)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: The Cylinder
To: cube <Cube-Lovers@ai.mit.edu>

I was checking the Rubik official website and I was
surprised not to find one product that I seem to find
here (I live in Colombia South America) fairly
easily. I am talking about the cylinder.
When I first saw it I bought it and thought it was
going to be some amazing and tricky to solve puzzle,
it ended up being a 3x3 cube with the corners cut, so
corner cubelets only have 2 colors and there are two
types of borders, 8 borders with the usual two colors
and 4 with only one.

Does it mean that this cube was "invented" by some
manufacturer other than Mr Rubik and that is not so
common?
===
Jorge E Jaramillo

[Moderator's note: I own such a puzzle, but I would call its shape
 an octagonal prism, rather than a cylinder.  On mine, the solved
 position is not an octagonal prism because one beveled face is
 rotated 90 degrees, forming a decahedron whose faces are six
 rectangles and four irregular hexagons. I don't remember whether
 it was originally manufactured this way or whether I altered the
 color tabs. ]

From cube-lovers-errors@mc.lcs.mit.edu  Thu Nov 19 17:36:55 1998
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Date: Thu, 19 Nov 1998 15:00:06 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re : The Cylinder
In-Reply-To: <19981117105414.18936.rocketmail@attach1.rocketmail.com>
To: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Cc: cube <Cube-Lovers@ai.mit.edu>
Message-Id: <SIMEON.9811191506.A@GN209A.PSTCC.CC.TN.US>

Jorge E. Jaramillo's message of 17 Nov 1998 included:

>[Moderator's note: I own such a puzzle, but I would call its shape
>  an octagonal prism, rather than a cylinder.  On mine, the solved
>  position is not an octagonal prism because one beveled face is
>  rotated 90 degrees, forming a decahedron whose faces are six
>  rectangles and four irregular hexagons. I don't remember whether
>  it was originally manufactured this way or whether I altered the
>  color tabs. ]

I also own such a puzzle, although I have never seen one in 
a store.  I got mine at a garage sale for $0.25.

I haven't played with it in a long time.  But my best
recollection is that it can be solved basically the same
way as a 3x3x3 cube, except that *I think* (don't remember
for sure) that the color scheme permits invisible swaps of
identically colored pieces which can make the puzzle seem
"impossible" to solve unless you realize that the
identically colored pieces must be swapped.  It is also my
best recollection that such a puzzle is mentioned briefly
and is pictured in one of Douglas Hofstadters's cube
articles in Scientific American back in the early 80's.  So
I don't think it is any kind of new invention.

----------------------------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us
Pellissippi State Technical Community College

From cube-lovers-errors@mc.lcs.mit.edu  Fri Nov 20 11:11:16 1998
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Date: Mon, 16 Nov 1998 23:01:39 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Fwd: Re: HOW TO SOLVE THE PROBLEM OF THE PROFESSOR CUBE 
To: cube-lovers@ai.mit.edu

I thought this would be of value to Rubik fans out there with
Professor Cubes (5x5x5)...

note: forwarded msgs attached.
_________________________________________________________

---Uwe Meffert <uwe@ue.net> wrote: [reprinted here with his permission]

> Dear Mr. Wen

> Thank you for your interest in my puzzles, I am sorry to hear that
> you are having problems with the Prof. Cube.

> You received one of the last ones from the last production batch and
> the next production is not for another month.

> What unfortunately happened is that when gluing the center small
> caps excess glue fixed the screw to the plastic centre piece that it
> should turn in.  So when you turn these sections it will tighten /
> loosen that one screw.

> If you are skilful enough you can try and carefully remove the
> centre label and then pry open and remove the centre cap of the blue
> and orange side.  Then try to remove the excess glue from around the
> screw with a sharp object and try turning the screw with a
> screwdriver firmly holding the plastic piece so you can break the
> glue bond.  Once the screw can freely turn inside the plastic part,
> re-tighten it to the same tension as it was originally, so as to
> allow smooth turning without any pieces falling out during play.
> Then carefully using only very little glue fix the centre cap back
> into place and re-attach the color label.

> Good Luck and Happy Puzzeling.

> Please let me know the outcome of this recommended procedure.


> With warm regards
> Uwe Meffert
________________________________________________________________

Date: Mon, 16 Nov 1998 22:57:20 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Re: HOW TO SOLVE THE PROBLEM OF THE PROFESSOR CUBE 
To: Uwe Meffert <uwe@ue.net>
Cc: Jing Meffert <jing@ue.net>

Hi,

My cube is all fixed.

Thank you for your prompt reply.  You were right, the glue used to fix
the caps also fixed the spindle screw! Actually, your instructions
gave me the perfect excuse to take your cube apart.  I was dying to
find out how the heck all these pieces are held together.  

Anyways, I popped of the caps carefully using a razor blade, scraped
off all the excess glue, greased the screw head and before screwing it
back together, I took all the pieces for one face out just to see and
understand the engineering holding all the pieces together. Wow, what
an amazing bit of engineering.  It's like a cube spindle inside
another cube spindle!  Amazing. 

Actually, the center caps don't really need glue.  They fit nice and
snug, and it also leaves me the option to adjust the screw again in
case it becomes loose.

Now that I understand the mechanism, I've decided to only rotate faces
clockwise to minimize the possibility that a counter-clockwise
rotation will actually loosen one of the spindle screws.

I still haven't messed the faces up though. I'm so close to finishing
the Megaminx.  I just have two edge pieces to swap on the last face! 
The other 11 sides were fairly straightforward to solve. I also got
the corners of the last face fairly quickly by using Sune's move to
twist corner pieces (a standard Rubik's cube move).  However, getting
those edge pieces was a different story.  I had to develop quite a few
moves to rotate and twist the edge pieces around.  I'm close... so
close..! :)

I hope you keep inventing and making new puzzles.  I eagerly click on
your new releases on your web page quite regularly, hoping to find a
worthy successor to the Megaminx or the Professor's Cube.  Maybe a
7x7x7?!! Or a Buckyball?  One can only imagine...

-Han-

From cube-lovers-errors@mc.lcs.mit.edu  Fri Nov 20 14:44:20 1998
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Message-Id: <3654ED01.E6D38EE8@hurstlinks.com>
Date: Thu, 19 Nov 1998 23:16:01 -0500
From: "Guy N. Hurst" <gnhurst@hurstlinks.com>
Organization: HurstLinks Sites On the Internet <http://www.hurstlinks.com>
To: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Cc: cube <Cube-Lovers@ai.mit.edu>
Subject: Re: The Cylinder
References: <19981117105414.18936.rocketmail@attach1.rocketmail.com>

I have seen one or both of these puzzles, and they were very different
from each other.  The cylinder, or prism, was actually the first cube
I learned to solve, when my cousin from Luxembourg visited back in
1981. I have pleasant memories of it, because it was very well made
and pleasing to view. It is harder to solve than the cube since the
four of the edges are "cut", so it is impossible to match edges to
centers - leaving the possibility of having to backtrack later and
figure out which "corner" (and matching edge) is in the wrong place!
But I had it down and could quickly readjust (usually had to swap
corners diagonally in the top two layers if I found a single flipped
edge left in the bottom layer when almost done solving it, if I
remember).
I liked it so much, I requested and obtained 4 more after my cousin
returned to Europe! I would take them to school, one at a time, until
(unfortunately) they all eventually disappeared. At least two were
stolen out of my (locked) locker on different occasions.  Someone else
liked them, too.
Anyway, I never found that puzzle in the US, and could only get it
from my cousin in Europe. (Who I think may have gotten them from
England).
But the other puzzle, as described by the moderator, was available in
the US back then, I think in the following year or so after my cousin
visited, since one of my friends had one. But it wasn't as nice
looking or well made.  So I didn't care for it. It was more like the
cube with its corners cut, forming rectangles and triangles in a
spherical symmetry, as opposed to the prism from Europe which has four
of its 3-piece-edges cut, forming only rectangles and having a
cylindrical symmetry.

Guy N. Hurst

From cube-lovers-errors@mc.lcs.mit.edu  Fri Nov 20 15:16:19 1998
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Date: Fri, 20 Nov 1998 08:46:30 +0100 (CET)
From: Bas de Bakker <basde.bakker@pica.nl>
To: Cube-Lovers@ai.mit.edu
In-Reply-To: <SIMEON.9811191506.A@GN209A.PSTCC.CC.TN.US> (message from Jerry
	Bryan on Thu, 19 Nov 1998 15:00:06 -0500 (Eastern Standard Time))
Subject: Re: The Cylinder
References:  <SIMEON.9811191506.A@GN209A.PSTCC.CC.TN.US>

>>>>> "Jerry" == Jerry Bryan <jbryan@pstcc.cc.tn.us> writes:

[About the octagonal "cube"]

    Jerry> I haven't played with it in a long time.  But my best
    Jerry> recollection is that it can be solved basically the same
    Jerry> way as a 3x3x3 cube, except that *I think* (don't remember
    Jerry> for sure) that the color scheme permits invisible swaps of
    Jerry> identically colored pieces which can make the puzzle seem
    Jerry> "impossible" to solve unless you realize that the
    Jerry> identically colored pieces must be swapped.

Your recollection is not exact.  There are no identically colored
pieces to swap, but you can swap complete columns consisting of two
"corners" (what would have been corners on the cube) and one "edge"
without noticing.

In fact, if you create an even permutation of those columns, there is
no problem.  But if you create an odd permutation, it will become
impossible to solve the upper layer.

Presuming you solve cubes in layers, the easiest way out is to not
start at one of the octagonal layers (which seems the most natural
way), but to start with a "side" layer.  If you do it this way, it
will always be possible to solve the last layer.

I hope I'm making myself at least somewhat clear,
Bas.

From cube-lovers-errors@mc.lcs.mit.edu  Fri Nov 20 17:41:06 1998
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Date: Fri, 20 Nov 1998 10:53:40 +0000
To: jbryan@pstcc.cc.tn.us
From: "Andrew R. Southern" <a.southern@ic.ac.uk>
Subject: Space Shuttle.
Cc: Cube-Lovers@ai.mit.edu, kingeorge@rocketmail.com

I got a similar puzzle in the same way. Mine was called the "Space
Shuttle". The lack of the name Rubik probably meant it wasn't from Rubiks.
But since they only have a patent in Hungary, and everywhere else they are
protected by copyrights on the name and the external appearence, this is
probably legally legitimate. The colour scheme allowed the puzzle to be
solved in more ways than the cube and so I reckon its easier.

Each of the chamfered sides was coloured in a colour which did not relate
to the rest of the puzzle, and so these were (within the boundaries of a
2-swap) possible to position ("correctly") in a few different positions.

The mid-edges of the chamfered edges were rotationally symmetrical every
180 degreees and so, since they were all one colour, it was possible to
have one (or three) of them rotated, and hence one of the other mid edges
rotated and it would look majorly FUBAR'd. Once you'd realised what
happened, the puzzle was easier than the cube though.


-Andy Southern.

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov 23 13:45:28 1998
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Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Nov 20 22:15:27 1998
Date: Fri, 20 Nov 1998 22:13:38 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Cc: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>,
        cube <Cube-Lovers@ai.mit.edu>
Subject: Re: Re : The Cylinder
In-Reply-To: <SIMEON.9811191506.A@GN209A.PSTCC.CC.TN.US>

Uwe Meffert had a printed color catalogue around 1985 that showed some
very interesting moving-piece "group theory" puzzles (is that a proper
term?). Although I have a copy safely stashed somewhere, I don't know
where.

I'm just about sure that one was a cylinder, possibly in three layers
like layer cake; it also, iirc, had maybe three more "cutting planes"
that were spherical sectors bounded by the cylinder. Rotating the pieces
would exchange top and bottom.


|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov 23 16:35:08 1998
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From: roger.broadie@iclweb.com (Roger Broadie)
To: "cube" <Cube-Lovers@ai.mit.edu>
Cc: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: Re: The Cylinder
Date: Sat, 21 Nov 1998 14:29:22 -0000

I was given a cylinder here in England in 1981.  I no longer have the
packaging, but I suspect it was Taiwanese, unless the Hungarians made
this variant.  It was my first cube puzzle, and its shape was so
unappealing when disturbed that I put it on one side and got a genuine
cube to learn on - well, almost genuine: it came from a street trader
in Regent Street.

The apparently impossible state is a monoflip of a top or bottom edge
piece.  There will be a matching flip of a middle-layer edge piece,
but that will be invisible, since the piece has only one face.

I wondered if it would be possible to get the puzzle into the solved
shape and then restore the positions of the pieces without losing the
shape, that is, only allowing turns from the group <U, D, F2, L2, B2,
R2, S, A>, where S and A are slice and anti-slice moves of the middle
layers (I needed them).  In fact it is not.  There may always be a
hidden flip in the middle layer and you can't correct that without
moving the piece out of the middle layer, which needs a turn like F,
and that destroys the shape.  But if you cheat a little and make sure
the flips are got right before the shape is finally restored, then it
can be done.

Andy  Southern has already made the point about the flips.  He also
pointed out that the configuration is not unique because columns
corresponding to the vertical edges on a normal cube can be swapped.
As a rider to that point, the pretty pattern stripes on the normal
cube is not distinguishable on the octagonal prism, because it's
striped already.

It should be possible to work out whether our moderator's puzzle came
in the form he now has by counting stickers.  The octagonal prism has
2 sets of 9 (the top and bottom) and 8 of 3 (the side columns).  If
I've grasped his configuration correctly, if it came in that form
originally it should have 4 sets of 6 and 6 of 3.

Roger Broadie

[ Yes, my decahedron's stickers are incompatible with an octagonal
  prism solution.  I just can't remember whether I replaced some of
  the stickers to make this new shape. --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Mon Nov 23 18:13:16 1998
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Date: Sun, 22 Nov 1998 23:19:31 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Method for Solving the Megaminx
To: Cube-Lovers@ai.mit.edu

Hi,

Well, it took me a 2-3 days, but I finally solved the Megaminx. Whew.
I know, big deal. It's been done.. many many times... over a decade
ago. But, I thought some of the moves I found may of interest to some
of the Megaminx aficionados out there.  So, here it is, I apologize
for its length:

Solving the Megaminx faces 1-11 are fairly straightforward.
Ironically, the larger number of faces makes it easier to solve than
the Rubik's cube, because they provide a lot more "free lanes" to move
pieces around.  There's actually just one move you need to remember to
solve these faces.  It's the same move when solving the middle layer
of the Rubik's cube, when you want to move edge pieces from the bottom
layer to their respective position in the middle layer.  Namely,

D'R'DRF'RFR'

Solving the last face, however, is another matter. The general
strategy I followed is the same as some of the standard methods for
solving the bottom layer of the Rubik's cube.  Namely, I first solve
the 5 corners, then I solve the 5 edge pieces.  To solve the corners,
I simply used Sune's move applied with slight modification to the
Megaminx.  For the remaining edge pieces, I had to develop moves that
only moved the edge pieces around, while leaving the corners
unchanged.  Noel Dillabough's Puzzler program was an invaluable tool
for helping me experiment with various edge moves.  Anyways, the
following are my notes describing some of the more useful moves I've
found.  I'm pretty sure they're not the most efficient method for
solving the Megaminx, but they're the best I could come up with.
___________________________________________
Notation for Solving the Last Face corner pieces:

F=Front Face, D=Lower Face, L=Left Face, R=Right Face
The F and D faces are adjacent

The last layer containing the corners you need to flip/permute should
be positioned at the D-face
____________________________________________
Move for Solving the Last Face corner pieces:

Name: Sune's Double-Swap
Description: Sune's Rubik's Cube move applied to the Megaminx
Number of pairs of corners swapped: 2
Number of corners twisted counterclockwise: 3
Move: R'D'RD'R'D'3R
____________________________________________
Strategy for Solving the Last Face corner pieces:

- Position the corners
- Twist the corners in place by applying Sune's Double-Swap
   move twice

============================================
Notation for Solving the Last Face edge pieces:

F=Front Face, U=Upper Face, L=Left Face, R=Right Face
The F and U faces are adjacent

X= L'R U2 LR' F2
X'=L'R U'2 LR' F'2
X2= X X = (L'R U2 LR' F2) (L'R U2 LR' F2)
Xa= L'R U2 LR' F'2
Y= LR' F2 L'R U2

The last layer containing the edges you need to flip/permute should be
positioned as the F-face or the U-face depending on the move described
below:
_____________________________________________
Moves for Solving the Last Face edge pieces:

Name: F Tricycle 1
Description: Permutes 3 adjacent edges clockwise on the lower left of
the F-face
No. Edges permuted: 3
No. Edges flipped: 2
Move: (Xa3 X'2)^2

Name: F Tricycle 2
Description: Permutes 3 adjacent edges clockwise on the upper half of
the F-face
No. Edges permuted: 3
No. Edges flipped: 2
Move: (Xa3 X2)^2

Name: U Tricycle 1
Description: Permutes 3 edges clockwise on the U-face
No. Edges permuted: 3
No. Edges flipped: 2
Move: F' X2 Y'2 F

Name: U Tricycle 2
Description: Permutes 3 edges counterclockwise on the U-face
No. Edges permuted: 3
No. Edges flipped: 2
Move: F X'2 Y2 F'

Name: Cross-country Tricycle
Description: Permutes 3 edges across the U and F faces
No. Edges permuted: 3
No. Edges flipped: 1
Move: (X2 X'2)^4

Name: U Bi-Flip 1
Description: Flips two opposite edges on the U-face
No. Edges permuted: 0
No. Edges flipped: 2
Move: (Xa3 X'2)^3

Name: U Bi-Flip 2
Description: Flips two adjacent edges on the U-face
No. Edges permuted: 0
No. Edges flipped: 2
Move: (X2 X2 X'2)^5

Name: Cross-country Bi-Flip
Description: Flips two edges, one on the U-face, one on the F-face
No. Edges permuted: 0
No. Edges flipped: 2
Move: (Xa3 X2)^3

Name: "W"-Cycle
Description: Permutes all edges on the F-face in a "W" pattern
No. Edges permuted: 5
No. Edges flipped: 2
Move: (X2 X2 X'2)^2

Name: "Figure 8"-Cycle
Description: Permutes all edges on the F-face in a "Figure 8" pattern
No. Edges permuted: 5
No. Edges flipped: 4
Move: (X2 X2 X'2)^4
____________________________________________
Strategy for Solving the Last Face edge pieces:

- You should only need to use F Tricycle and the Bi-Flip moves to
  completely solve the edges.  The F Tricycle move usually needs
  to be applied twice.

If anything is vague/unclear please feel free to request clarification.

-Han Wen-

From cube-lovers-errors@mc.lcs.mit.edu  Tue Nov 24 16:09:26 1998
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: The Cylinder
Date: 24 Nov 1998 18:53:56 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <73evc4$cq0@gap.cco.caltech.edu>
References: <cube-lovers.001e01be155b$9a456940$6bc4b0c2@home.icl.web>

roger.broadie@iclweb.com (Roger Broadie) writes:
>I was given a cylinder here in England in 1981.  I no longer have the
>packaging, but I suspect it was Taiwanese, unless the Hungarians made
>this variant.  It was my first cube puzzle, and its shape was so
>unappealing when disturbed that I put it on one side and got a genuine
>cube to learn on - well, almost genuine: it came from a street trader
>in Regent Street.

There is a Taiwanese manufacture of the octagonal prism.  I have
part of one in my collection.  (Got it when I was 10, and many
cubies have disappeared since then.)

I also have one of the "truncated cubes" mentioned earlier in this
thread.

I find the discussion on these two quite strange, since I always
thought of these as cubes with weird cubies -- no more special than,
say, that spherical "cube" they had a few years back.

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
StethoPHONE, not stethoSCOPE.  What do doctors SEE in those things anyway?

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  1 14:27:08 1998
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To: Cube-Lovers@ai.mit.edu
From: "Andrew R. Southern" <a.southern@ic.ac.uk>
Subject: Uwe Meffert's Re-issueing of Prof. Cube
Message-Id: <E0zkc71-00072b-00.1998-11-30-22-48-11@mail2.svr.pol.co.uk>
Date: Mon, 30 Nov 1998 22:48:11 +0000

Dear Cube Lovers,

I have written a website for Uwe Meffert (with input from both W. David
Joyner and David Byrden) that can be found at:

http://www.ue.net/mefferts-puzzles/

and was speaking with him earlier today.

Uwe is going to make another batch of Professor Cubes (5x5x5) in the next
week or so, and is taking orders through his site.

This is a subject that is often raised on the newsgroup, and I hope people
don't think of this as taking too much of a liberty.

The website contains a credit card order page, information about the puzzles
(including a solution to all of his popular puzzles) and multiple links to
other pages. Whilst I am not involved with the day to day running of the
website, if people would like their pages added to the links, please forward
the URL to this address WITH A SHORT SUMMARY that will appear with the link.

Puzzle available include some of the more recent ones (Orbix, Pyramorphix,
Megaminx, Prof Cube).

I am told that orders *may* still be on time for delivery before Chirstmas.

I hope this has been of use to you guys,

Andy Southern.
a.southern@ic.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  1 15:45:22 1998
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Date: Mon, 30 Nov 1998 22:09:18 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Method for Solving the Professor's Cube (5x5x5)
To: Cube Lovers <Cube-Lovers@ai.mit.edu>
Cc: Charles Lin <clin@ibm.net>, Keith Miller <Keith_A_Miller@amat.com>

Hi,

Okay, another so what, big deal.  I finally solved the Professor's
Cube. For those who may not be familiar, the Professor's Cube is a
5x5x5 Rubik's cube. Whew that was hard.  It took me a good 4 days to
figure out all the moves.  Gees, it made the Megaminx seem like
child's play in comparison.  Once again, Noel Dillabough's Puzzler
program was an invaluable tool to visualize and experiment with
various moves.  Thanks Noel!

For those brave souls who would like to conquer this beast, the
following solution may provide some enlightenment.  It's a layers
solution, in contrast to the corners-first solution that I have seen
posted on various web sites. Good luck to you.  The Professor's Cube
is a truly challenging puzzle.
______________________________________________________
Method for Solving the Professor's Cube (5x5x5)

I will use Noel Dillabough's system for referring to various slices or
layers, as described in his Puzzler's F1 help.
________________________________________
Notation:

U - The upper slice
u - One slice away from the upper slice
e - The equator slice
d - One slice away from the lower slice
D - The lower slice

L - The leftmost slice
l - One slice away from the leftmost slice
m - The middle slice
r - One slice away from the rightmost slice
R - The rightmost slice

F - The facing slice
f - Once slice away from the facing slice
M - The facing middle slice
b - One slice away from the back face
B - The back slice

I will use the words "slice" and "layer" synonymously. A "face" is one
of the six outer slices; namely, U, D, L, R, F or B.  Rotations of the
middle slices e, m or M will be in the same direction as the U, R and
F faces, respectively.

Let y denote one of the slices.
y - represents a clockwise 1/4 turn of the y-slice
y' - represents a counterclockwise 1/4 turn of the y-slice
y2 - represents a clockwise 1/2 turn of the y-slice
(For example, Rrm represents clockwise 1/4 turns using the RIGHT-hand
of the R, r and m slices.  Ll represents clockwise 1/4 turns using the
LEFT-hand of the L and l slices.)

Finally, let's consider the pieces or cubelets on any given face.
There are four types of cubelets: corners, edges, centrals and a
center.  For a given face, there are 4 corners, 12 edges, 8 centrals
and 1 center.  With these four types and the intersection of any two
slices using Dillabough's notation, we can specify the location of any
cubelet.  For example, consider the F-face:

LU-corner: the corner cubelet on the upper left-hand corner of the
F-face
re-central: the central cubelet adjacent and to the right of the
center cubelet of the F-face
_______________________________________
First Layer (U slice):

Solving the first layer is fairly straightforward.  Basically the same
as solving the Rubik's cube.  The central pieces are the only thing
really different.
_______________________________________
Second Layer (u slice):

1. First, solve for the mid-central pieces (F-face mu, B-face mu,
L-face Mu, R-face Mu).  Get one of the mid-central piece on the same
color face, and then rotate it into position by using the "free lane"
from the opposite face.  For example, let's say we want have a
mid-central piece at the re position of the F-face.  Use the D-"free
lane" of the B face to position the mid-central piece without
affecting your newly completed U slice, by moving: B2 U2 F' U2 B2.

2. Now, solve for the left and right central pieces (F-face lu, ru,
L-face bu, fu, etc).  Here's where we'll use a genuinely new move.
Position one of the left/right central pieces on the D-face so that it
and the position you want to move the cubelet into lie in the save
vertical slice.  For example, let's say we want to move the left
central cubelet into the F-face lu position.  Position the left
central cubelet at the D-face lb position and perform the following
u-layer DF Swing move:

>From the D-face lb position: l d' l' d' l d2 l'

See how that works?  The corresponding move at the D-face rb position
is:

>From the D-face rb position: r' d r d r' d2 r

This same concept is used to move the left/right central pieces into
position for both the Second (u-slice) and Fourth (d-slice) layers.

"Hey, what if my left/right central piece is on the F face? How do I
move the piece to the D face so that I can apply this move?"  Good
question.  Position the piece on the F-face ld or rd position and
apply the corresponding move described above.  That should move the
cubelet to the D face where you can then apply the move again to move
it into the correct left/right central position.

3. Finally, solve for the left and right edges (F-face and B-face Lu,
Ru).  Use the classic Rubik's cube move to rotate an D-edge piece into
one of the middle layer edge positions.  Namely, if the cubelet is at
the F-face rD or lD position and the destination position is F-face Ru
or Lu then perform the following:

F-Edge Swing Moves:
Destination position F-face Ru: D' R' D R F' R F R'
Destination position F-face Lu: D L D' L' F L' F' L
_______________________________________
Third Layer (e slice):

1. Solve for the left/right central pieces (F-face le, re, L-face be,
fe, etc).  You'll notice that the DF Swing moves will not work here.
Darn.  Instead, we'll use the F-Edge Swing move adapted for the l and
r slices.  Position the cubelet at the F-face md position then perform
the following:

F-Central Swing Moves:
Destination position F-face Re: d'r'dD rR f'F' r fF r'R'
Destination position F-face Le: d l d'D' l'L' fF l' f'F' lL

"Hey,  what if my left/right central piece is on the D face? How do I
move the piece to the F face so that I can apply this move?"  Same
problem.  Position the cubelet at the D-face rM position then apply
the Re F-Central Swing move.

2. Solve for the left and right edges (F-face and B-face Le, Re).
Again, a slight variation of the F-Edge Swing move will do.  Position
the edge piece on the F-face mD position and perform the following:

e-Layer F-edge Swing Moves:
Destination position F-face Re: D' R' D rR F' R F r'R'
Destination position F-face Le: D L D' L'l' F L' F' Ll
______________________________________
Fourth Layer (d slice):

1. First, solve for the mid-central pieces (F-face md, B-face md,
L-face Md, R-face Md). This is one of the most difficult steps.  The
mid-central pieces will be on either the d-slice or on the D-face. To
move them into there correct positions, you'll need to use a few
modified Rubik's cube moves:

Place the D-face as the U-face when applying these moves:

The following sets of cubelets are affected by these moves:
cL = (central L-face Lu, edge U-face LM and central U-face lM)
cR = (central R-face Ru, edge U-face RM and central U-face rM)
cF = (central F-face mu, edge U-face mF and central U-face mf)

Mid-central Tricycle:
move: T2(U) = F2 f2 Uu Ll r'R' F2 f2 L'l' rR Uu F2 f2
action: Permutes the three sets of cubelets (cL, cR, cF) clockwise:

Mid-central Bi-Flip Tricycle:
move:  S2(B) = L'l' rR bB Ll r'R' U2u2 L'l' rR Bb Ll r'R'
action: Permutes the three sets of cubelets (cL, cR, cF) clockwise and
flips the cR and cF sets.  Let's clarify "flipping".  Let's say for
the cR set you have the colors: blue, (blue, yellow), yellow
corresponding to the three cubelets. After flipping the cR set you'll
have the colors: yellow, (yellow, blue), blue.

Use these two moves to position all the mid-central pieces for the
Fourth Layer.  Now, if you're lucky, and Murphy's Law says that you
will be, you may end up in a configuration where you'll have three of
the mid-central pieces positioned properly, but the fourth mid-central
position will be on the D-face.  Okay, now we're going to start having
fun.  Position the central cubelet at the D-face lM position (i.e. on
the left-hand side).  Place the D-face as the U-face and then apply
the following sequence of moves:

S2(B) T2(U') U2 T2(U) S2(B') U' S2(B')

Yes, all that trouble just to move one mid-central cubelet from the
U-face to the F-face.

2. Whew, congratulate yourself if you've made it this far.  Now, solve
for the left/right central cubelets, (F-face ld, rd, L-face bd, fd,
etc).

Position the left central cubelet at the D-face lf or rf position and
perform the following d-layer DF Swing move:

>From the D-face lf position: l d l' d l d2 l'
>From the D-face rf position: r' d' r d' r' d2 r

3. Solve for the left and right edges (F-face and B-face Ld, Rd).
Again, a slight variation of the F-Edge Swing move will do.  Position
the edge piece on the F-face lD or rD position and perform the
following:

d-Layer F-edge Swing Moves:
Destination position F-face Rd: D' R' D mrR F' R F m'r'R'
Destination position F-face Ld: D L D' L'l'm F L' F' Llm'
______________________________________
Fifth Layer (D slice):

1. Solve for the corner cubelets using standard Rubik's cube moves.
First, position the corners in their correct locations using the usual
corner swappers:

Adjacent corners swap: R' D' R F D F' R' D R D2
Diagonal corners swap: R' D' R F D2 F' R' D R D

And then rotate or twist the corners in position using Sune's move:

Sune's 3-corner twister: : R' D' R D' R' D2 R D2

2. Solve for the mid-edges (mF, RM, mB, LM) using a slight
modification to the Tricycle moves.

Place the D-face as the U-face when applying these moves:

Mid-edge Tricycle:
move: F2 U Ll r'R' F2 L'l' rR U F2
action: Permutes the three edges (LM, RM, mF) clockwise:

Mid-edge Bi-Flip Tricycle:
move: L'l' rR B Ll r'R' U2 L'l' rR B Ll r'R'
action: Permutes the three edges (LM, RM, mF) clockwise and flips the
RM and mF.

3. Solve for the left/right edges (lF,rF, Rf, Rb, lB, rB, Lf, Lb).
Now, we're going to have some serious fun. The hardest part of this
step is not getting lost while performing the long sequence of moves.
Also while spinning all these slices, another difficulty is preventing
the cube from exploding and keeping the central pieces from twisting
around.

Again, place the D-face as the U-face with applying these collection
of moves:

LR-edge Tricycle:
move: F2 U Lm'R' F2 L'mR U F2
action: Permutes the three pairs of edges ((Lf,Lb), (Rf,Rb), (lF,rF))
clockwise:

LR-edge Bi-Flip Tricycle:
move: L'mR B Lm'R' U2 L'mR B Lm'R'
action: Permutes the three pairs of edges ((Lf,Lb), (Rf,Rb), (lF,rF))
clockwise and flips the Rf, Rb, lF and rF.

To get those last remaining cubelets in place, a few more exotic moves
are necessary:

Definitions:
T(x) = F2 U x F2 x' U F2
x1 = L r' R'
x2 = L l R'
x3 = L m' R'
(T(x) is a generalized form of the Mid-edge Tricycle)

X1 = T(x1) T(x1) T(x1)
X2 = T(x2) T(x2) T(x2)
X3 = T(x3)

Name: Double pair F swap
Description: Swap two pairs of edges: (lF - Lb)  and (rF - Rb)
Move: X2 X1

Name: Double pair F cross swap
Description: Swap two pairs of edges: (lF -Lf ) and (rF -Rf )
Move: X1 X2

Name: Double pair R swap
Description: Swap two pairs of edges: (Rb - Lf) and (Rf - lF)
Move: X2 X1 X3

Name: Double pair R cross swap
Decription: Swap two pairs of edges:  (Rb - rF) and (Rf - Lb)
Move: X1 X2 X3

Name: Double pair L swap
Description: Swap two pairs of edges: (Lb - Rf) and (Lf - rF)
Move: X3 X2 X1

Name: Double pair L cross swap
Description: Swap two pairs of edges: (Lb - lF) and (Lf - Rb)
Move: X3 X1 X2

Name: LRL-edge Bi-Flip Tricycle
Description: Permutes (lF, Lf, Rf) edges clockwise and flip lF and Lf
edges
Move: X3 X1

Name: LLR-edge Bi-Flip Tricycle
Description: Permutes (lF, Lb, Rb) edges clockwise and flip Lb and Rb
edges
Move: X1 X3

Name: RRL-edge Bi-Flip Tricycle
Description: Permutes (rF, Lf, Rf) edges clockwise and flip Lf and Rf
edges
Move: X2 X3

Name: RLR-edge Bi-Flip Tricycle
Description: Permutes (rF, Lb, Rb) edges clockwise and flip rF and Lb
edges
Move: X3 X2

With these collection of moves, you should be able to finish off the
Professor's Cube!   *Sigh*

-Han-

P.S. Thanks "Professor" Meffert.  For those folks like myself who have
wrestled and completed your 5x5x5 cube, we can only ask and plead,
"What's Next?!!"  :)

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  1 19:18:47 1998
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Date: Mon, 30 Nov 1998 23:13:51 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812010413.XAA11878@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: new types of cyclic shifters

a few months ago, i introduced the position  superflip composed with
four spot  and showed that it had a new type of cyclic shifting
property.  i've now found some new ways to generalize cyclic shifting,
and this in turn suggests some new positions to consider.

first, some brief review.

the position  superflip  is central, so it commutes with all turns.
therefore, if  x y  produces superflip, so does  y x .  we can shift
one turn at a time, removing the first turn of the sequence and
shifting it to the end.  in other words, if  m  produces superflip,
then  x m = m x  for all turns  x .  clearly, any position  m  with
the property that  x m = m x  for all turns  x ,  is central, so the
only such positions are  superflip  and  start.

i showed in an earlier message, that if  x  is any turn and  m  is
superflip composed with four spot, then  x m = m y , where  y  is
x  conjugated by the cube rotation  C_U2 .  more generally, we can
ask for positions  m  such that for any turn  x , there is another
turn  y  satisfying  x m = m y .  for such a position, we can
cyclically shift any maneuver, one turn at a time, by replacing
the turn  x^(-1)  at the beginning with the corresponding  y^(-1)
at the end.  some other positions with this property are:  four
spot, six spot, six spot composed with superflip.

a new way to generalize this is to consider positions  m  such that
for any turn  x ,  we have  x m = n x , where  n  is the same
pattern as  m , but perhaps in a different orientation.  for such
a position, we can cyclically shift any maneuver, by shifting the
first turn to the end, and then conjugating by the appropriate cube
symmetry.  for example, consider the position in which the UFR corner
is twisted clockwise, and the other seven corners are twisted
counterclockwise.  (i'll call this "1-7-twist" for now, but this
pattern needs a better name.)  this position is created by

     U  F2 B' U  B  U  D2 R2 U2 B  U' D2 B  U  F2 B  L2 B2

now, cyclically shift the  U  at the beginning to the end to get

     F2 B' U  B  U  D2 R2 U2 B  U' D2 B  U  F2 B  L2 B2 U

which produces a different orientation of the same position; this
time, the ULF corner is twisted clockwise.  now conjugate this
maneuver by  C_U  to get

     R2 L' U  L  U  D2 B2 U2 L  U' D2 L  U  R2 L  F2 L2 U

which produces the original position, in its original orientation.
actually, there are 3 cube symmetries by which one could conjugate,
since the position has 3-fold symmetry.  another position with
this type of cyclic shifting property is  1-7-twist composed with
superflip.

we can combine both types of generalizations, and ask for positions
m  that have the property that for any turn  x , we have  x m = n y ,
where  y  is another turn, and  n  is the same pattern as  m , but
perhaps in a different orientation.  for such positions, we can
cyclically shift any maneuver by replacing  x^(-1)  at the beginning
of the maneuver by the corresponding  y^(-1)  at the end, and then
conjugating the whole maneuver by the appropriate cube symmetry.
two such positions are:  1-7-twist composed with four spot, and
1-7-twist composed with four spot composed with superflip.

here's all the examples of cyclic shifters that i know, along with
minimal maneuvers:

1. central positions

start  (0q*, 0f*)

superflip
 R' U2 B  L' F  U' B  D  F  U  D' L  D2 F' R  B' D  F' U' B' U  D' (24q*, 22f)
 F' B' D2 L' B2 L2 F2 U' D  B' D2 R  L  D' F2 U' L2 D' F2 D'  (20f*, 28q)

2.
four spot   F2 B2 U  D' R2 L2 U  D'  (12q*, 8f*)
six spot    F  B' U  D' R  L' F  B'  (8q*, 8f*)

four spot composed with superflip
 U2 D2 L  F2 U' D  R2 B  U' D' R  L  F2 R  U  D' R' L  U  F' B'  (26q*, 21f)
 F  U2 R  L  D  F2 U  R2 D  F2 D  F' B' U2 L  F2 R2 B2 U' D   (20f*, 28q)

six spot composed with superflip
 R' U  D  R' U  F' D  R' B  U' L' U' F' D  F' B' D' R' F  D  F  D' R2
                                                               (24q*, 23f)
 U2 F  B' R  F  L2 F2 D  B2 D2 R2 B' L2 F' D2 R2 D' B  R  B2  (20f*, 30q)

3.
1-7-twist
 F  R' U' L' F' U' B' L' U' R2 F  L' D' R' F' D' B' R' D' L2  (22q*, 20f)
 F  R2 L' F  L  F  B2 U2 F2 L  F' B2 L  F  R2 L  D2 L2  (18f*, 26q)

1-7-twist composed with superflip
 F  R  F' R  U  B  L  D  B  D' L' D  L  F  B' R  B  D  F  R'  (20q*, 20f)
 F  R  L' B  L  D  F' L2 B  D2 B' L' F2 B' D  B  U  L  B   (19f*, 22q)

4.
1-7-twist composed with four spot
 F  B2 L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D  B' U'  (22q*, 20f*)

1-7-twist composed with four spot composed with superflip
 F  U' R' L  F' U  R  L' U2 D' B  R  L  F' R  D' R' F2 L  U'  (22q*, 20f*)

as usual, i give a maneuver which is minimal in both metrics whenever
this is possible.  i don't claim that i've found all positions in
these categories, but these are all that i know.  if you find any others,
they'd be good candidates for positions far from start.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  2 14:51:54 1998
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Date: Mon, 30 Nov 1998 23:39:25 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812010439.XAA11918@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: asymmetric local maxima

based on the previous analysis, i can now give 2 asymmetric positions
that are local maxima, namely

     1-7-twist composed with four spot, and
     1-7-twist composed with four spot composed with superflip.

all previous examples of local maxima had some symmetry (although
jerry bryan recently gave a bunch of new local maxima at distance 12q;
perhaps these contain some other examples.)

to show that a position is locally maximal, i must give a minimal
maneuver that ends with each possible quarter turn.  bear in mind
that  F2 = F F = F' F'.

1-7-twist composed with four spot:
 F  L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D  B' U' F2        (22q*)
 U' R' D  L  F' U' L' B  U' B' U  F2 U' D  F  R' U' F' L  R2        (22q*)
 D' L  B' D' L' F  B2 D' F' U  B  R' D' B' L  D' L' D  R2 U         (22q*)
 F  B2 L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D  B' U'        (22q*)
 R' U  L' F  B  D' F  L' D  B' L  F' L' F  D' B  D' B  L' U  R' B   (22q*)
 U' F  L' U' B' L  D  R' U' R  L2 F' U' L' F  U' D  F2 U  B'        (22q*)
 U' R' D  L  F' U' L' B  U' B' U  F2 U' D  F  R' U' F' R2 L         (22q*)
 U' F' L  U' L' U  R2 U' D  R  B' U' R' F  B2 U' F' D  B  L'        (22q*)
 F' U' F  B2 L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D         (22q*)
 L  B' D' L' F  B2 D' F' U  B  R' D' B' L  D' L' D  R2 U  D'        (22q*)

1-7-twist composed with four spot composed with superflip:
 R' D' R  B' R  L  F  U2 D' R  L' U  B' R' L  U' B  U' R  F2     (22q*)
 B  U' R  U' F  B' R' U  F' B  U2 D' L  F  B  R' B  D' B' R2     (22q*)
 B  D  R  B  R  L' U  F' B' L' F  L  D' B2 U  R' B  L  B  D' U   (22q*)
 F  U' R' L  F' U  R  L' U2 D' B  R  L  F' R  D' R' F2 L  U'     (22q*)
 U' D2 L  F  B  R' F  U' F' R2 B  D' L  D' F  B' L' D  F' B      (22q*)
 U' F  L' F  B  R  U' D2 F  B' D  L' F' B  D' L  D' F  R2 B'     (22q*)
 U' F  L2 B' D' B  L' F  B  R  U2 D' F' B  U  L' F  B' U' L      (22q*)
 B' D  R  L' U' D2 F  R  L  B' R  U' R' B2 L  D' F  D' R  L'     (22q*)
 R  L' D  F' R' L  D' F  D' R  B2 L' U' L  B' R  L  F  U' D2     (22q*)

these positions are also strong local maxima in the face turn metric.

1-7-twist composed with four spot:
 U  D' B2 D  F' D' F  R' D' B' R  U  L' D' R2 L  F' D' R' F   (20f*)
 F  L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D  B' U' F2  (20f*)
 U' R' B  D  F' U' F  B2 L' U' B' L  U' D  L2 U  R' U' R  F'  (20f*)
 F' U' L' B  U' B' U  F2 U' D  F  R' U' F' R2 L  U' L' D  R   (20f*)
 D  R' D' R  B' D' L' B  U  F' D' F  B2 R' D' B' R  U  D' R2  (20f*)
 F  B2 D' F' U  B  R' D' B' L  D' L' D  R2 U  D' R  F' D' R'  (20f*)
 D' L  B' D' L' F  B2 D' F' U  B  R' D' B' L  D' L' D  R2 U   (20f*)
 D' B  D  F2 B  R2 B2 D  F2 B' D  B  D' R2 U2 B2 U2 D2 F' U2  (20f*)
 F  B2 L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D  B' U'  (20f*)
 L2 U  R2 B  D2 B  L2 U' F' B2 D  F  D' R2 L2 D2 B2 L2 D' B   (20f*)
 B  L2 U  F' L2 U' F' B  U  F  D' L2 U  F2 D  F2 D' R2 U2 B2  (20f*)
 U' F  L' U' B' L  D  R' U' R  L2 F' U' L' F  U' D  F2 U  B'  (20f*)
 U' R' D  L  F' U' L' B  U' B' U  F2 U' D  F  R' U' F' R2 L   (20f*)
 D2 F2 U' L2 U  L2 D  B2 U' L  D  R  L' D' B2 L' D  B2 R  L2  (20f*)
 U' F' L  U' L' U  R2 U' D  R  B' U' R' F  B2 U' F' D  B  L'  (20f*)
 F' U' F  B2 L' U' B' L  U' D  L2 U  R' U' R  F' U' L' F  D   (20f*)
 F2 B  R2 D  F' R2 D' F' B  D  F  U' R2 D  F2 U  F2 U' L2 D2  (20f*)
 L  B' D' L' F  B2 D' F' U  B  R' D' B' L  D' L' D  R2 U  D'  (20f*)

1-7-twist composed with four spot composed with superflip:
 R  L  F' L  U' L' F2 R  D' B  D' R' L  B' D  R  L' U' D2 F   (20f*)
 R' D' R  B' R  L  F  U2 D' R  L' U  B' R' L  U' B  U' R  F2  (20f*)
 R  L  F  U' D2 R' L  D  B' R  L' D' B  D' L  F2 R' U' R  F'  (20f*)
 R  F' L  U' L' F2 R  D' B  D' R' L  B' D  R  L' U' D2 F  R   (20f*)
 B  U' R  U' F  B' R' U  F' B  U2 D' L  F  B  R' B  D' B' R2  (20f*)
 F' B  D' L  D' F  R2 B' U' B  R' F  B  L  U' D2 F' B  D  R'  (20f*)
 F  D' B' R2 D' F  R  U  B' L  D2 B  R2 U2 B2 U' R  L2 B  U   (20f*)
 R2 U' F  B2 R  U  R  D' L' B2 D' R  B  U  L' F  D2 L  B2 U2  (20f*)
 F  U' R' L  F' U  R  L' U2 D' B  R  L  F' R  D' R' F2 L  U'  (20f*)
 U' D2 L  F  B  R' F  U' F' R2 B  D' L  D' F  B' L' D  F' B   (20f*)
 B  L  U  L  D' R' F2 D' L  F  U  R' B  D2 R  F2 U2 R2 U' B2  (20f*)
 U' F  L' F  B  R  U' D2 F  B' D  L' F' B  D' L  D' F  R2 B'  (20f*)
 U' F  L2 B' D' B  L' F  B  R  U2 D' F' B  U  L' F  B' U' L   (20f*)
 D' F2 B  R  D  R  U' L' F2 U' R  F  D  L' B  U2 L  F2 D2 L2  (20f*)
 B' D  R  L' U' D2 F  R  L  B' R  U' R' B2 L  D' F  D' R  L'  (20f*)
 R  L' U' D2 F  R  L  B' R  U' R' B2 L  D' F  D' R  L' F' D   (20f*)
 R  L' D  F' R' L  D' F  D' R  B2 L' U' L  B' R  L  F  U' D2  (20f*)
 F  D' R  B2 L' U' L  B' R  L  F  U' D2 R' L  D  B' R  L' D'  (20f*)

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  2 20:31:57 1998
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Date: Wed, 2 Dec 1998 00:49:45 -0500
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From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject: Re: Method for Solving the Professor's Cube (5x5x5)
To: Cube Lovers <Cube-Lovers@ai.mit.edu>

It is difficult.  I also took a long time to solve it, but I used a very
different solution.  Check out www.wunderland.com/WTS/Jake/5x5x5.html for
my solution.  Perhaps you can combine the best moves of both solutions to
find a way of solving this interesting puzzle that you find most pleasing.
If you do use my solution and have any comments about how I can make it
better, either in my writing or my moves, please let me know.

-Jacob

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec  3 15:12:43 1998
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Method for Solving the Professor's Cube (5x5x5)
Date: 2 Dec 1998 18:21:22 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <7440f2$q5v@gap.cco.caltech.edu>
References: <cube-lovers.19981201060918.6975.rocketmail@send104.yahoomail.com>

Han Wen <hansker@yahoo.com> writes:
>For those brave souls who would like to conquer this beast, the
>following solution may provide some enlightenment.  It's a layers
>solution, in contrast to the corners-first solution that I have seen
>posted on various web sites. Good luck to you.  The Professor's Cube
>is a truly challenging puzzle.
>______________________________________________________
>Method for Solving the Professor's Cube (5x5x5)

[snip]

Well, since we're sharing solutions, here's my solution to the 5x5x5:

First, a preliminary exercise that
should be mastered before the solution is attempted:

Let's ignore all the corners and all the cubies adjacent or
diagonally adjacent to the corners.  (In other words, ignore the
"supercorners," where "super" is a prefix meaning "two layers deep.")
Ignore the centers, too.  Paint all of them black, if you want.  :-)

Now, all we have left are the 12 "superedges."  Each superedge is
composed of a normal edge piece and two attached edge centers.  Or,
in other words, each of the 24 edge centers are attached to
an edge piece face.  In a normal messed-up cube, these edge
centers will not match their edge piece faces.  Our goal in this
exercise will be to match all the edge centers with their edge piece
faces.

Note that superface turns never destroy an edge center pairing.

Now, consider the following sequence of moves:
1.  Rotate any face (NOT superface) 180 degrees.
2.  Turn any center slice (as much as you want).
3.  Rotate the same face in step 1 180 degrees.
    (i.e., perform the inverse move of step 1.)

Now, if you chose the center slice to be parallel to the face,
obviously this sequence doesn't do anything.  Ditto for when you
turned the center slice some multiple of 360 degrees.

In all other cases, this will essentially perform two swaps of
edge centers.  Step 1 swaps two pairs of edge centers all around the
face, but one of those swaps gets undone by Step 3.  Step 2
moves the other pair out of the way and puts another pair in its
place to be swapped again.  So, if you choose wisely, you can
increase the number of correctly matched edge center pairs by this
move.

Many of these moves, interspersed with superface turns, will
allow you to match all the edge center pairs.  Practice this on
your cube.

Thus ends the preliminary exercise.  Note that all the moves in
the exercise do not disturb the individual supercorners (well, one
move does for a bit, but then it undoes the damage) but does change
their orientation with respect to each other.

Now, the solution!
Step 1.  Ignore the superedges and the centers.  You now have what
  is equivalent to a 4x4x4.  Solve it.
Step 2.  Match the edge centers with the edges as detailed in the
  preliminary exercise.
Step 3.  You now have a cube with correct supercorners (as done in
  step 1) and correct superedges (as done is step 2).  This means
  that your cube is equivalent to a 3x3x3, using only superface
  turns.  Solve it.
Step 4.  Tada!  Your 5x5x5 is now solved.

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
O*e T*o: "Thre* *our fi*e s*x; se*en *ight *ine, *en!"

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec  3 16:32:04 1998
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Date: Wed, 2 Dec 1998 23:15:57 -0700
To: Cube Lovers <Cube-Lovers@ai.mit.edu>
From: Steve LoBasso <slobasso@dtint.com>
Subject: Re: Method for Solving the Professor's Cube (5x5x5)

Although I use a different method, centrals first, edge combinations, edge
parity corrections, finish using 3x3x3 solution.

I was playing with a layered solution last week also, amazing coincidence.
Much of my solution is the same as Han's.

Most of the differences are in the 4th and 5th layers.
To move the 4th layer mid centrals into place:

central D-face bm to central F-face dm: F l D l' D' F'

If there are no central pieces in the bottom central area, simply move a
bottom central up causing another central to go down.

To move 4th layer edges into place:

edge L-face Db to edge F-Rd: R' D' r D R D' r'
edge R-face Db to edge F-Ld: L D l' D' L' D l

My 5th layer edge moves are a bit different but I haven't had time to write
them down with this terminology.

--
Steve LoBasso                          mailto:slobasso@dtint.com
Digital Technology International    or mailto:slobasso@hotmail.com
500 West 1200 South, Orem, UT, 84058   http://members.tripod.com/~slobasso
(801)226-6142  ext.265                 FAX (801)221-9254

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec  3 20:49:15 1998
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Date: Thu, 3 Dec 1998 17:47:33 -0500
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From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject:  (5x5x5) edge parity corrections
To: Cube Lovers <Cube-Lovers@ai.mit.edu>

I don't like the edge parity correction move that I use in my solution, and
I'm hoping that someone can give me a better one.

The parity problem is found in 5x5x5 cubes (and 4x4x4 cubes, I understand)
when two of the edges right next to the corners (which I call "wings") are
switched.  Some fairly simple moves can get all three edges in line with
each other, but half the time two wings need to be switched.  By the time I
figure this out when doing a 5x5x5 cube, I've solved most of it, and my
parity fixing move messes up many of the edges I've been working on.

How do other people fix this problem?

-Jacob

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 11:30:15 1998
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Message-Id: <19981204043241.26555.rocketmail@send106.yahoomail.com>
Date: Thu, 3 Dec 1998 20:32:41 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Re: Method for Solving the Professor's Cube (5x5x5)
To: Steve LoBasso <slobasso@dtint.com>
Cc: Cube-Lovers@ai.mit.edu

Hi,

> To move the 4th layer mid centrals into place:
>
> central D-face bm to central F-face dm: F l D l' D' F'
>
This is a stunningly elegant move.  You've reduce the difficulty in
solving the 4th layer by an order of magnitude.  I tried this move
out, actually it swaps two centrals:

F-face md <-> D-face lM (mid central swap)
F-face rd <-> D-face lf (right central swap)

Beautiful move.

There is one particular move that I haven't figured out yet.  It pops
up occasionally when I solve the edges of the D-layer.  Sometimes I
end up with every cubie in place except for two right centrals on
adjacent faces. For example: F-face rD and L-face Lf.  The two pieces
only need to be swapped.  No flipping is needed.  Does anyone know how
to perform this move?  I've been beating the Puzzler program for a
while, but I have been unsuccessful so far.
______________________________________________
Han Wen
Applied Materials
3050 Bowers Ave, MS 1145
Santa Clara, CA  95054
e-mail: Han_Wen@amat.com / hansker@yahoo.com

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 13:17:06 1998
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Date: Thu, 3 Dec 1998 23:58:19 -0500 (EST)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199812040458.XAA26116@Twig.Rodents.Montreal.QC.CA>
To: Cube-Lovers@ai.mit.edu
Subject: Re: (5x5x5) edge parity corrections

> The parity problem is found in 5x5x5 cubes (and 4x4x4 cubes, I
> understand) when two of the edges right next to the corners (which I
> call "wings") are switched.

Yes, it does occur equally on the 4-Cube.  Though I have never seen
one, I feel certain that similar parity problems will occur on all
higher-order Cubes as well, though above order 5 there will be multiple
distinct types of "wings", each of which will have its own comparable
potential problem.

Note that the problem goes away entirely if cube faces are marked such
that symmetrically placed face cubies are not visually
indistinguishable, because the parity problem in question always occurs
in conjunction with a similar parity problem on face cubies, but the
latter is invisible on most cubes.

> [...] half the time two wings need to be switched.

> How do other people fix this problem?

Most briefly, how I do it is to make a single quarter-turn of a slice
containing one of the wing pieces involved, then fix up the damage by
moving wings back into place using commutators rather than slice moves.

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 14:26:23 1998
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Method for Solving the 4x4x4
Date: 4 Dec 1998 17:26:00 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <7495v8$3nv@gap.cco.caltech.edu>
References: <cube-lovers.7440f2$q5v@gap.cco.caltech.edu>

I have been told that my solution for the 5x5x5 includes
knowing how to solve the 4x4x4, which is of course not
trivial.  With the post asking about the parity problem, I
thought I might as well post my solution to the 4x4x4.

Yes, the biggest barrier is the parity problem where two
adjacent edge cubies are flipped.  My earliest attempt at a 4x4x4
solution was the following:
 1.  Match all the centers.
 2.  Match all the edges.
 3.  You now have a 3x3x3.  Solve.
Unfortunately, with the parity problem you can often
end up with an unsolvable 3x3x3 by the time you get to step 3.
Any simple moves that fix the parity problem tend to mess up the
rest of the cube quite badly -- I wrestled with this problem
a long time until I realized one thing:

Most solutions of the 3x3x3 treat the centers as static, using them
as "anchors" for the entire cube.  But this is entirely unnecessary!
If you solve the 3x3x3 while IGNORING the centers, you will eventually
get a solved cube where the centers are either in the "6 dots" or
"4 dots" situation well known to cubists -- and these have rather
simple solutions, essentially consisting of a slice turn conjugated
with another slice turn.

So, my most favorite 4x4x4 solution is now:
 1.  Match all the edges.
 2.  Solve the parity problem, if necessary (postpone until after
      step 3 if desired).
 3.  Ignore the centers and treat the cube as a 3x3x3.  Solve.
 4.  Solve the centers.

Okay.  Now to qualify the solution.  Part 1 is simple and can be
 done anyway you wish (the move rF2r'F2 will be rather useful in
 the later stages).  Part 3 is simple, with the caveat that you
 may be treating the "centers" in the wrong manner!

Part 2 stems from the fact that the cube apparently has an "even"
permutation (a 2-cycle involving two edge pieces), an apparent
paradox since 2-cycles should not exist (e.g., on the 3x3x3 it is
impossible to swap exactly two edges).  The reason this is only
an "apparent" paradox, however, is because of the misassumption that
the centers of the 4x4x4 are static, which they certainly are not!
In fact, just rotate one slice incident on your 2-cycle, and you
have magically turned the 2-cycle into a 5-cycle, which is perfectly
solvable!

Personally, I solve the 5-cycle by two or more 3-cycles, which generally
take on the form:
  FR'F' r FRF' r'
This move performs a cycle on the three edges fUR, FUr, and FDr,
without disturbing the corners, but doing rather annoying things to
the centers.  (This move is an extension of the perhaps-not-so-well-known
sequence for the 3x3x3: FR'F'LR'DRD'L'R that rotates 3 edges.)
  For FR'F' you may substitute any sequence of moves that brings
your desired edge piece (in this case fUR) to the FUr position,
as long as it does not disturb any other edges on the r slice (specifically,
the edges FDr, BUr, and BDr).  You will also have to substitute
the inverse of your sequence for FRF'.  As an example, the move
  F'L2F r F'L2F r'
cycles BdL, FUr, and FDr.  You may also use r2 instead of r and r', which
means that BDr is affected instead of FDr.

And finally, step 4: the centers.  This is solved by a
generalization of the "6-dots" rule.
This move creates "6 dots":
  u'r'ur
This permutes bUr, Fur, and buR, as well as their "opposites"
fDl, Bdl, and fdL, while affecting no other cubies.  This is two 3-cycles
on 6 faces, which is rather unwieldy, so I conjugate (is that the
right word?) it with a simple face turn to get
  u'r'ur F r'u'ru F'
which permutes Fur, Fdr, and buR in a simple 3-cycle.  Both Fur and Fdr
are on the same face, which makes this move rather easy to deal with.
Especially, if one of those is the right color already, it can be
involved in the 3-cycle without "increasing error."

I think I may have extended my personal jargon a bit more into this
post -- if you wish to understand anything in this post or, conversely,
would like to teach me more "Standard" jargon, please e-mail me.

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
O*e T*o: "Thre* *our fi*e s*x; se*en *ight *ine, *en!"

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 15:28:52 1998
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Message-Id: <c=US%a=_%p=INEA_Corporation%l=INEADEV-981204174407Z-7528@proxy.ineacorp.com>
From: Noel Dillabough <Noel.Dillabough@ineacorp.com>
To: "'Cube Lovers'" <Cube-Lovers@ai.mit.edu>
Cc: "'Jacob_Davenport@scudder.com'" <Jacob_Davenport@scudder.com>,
        "'noel@mud.ca'" <noel@mud.ca>
Subject: RE: (5x5x5) edge parity corrections
Date: Fri, 4 Dec 1998 12:44:07 -0500

The parity problem can be solved on a 4x4x4 or 5x5x5 by using the
following move (can be pasted into puzzler's move macro):

r2D2l1D2l1D1l3r3d2l1r1D3l3r3d2B2r1B2l3B2l1B2r2

For the 4x4x4, this is all that is needed, but for the 5x5x5, two
crosses (centre edges) are swapped.  So you'll need to use the following
to solve the crosses:

First, get the crosses across from each other with:

F2l3F2e1l2e3l2F2l1F2

Now swap the opposite crosses with:

R2e1l2e3l2R2e1l2e3l2

Parity problem solved...

If anyone has a better solution to this rather long one, let me know,
I'm sure some moves could be shaved off.

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 16:11:19 1998
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Message-Id: <199812041806.NAA21024@pike.sover.net>
Date: Fri, 04 Dec 1998 13:07:36 -0500
To: Jacob Davenport <Jacob_Davenport@scudder.com>
From: Nichael Lynn Cramer <nichael@sover.net>
Subject: Re: (5x5x5) edge parity corrections
Cc: Cube Lovers <Cube-Lovers@ai.mit.edu>
In-Reply-To: <00096300.C22092@scudder.com>

Jacob Davenport wrote:
>I don't like the edge parity correction move that I use in my solution, and
>I'm hoping that someone can give me a better one.
>
>The parity problem is found in 5x5x5 cubes (and 4x4x4 cubes, I understand)
>when two of the edges right next to the corners (which I call "wings") are
>switched.  Some fairly simple moves can get all three edges in line with
>each other, but half the time two wings need to be switched.  By the time I
>figure this out when doing a 5x5x5 cube, I've solved most of it, and my
>parity fixing move messes up many of the edges I've been working on.
>
>How do other people fix this problem?
>
>-Jacob

Hi Jacob

In both cases (4X and 5X) I solve this problem in the following way:

1] I solve the rest of the cube, leaving me with the two "switched wings"
(in your terminology).

2] I then arrange things so both  "wings" are on the same
"off-center-slice".  (Also it will always be the case that both of these
winds are now on the same face.)

This will be easy to do using the 3-wing swapping operators.

3] At this point I now rotate the "off-center-slice" containing the
"switched wings" by a quarter turn.

As a result of this move it will be the case that that the
"off-center-slice" now has one of the previously "switched wings" in its
"correct cubicle".  The other three "wings" will be now be in a cyclic
permutation.

4] Since --from your note above-- I assume you understand how to cycle
three "wings", all you have to do now is put the "wings" in the right place
and replace the damage to the off-center central faces that were messed up
during that initial quarter-turn above.  (And since they are in "paired"
clusters, this should be pretty straightforward.)

(In short, the quarter-turn of the non-central slice puts the cube back in
the proper "orbit" for finishing up.)


Now clearly this is far from maximal.  And it's certainly not terribly
fast.  But I find it a very simple, and an easy (and easy-to-remember [and
easy-to-explain]) way to clean up this potentially messy situation.

Hope this helps
Nichael
--
Nichael Cramer
nichael@sover.net                        deep autumn--
http://www.sover.net/~nichael/                   my neighbor what does she do

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 16:49:43 1998
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Date: Fri, 4 Dec 1998 11:26:10 -0700
To: Jacob_Davenport@scudder.com (Jacob Davenport)
From: Steve LoBasso <slobasso@dtint.com>
Subject: Re: (5x5x5) edge parity corrections
Cc: Cube-Lovers@ai.mit.edu

This should solve the edge parity problem by swapping the
edge F-Ru and edge F-Rd pieces.

R2 d L2 d L2 d' R2 u' F2 u2 F2 u' F2 L2 F l' F' L2 F l F

This move swaps only these two pieces and some centrals, but only within
their face. A variant of this move should be scalable to solve parity
issues in any NxNxN cube.

The only way I can think of to not have the parity problem, or at least not
require such a long series, is to solve centrals last. Another other idea
would be to spot the parity problem much earlier by counting edge flips.
Not very easy for a person to do, but I have seen it done in software for
normal 3x3x3 cubes. If it were were known very early in either the centers
first or layered solution, it would be trivial to fix.

>I don't like the edge parity correction move that I use in my solution, and
>I'm hoping that someone can give me a better one.
>
>The parity problem is found in 5x5x5 cubes (and 4x4x4 cubes, I understand)
>when two of the edges right next to the corners (which I call "wings") are
>switched.  Some fairly simple moves can get all three edges in line with
>each other, but half the time two wings need to be switched.  By the time I
>figure this out when doing a 5x5x5 cube, I've solved most of it, and my
>parity fixing move messes up many of the edges I've been working on.
>
>How do other people fix this problem?
>
>-Jacob



--
Steve LoBasso
Digital Technology International    mailto:slobasso@dtint.com
500 West 1200 South              or mailto:slobasso@hotmail.com
Orem, UT    84058                   http://members.tripod.com/~slobasso
(801)226-6142 ext.265               FAX (801)221-9254

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec  4 17:13:44 1998
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Message-Id: <36683B51.A50833E4@switchview.com>
Date: Fri, 04 Dec 1998 14:43:13 -0500
From: Michael Swart <Michael.Swart@switchview.com>
Organization: Switchview
To: Cube-Lovers@ai.mit.edu
Subject: Re: (5x5x5) edge parity corrections
References: <199812040458.XAA26116@Twig.Rodents.Montreal.QC.CA>

I got this from the archives, it may be relevant to repost it. It's a
way of solving the parity problem:
r2 U2 r l' U2 r' U2 r U2 r l U2 l U2 r U2 l r2 U2
I'm confident you can't do too much better than this.
Mike

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 11:22:45 1998
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Message-Id: <199812042200.RAA02240@pike.sover.net>
Date: Fri, 04 Dec 1998 16:40:51 -0500
To: der Mouse <mouse@rodents.montreal.qc.ca>
From: Nichael Lynn Cramer <nichael@sover.net>
Subject: Re: (5x5x5) edge parity corrections
Cc: Cube-Lovers@ai.mit.edu
In-Reply-To: <199812040458.XAA26116@Twig.Rodents.Montreal.QC.CA>

der Mouse wrote:
>> The parity problem is found in 5x5x5 cubes (and 4x4x4 cubes, I
>> understand) when two of the edges right next to the corners (which I
>> call "wings") are switched.
>
>Yes, it does occur equally on the 4-Cube.  [...]

The appearance is particularly striking on the 4X cube.  Especially in the
situation where the two out-of-place "wings" are side-by-side.

It looks very similar to a solved 3X cube with a single edge-cubie flipped.
This is an interesting state to leave your cube in, when it is just lying
around your office, for visitors to find.

N

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 12:38:33 1998
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Message-Id: <199812051241.VAA08521@soda3.bekkoame.ne.jp>
Date: Sat, 5 Dec 1998 21:47:39 +0900
To: Cube-Lovers@ai.mit.edu
From: Ishihama Yoshiaki <ishmnn@cap.bekkoame.ne.jp>
Subject: 4DRubik Cube

I have simulated 4DRubikCube for Macintosh.
It is madeup of 2x2x2x2 hypercubes.
It is on my HomePage.


//----------------------------------------//
Ishihama Yoshiaki
Tokyo Chofu

E-mail: ishmnn@cap.bekkoame.or.jp (Until 1999/3/31)
            ishmnn@cap.bekkoame.ne.jp ( This is correct address)
HomePage : http://www.asahi-net.or.jp/~hq8y-ishm/
//--------------------------------------//

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 13:31:28 1998
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Message-Id: <366C1ED9.C11@hrz1.hrz.tu-darmstadt.de>
Date: Mon, 07 Dec 1998 19:30:49 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: cube-lovers@ai.mit.edu
Subject: Optimal Cube Solver

New Optimal Cube Solver

I wrote an optimal Cube Solver and experimented with coordinates
different of those I use in my Cube Explorer program or of those in Mike
Reid's Optimal Cube Solver. Its pruning tables are not very large (about
25MB), so the performance is relatively low (at least in comparison with
Mike's program), but I think it is worth to give you some information
about it.

Some general considerations on the use of "coordinates" in cube solving
algorithms first. Instead of representing a state of the cube by the
positions of corners or edges, the use of coordinates not only increases
the speed of computing a face-turn but also serves as an index for the
pruning tables.

If we have an arbitrary subgroup H of the Cube Group G, we map the right
cosets Ha to natural numbers from  0 to ord(G)/ord(H)-1). A face-turn T
(which also is an element from G) now induces a map on these numbers,
which can be implemented as a simple lookup-table. For this to work we
have to ensure that if x=h1*a and y=h2*a are in the same coset Ha, then
x*T and y*T are in the same coset Hb. But this is true because
(x*T)*(y*T)^-1 = (h1*a*T)*(h2*a*T)^-1 = h1*h2^-1 is in H.

If  we take for example H1={all g from G with corner orientations 0,
corner permutations and edges arbitrary} the resulting coordinate
(0<=x<2187) represents the orientation of the corners.

It also should be possible to reduce the size of the coordinates by the
48 symmetries of the cube (or at least by a subgroup of the symmetry
group M). This is done by defining equivalence classes on the cosets.
Two cosets Ha and Hb are called equivalent, if there is an m from M with
Hb = m*Ha*m^-1. But to make this definition work we have to ensure, that
the elements of a coset Ha are really all mapped to the same coset Hb by
the conjugation with m. This only is true, if

(1) mHm^-1=H

The subgroup H1 from above  for example does have this property only for
symmetries which do not change the UD-axis in the way the orientations
of the corners are usually defined. So the corner orientation coordinate
can only be reduced by 16 symmetries. Is it possible to define the
corner orientations in another way, so that (1) holds for all 48
symmetries? I do not believe it, but I do not know how to prove this.
For the analogous case of the edge orientations there is a possibility
to define the orientations in a way (different to the way usually used)
which allows reduction by all 48 symmetries: every quarter turn changes
the orientation of any involved edge.

In my program I use 3 coordinates. The first (let's call it the
X2-coordinate) is defined by the subgroup, where the edges are arbitrary
and the corners are generated by <U2,D2,R2,L2,F2,B2>. There are 918540
different cosets. Because (1) holds for all m, they can be reduced by
all 48 symmetries and we get 19926 equivalence classes.

The second coordinate is the edge orientation defined by the subgroup
{all g from G with edge orientations 0, edge permutations and corners
arbitrary}. There are  2048 cosets. I do not reduce them by symmetries
because the number is relative small.

The third coordinate describes the edge permutation. Because there are
12! coordinate values, even reduction by 48 symmetries still gives too
many coordinate values. So for use in a turntable we define two edge
permutations a and b equivalent, if a=m1*b*m2, were m1 and m2 are in M.
In this way we get 208816 equivalence classes c. If now m1*c*m2 is a
(not necessarily unique) representation of an edge permutation applying
a faceturn T is done like that:

(m1*c*m2)*T = m1*c*[m2*T*m2^-1]*m2 = m1*[c*T']*m2=
[m1*m1']*c'*[m2'*m2]=m1''*c'*m2''

The operations in square brackets are done by table lookups:
[m2*T*m2^-1]:=T',  [c*T']:=m1'*c'*m2', [m1*m1']:=m1''  and [m2'*m2]:=
m2''.

A cube, which has all three coordinates zero, is in a subgroup with 96
elements, were the edges are in place and the corner orientations are
correct. To find such states, I use two pruning-tables. The first
combines the X2-coordinate and the edge-orientation coordinate which
takes 19926*2048/2=20404224Bytes of memory (we only need 4 bit per
entry). The maximal table entry is 12, with an average of about 9.5. The
second is a pruning table for the edge-permutation. It takes
208816*48/2=5011584Bytes, the maximal table entry is 10 (so it takes not
more then 10 faceturns to position all edges ignoring the orientations).

The program produces about 1 million nodes per second on a P350 and a
depth 15 search is done in about 4 minutes (depending on the situation).
So a complete depth 18 search will need a few days which of course is
not very satisfying. A possible improvement could be to use the subgroup
<U2D2,R2L2,F2B2> instead of <U2,D2,R2,L2,F2,B2> for the first
coordinate. The subgroup has only 4 elements, so the coset-space has 24
times the size. The pruning table will need about 480MB instead of 20MB
which is above that what is possible for me in the moment. But a
complete depth 18 search should be done in about 1/24 of the time which
will be a few hours then.

Herbert

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 14:34:56 1998
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From: jmb184@frontiernet.net (John Bailey)
To: ishmnn@cap.bekkoame.or.jp (Ishihama Yoshiaki)
Cc: Submissions Cube-Lovers <Cube-Lovers@ai.mit.edu>
Subject: Re: 4DRubikCube
Date: Sat, 05 Dec 1998 13:10:10 GMT
Message-Id: <36692f00.213266943@mail.frontiernet.net>
References: <ishmnn-0512981857030001@p849359.mitk.ap.so-net.ne.jp>

On Sat, 05 Dec 1998 18:57:03 +0900, in rec.puzzles you wrote:

>I have created 4Dimension Rubik Cube for Macintosh.
>URL: http://www.asahi-net.or.jp/~hq8y-ishm/

I went there the instant I read your post.  Unfortunately, I am
running a Pentium based machine.  Could you put a gif image of your
cube on the page? Maybe even a screen copy bitmap of the user
interface. We IBM-PC types can stare and drool.
If you haven't checked out my 2x2x2x2 cube, it's at
http://www.ggw.org/donorware/4D_Rubik

John
http://www.frontiernet.net/~jmb184

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 16:03:32 1998
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Date: Fri, 04 Dec 1998 17:00:42 -0500
To: cube-lovers@ai.mit.edu
From: Nichael Lynn Cramer <nichael@sover.net>
Subject: Re: Method for Solving the Professor's Cube (5x5x5)
In-Reply-To: <7440f2$q5v@gap.cco.caltech.edu>
References: <cube-lovers.19981201060918.6975.rocketmail@send104.yahoomail.com>


>>Method for Solving the Professor's Cube (5x5x5)
>
>[snip]

This is not a formal solution, but --say when I want to kill some time-- I
often find it entertaining to solve the 5X cube in "ascending spirals".

By which I mean:  Start with the center face on a particular color (I
always start with blue).  Next solve the non-center face cubies, one by
one, in order moving clockwise around the "loop".  When that loop is done,
then solve one of the blue-faced corners and then solve the remaining
blue-sided edge cubies (in order).  Then move up, solving each
parallel-to-the-blue-face internal slice in order; and so on.

Needless to say, this is hardly an optimal solution (in either time or
number of moves).  But think of it as a way to "practice scales"  (Or as I
say, just a good way to kill some time.  ;-)

There are obvious variations on this.  For example, solve the individual
faces in "ascending spirals" like the above, but instead of starting on a
center face cubie, start on a corner cubie and work your way diagonally, in
slices, across the cube toward the opposite corner.

Or, for the truly masochistic, solve the cube --again a cubie at a time--
in a checkboard pattern (i.e. the result of putting the 5X cube through the
Pons Asinorum transformation) doing first the half of the cubies in the
first "phase" and then the cubies in the other.
--
Nichael Cramer
work: ncramer@bbn.com
home: nichael@sover.net
http://www.sover.net/~nichael/

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec  8 19:08:13 1998
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Date: 	Tue, 8 Dec 1998 18:37:02 -0500 (EST)
From: Alchemist Matt <monroem@email.unc.edu>
Reply-To: Alchemist Matt <monroem@email.unc.edu>
To: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Optimal Cube Solver
In-Reply-To: <366C1ED9.C11@hrz1.hrz.tu-darmstadt.de>
Message-Id: <Pine.GSO.3.95L.981208183158.13948B-100000@sunny.isis.unc.edu>

This question is directed to both Herbert and Mike Reid in case he's
reading this list:  With all this discussion of the "Professor Cube"
lately, how hard would it be to extend either Optimal cube solving program
to handle 4x4x4 and 5x5x5 cubes in addition to the traditional 3x3x3?
Considering reasonable table files (50 - 100 mb), how much longer would
the computation time be extended by.  If either of you would find the time
to implement this modification, I would be very interested in trying out
the program.

		Matt

-----------------------------------------------------------------------
Matthew Monroe				Monroem@UNC.Edu
Analytical Chemistry      		http://www.unc.edu/~monroem/
UNC - Chapel Hill, NC                   This tagline is umop apisdn
-----------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  9 12:45:11 1998
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Date: Tue, 8 Dec 1998 23:24:09 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812090424.XAA00740@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Optimal Cube Solver

matt monroe asks

> This question is directed to both Herbert and Mike Reid in case he's
> reading this list:  With all this discussion of the "Professor Cube"
> lately, how hard would it be to extend either Optimal cube solving program
> to handle 4x4x4 and 5x5x5 cubes in addition to the traditional 3x3x3?
> Considering reasonable table files (50 - 100 mb), how much longer would
> the computation time be extended by.  If either of you would find the time
> to implement this modification, I would be very interested in trying out
> the program.

i think it's reasonable to say that an optimal solver for the 4x4x4
(or 5x5x5) is currently far out of reach.  one could write a program
that theoretically finds optimal solutions after running for enough
time.  but it would be feasible only for positions a few turns from
start; other positions would take years, centuries, millenia, ...

on the other hand, a sub-optimal solver is certainly possible.
just teach the computer your favorite method.  this would be more
"sub" than it is "optimal", so next we'd ask to make it as good as
possible.  the real question is: are computer methods superior to
human methods for the larger cubes?  so far, probably not, but not
much work has been done on sub-optimal solvers for larger cubes.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  9 15:20:18 1998
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Date: Tue, 8 Dec 1998 23:36:23 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812090436.XAA00765@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: meffert's web site and puzzles

i'm glad to hear that uwe effert is still making puzzles.
i hope this means he'll make that master pyraminx (which
was once planned) in which edges can also turn!

and while i'm dreaming ... how about a higher order pyramid,
preferably also of the edge-turning as well as peak-turning
variety?  david byrden's web site has some puzzles i'd like
to see made: the icosahedron, the octahedral "oddity", ...

any chance of making any of these?

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  9 16:00:55 1998
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Message-Id: <199812090645.BAA17397@terminus.idirect.com>
From: "Mark Longridge" <cubeman@idirect.com>
To: <cube-lovers@ai.mit.edu>
Subject: My rubik's cube webpage
Date: Wed, 9 Dec 1998 01:49:33 -0500

Hello cube-lovers,

My site has moved to: http://www.snipercade/com/cubeman/index.html

the old site: http://web.idirect.com/~cubeman will be up for a few days
still.

web masters please update your links!

Thanks,

-> Mark

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec  9 17:00:48 1998
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Creative ways of solving the cube
Date: 9 Dec 1998 15:48:18 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <74m642$lb5@gap.cco.caltech.edu>
References: <cube-lovers.199812042200.RAA02263@pike.sover.net>

Nichael Lynn Cramer <nichael@sover.net> writes:
>This is not a formal solution, but --say when I want to kill some time-- I
>often find it entertaining to solve the 5X cube in "ascending spirals".

Although I don't play with my 5x5x5 much, I do play with the 3x3x3 a lot
and have entertained myself by solving it in many different ways.

The canonical methods:
  1.  First level, second level, third level
  2.  Centers, corners, edges

After much more understanding, however, I now try different techniques
for entertainment.  In order of approximate difficulty:
  0.  Solve to a particular state (pons asinorum, super-flip)
  1.  Corners, edges, centers
  2.  Edges, corners, centers (rather disorienting)
  3.  First level, third level, center slice
  4.  One face at a time, with no regard to correct cubie placement
        as long as the color is correct (this is fun)
  5.  Solve to a particular subgroup (half-turn group, anti-slice group)
        then stay in that subgroup

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"I'd like to have the same quest again, sir."

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 14 12:44:15 1998
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Message-Id: <199812111354.IAA18055@terminus.idirect.com>
From: "Mark Longridge" <cubeman@idirect.com>
To: <cube-lovers@ai.mit.edu>
Subject: New URL Correction
Date: Fri, 11 Dec 1998 08:59:00 -0500

Hello folks...

Sorry, but the URL I posted for my new web page is wrong.

The correct URL is: http://www.snipercade.com/cubeman

The old site http://web.idirect.com/~cubeman will be up for a few days yet.

The virtual URL http://welcome.to/cubeman should always point to the
current site! :-)

More interesting stuff to follow

-> Mark

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 14 13:45:36 1998
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Message-Id: <199812111354.IAA18055@terminus.idirect.com>
From: "Mark Longridge" <cubeman@idirect.com>
To: <cube-lovers@ai.mit.edu>
Subject: New URL Correction
Date: Fri, 11 Dec 1998 08:59:00 -0500

Hello folks...

Sorry, but the URL I posted for my new web page is wrong.

The correct URL is: http://www.snipercade.com/cubeman

The old site http://web.idirect.com/~cubeman will be up for a few days yet.

The virtual URL http://welcome.to/cubeman should always point to the
current site! :-)

More interesting stuff to follow

-> Mark

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 14 15:00:05 1998
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Date: Thu, 10 Dec 1998 23:04:10 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812110404.XAA03343@cauchy.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: fixing edge parity on 4x4x4

several people have posted maneuvers for "fixing" the edge parity
on rubik's revenge.  i haven't seen any maneuvers as short as
mine (although there might be some disagreement about "length").
recall that i am using the notation  _R_  (R underscored) to mean
turn the outer two layers together.

to switch the two UF edges:

_R2_  B2 L U2  l  U2  r'  U2  r  U2 F2  r  F2  _L'_  B2  _R2_

side effects: rotates the set of 4 U centers by 180 degrees.
also makes a 4-cycle of internal (0 faces visible) cubies.

if you're not concerned about moving centers, use

_(R2 B)_  u  _(B' D2 B)_  u'  _B_  l  _(B2 D2 R2)_

here,  _( ... )_  means the whole thing inside the parentheses
is underlined.

the maneuver that i normally use, since it's appropriate for
my solving method, is

U2  (r  U2)^5

which makes a 4-cycle of edges in the  r-slice  (and also
rotates the set of 4 U centers by 180 degrees).  this one
is short and easy to remember!

these maneuvers all work well on the 5x5x5 also.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 15 08:40:06 1998
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Sender: bosch@sgi.com
Message-Id: <3675637C.6231@sgi.com>
Date: Mon, 14 Dec 1998 11:14:04 -0800
From: Derek Bosch <bosch@sgi.com>
To: Cube-Lovers@ai.mit.edu
Subject: re-assembling a 2x2x2?

Well, I accidentally managed to pop apart my Rubik's Mini-Cube,
aka the 2x2x2...  Are there any easy instructions on getting it
back together?  I'd rather not force it...

D
-- 
Derek Bosch        "A little nonsense now and then
(650) 933-2115      is relished by the wisest men"... W.Wonka
bosch@sgi.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 15 10:30:58 1998
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Date: Tue, 15 Dec 1998 17:30:24 +0900 (JST)
Message-Id: <199812150830.RAA03209@soda2.bekkoame.ne.jp>
To: cube-lovers@ai.mit.edu
From: Ishihama Yoshiaki <ishmnn@cap.bekkoame.ne.jp>
Subject: 4D Rubik Cube(2x2x2x2) Java

I have converted my "4DRubikCube" (for Macintosh) to java applet.
I have uploaded it to my java page.
I have not yet added direct drag mode, only rotate cubes by buttons.
Please check this applet.



//----------------------------------------//

Ishihama Yoshiaki ( Tokyo Japan)
E-mail: ishmnn@cap.bekkoame.ne.jp
HomePage : http://www.asahi-net.or.jp/~hq8y-ishm/

//----------------------------------------//

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 15 12:13:33 1998
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Date: Tue, 15 Dec 1998 11:35:13 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Derek Bosch <bosch@sgi.com>
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: re-assembling a 2x2x2?
In-Reply-To: <3675637C.6231@sgi.com>
Message-Id: <Pine.BSF.3.96.981215112038.5993D-100000@shell2.tiac.net>


 I've pulled mine apart a *few* times. Imho, it's probably impossible to
reassemble without some forcing. If it were made of cheap plastic, I
very much doubt that it could be assembled. Study the structure, so
you won't try to assemble it wrong; you probably wouldn't make such a
mistake, though. Hope you didn't lose any pieces! (Be *sure* to match
colors properly before assembling; of course, you know that, too.)

 My hopeful guess is that you'll succeed, but be rather amazed by the
force it takes, and also that the plastic can take such stress. I have
seen 2^3s on sale fairly recently, btw, so there has been a stock of
them, possibly a new production run.

 I've been a mechanical tech. at times for several decades, so I'm
reasonably sure of what I say. I've pulled apart many movable-part
puzzles, and the 2^3 is surely the most intractable of all I've dealt
with.  Alexanders' Star and the 4^3 have some parts that are easy to
break.

 I'd love to know how it's done at the factory. I hope it's not some
subtle ultrasonic welding. (Maybe someone could ask Dr. Christoph
Bandelow, Dr. Uwe Meffert, or (Dr.?*) David Singmaster.)

*Sorry if I forget!

Best regards, and good luck!

|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec 16 13:02:37 1998
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Date: Tue, 15 Dec 1998 18:41:19 -0500
To: Cube-Lovers@ai.mit.edu
From: Charlie Dickman <charlied@erols.com>
Subject: Re: re-assembling a 2x2x2?

>Well, I accidentally managed to pop apart my Rubik's Mini-Cube,
>aka the 2x2x2...  Are there any easy instructions on getting it
>back together?  I'd rather not force it...

You should use a small Phillips screwdriver to remove one of the triangular
flanges that forms the tracks that the "cubies" ride in. Then, put the
"cubie" pieces in place and then g_e_n_t_l_y spread the space between the 4
"cubie" surfaces that hide the stump that holds the triangular flange and
put the screw back in.

Be careful not to break the shaft between the cubie face and its anchor.


Charlie Dickman
charlied@erols.com

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec 17 12:58:39 1998
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Date: Wed, 16 Dec 1998 23:46:46 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Charlie Dickman <charlied@erols.com>
Cc: Cube-Lovers@ai.mit.edu
Subject: Newer mechanism? (Was: Re: re-assembling a 2x2x2?)
In-Reply-To: <v04020a00b29ca31d7c8f@[207.172.132.155]>
Message-Id: <Pine.BSF.3.96.981216232654.16519H-100000@shell2.tiac.net>


 Charlie, I'm just about dead sure my 2^3s (from about 15 (?) years go)
have no screws. I studied your description, and it seems that the
mechanism has been redesigned! I described the mechanism of mine in
considerable (if not painful!) detail, maybe a year and a half ago; it's
probably in the archives. The keyword "jack" should help to locate the
post.

 Perhaps a continuing market combined with the difficulty of assembling
the original design created a need for a new one.

 Would really *love* to know whether there is a newer and different
mechanism. As a somewhat casual student of these mechanisms, I've come
to realize that for all "sizes", more than one mechanism is possible.

 I have great admiration for the designers who create these marvelous
mechanisms. I love the 5^3 as much for its innards (which I regard as
thoroughly astonishing) as for its essential function.

 I also admire the mathematicians, programmers, and practical users of
group theory on this List; I have only a faint awareness of what they're
talking about, but their amazing posts keep my mind properly stretched.
I feel a bit like a dog listening to his human family discussing, say, a
trip to Australia (the dog didn't go). However, that's perfectly OK with
me! My mind is quite good, and to some degree it's circumstance that I'm
not "with it".

My regards to all,

|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec 17 14:01:55 1998
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Date: Thu, 17 Dec 1998 09:32:42 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re : Optimal Cube Solver
In-Reply-To: <366C1ED9.C11@hrz1.hrz.tu-darmstadt.de>
To: kociemba@hrz1.hrz.tu-darmstadt.de
Cc: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9812170942.C@GN209A.PSTCC.CC.TN.US>

On Mon, 07 Dec 1998 19:30:49 +0100 Herbert Kociemba
<kociemba@hrz1.hrz.tu-darmstadt.de> wrote:

>
> The third coordinate describes the edge permutation. Because there are
> 12! coordinate values, even reduction by 48 symmetries still gives too
> many coordinate values. So for use in a turntable we define two edge
> permutations a and b equivalent, if a=m1*b*m2, were m1 and m2 are in M.
> In this way we get 208816 equivalence classes c. If now m1*c*m2 is a
> (not necessarily unique) representation of an edge permutation applying
> a faceturn T is done like that:
>
> (m1*c*m2)*T = m1*c*[m2*T*m2^-1]*m2 = m1*[c*T']*m2=
> [m1*m1']*c'*[m2'*m2]=m1''*c'*m2''
>

This is remindful of a technique I used many years ago to
reduce the size of the search space for the 2x2x2 problem,
and the issue would apply to any cube such as the 4x4x4
with an even number of cubies per edge.  That is, in the
(2n)x(2n)x(2n) problem you can treat as equivalent any
positions of the form (m1)*x*(m2) for m1 and m2 in M,
provided only that both of m1 and m2 are rotations or that
both of m1 and m2 are reflections.

Another (and in some ways better) way to model a
(2n)x(2n)x(2n) problem and to reduce the size of the search
space is to fix one of the corners and to use the symmetry
group which preserves the major diagonal axis which includes
the corner which is fixed, but that is a different issue.

Dan Hoey showed that (m1)*x*(m2) is equivalent to m'xmc for
suitable choices of m and c, for m in M and for c in C (the
set of 24 rotations).  Requiring that m1 and m2 both be
rotations or both be reflections is necessary because you
really can't turn the corners inside out on a physical
cube.

Herbert does not impose the same restriction on both of m1
and m2 being rotations or reflections because his third
coordinate applies only to the edges, and the edges can
indeed be turned inside out on a physical cube, namely with
the Pons Asinorum maneuver. So for this case, (m1)*x*(m2)
is equivalent to m'xmc if m1 and m2 are both rotations or
both reflections, and is equivalent to m'xmcv if they are
not, where v is the central inversion of the edges
(essentially, the Pons Asinorum applied to the edges).

I used to talk about 1152-fold symmetry for the 2x2x2
(1152=24*48).  Herbert's approach for the third coordinate
yields a 2304-fold reduction in the search space
(2304=48*48).  However, the reductions in the search space
in the two cases are not really dealing with quite the same
issue. In the case of 1152-fold symmetry for the 2x2x2,
there are (up to) 1152 equivalent positions which are the
same distance from Start. If I am understanding Herbert's
technique correctly, when positions are equivalent in the
third coordinate, there are (up to) 2304 positions of
the edges for which the distance from Start has the same
lower bound.  (Maybe I should say "the same non-trivial
lower bound", because (for example) zero would be a lower
bound for all positions.)

----------------------------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us
Pellissippi State Technical Community College

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec 17 15:55:31 1998
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Date: Thu, 17 Dec 1998 10:02:24 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re : Re: Optimal Cube Solver
In-Reply-To: <Pine.GSO.3.95L.981208183158.13948B-100000@sunny.isis.unc.edu>
To: Alchemist Matt <monroem@email.unc.edu>
Cc: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9812171024.D@GN209A.PSTCC.CC.TN.US>

On Tue, 08 Dec 1998 18:37:02 -0500 (EST) Alchemist Matt
<monroem@email.unc.edu> wrote:

> This question is directed to both Herbert and Mike Reid in case he's
> reading this list:  With all this discussion of the "Professor Cube"
> lately, how hard would it be to extend either Optimal cube solving program
> to handle 4x4x4 and 5x5x5 cubes in addition to the traditional 3x3x3?
> Considering reasonable table files (50 - 100 mb), how much longer would
> the computation time be extended by.  If either of you would find the time
> to implement this modification, I would be very interested in trying out
> the program.

Mike Reid has already answered this question in the
negative with respect to optimal solvers, based on the
huge size of the search spaces that would be involved. For
several years, I have wondered about the same thing with
respect to a God's Algorithm search of a Start rooted tree
(how many positions are one move from Start, how many are
two moves from Start, etc.).

You could obviously get a few moves from Start, but I don't
think you would get very far.  For example, with my
existing program, I think maybe I could get five or six
moves from Start with the 4x4x4 or the 5x5x5.

However, I have been reluctant to deal with either the
4x4x4 or the 5x5x5 for several reasons.  One is that the
programming is not quite as easy as it might seem, or at
least not for my program the way it is written. In
principle, all I would have to do is replace the existing
tables for the permutations which generate the 3x3x3 with
the corresponding tables for the 4x4x4 and the 5x5x5 and
everything should just work.  However, my program contains
optimizations previously described on Cube-Lovers which are
very dependent on the edge and corner structure of the
3x3x3.  For the larger problems, I would have to add a bit
(not a lot, but a bit) of new code to deal with new kinds
of pieces.

Secondly, in the case of the 4x4x4 I would have to deal
with might be called rotational equivalences, for example
that RrL'l' (capital letters denote moving the outer layers
and lower case letters denote moving the inner layers)
would normally treated as being equivalent to the Start
state.  Both ways I know how to do it would require some
reprogramming, especially in light of the same existing
optimizations I talked about before with respect to the
3x3x3.  Namely, I could treat rotations as being
equivalent, so that x is equivalent to all positions of the
form xc for c in C.  Or I could fix one of the corners.

Thirdly, I would have to deal with what might be called
invisible equivalences, where pieces can be moved without
the movement being visible on a physical cube.  In the case
of the 4x4x4 (for example), the four "face center" facelets
on each face can move with respect to each other (subject
to parity constraints) without the movement being visible. 
I would think that you would want to treat such differences
as being equivalent.

Actually, I think that an optimal solver for the 4x4x4 or
the 5x5x5 would need to deal with some of the same issues,
in addition to the huge size of the search spaces that was
pointed out by Mike Reid in his original response to this
question.

----------------------------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us
Pellissippi State Technical Community College

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec 17 19:18:53 1998
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Date: Thu, 17 Dec 1998 21:39:14 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Cc: Cube Lovers <cube-lovers@ai.mit.edu>
Subject: Re: Optimal Cube Solver
References: <SIMEON.9812170942.C@GN209A.PSTCC.CC.TN.US>

Jerry Bryan wrote:
>
> On Mon, 07 Dec 1998 19:30:49 +0100 Herbert Kociemba
> <kociemba@hrz1.hrz.tu-darmstadt.de> wrote:
>
> >
> > The third coordinate describes the edge permutation. Because there are
> > 12! coordinate values, even reduction by 48 symmetries still gives too
> > many coordinate values. So for use in a turntable we define two edge
> > permutations a and b equivalent, if a=m1*b*m2, were m1 and m2 are in M.
> > In this way we get 208816 equivalence classes c. If now m1*c*m2 is a
> > (not necessarily unique) representation of an edge permutation applying
> > a faceturn T is done like that:
> >
> > (m1*c*m2)*T = m1*c*[m2*T*m2^-1]*m2 = m1*[c*T']*m2=
> > [m1*m1']*c'*[m2'*m2]=m1''*c'*m2''
> >

> I used to talk about 1152-fold symmetry for the 2x2x2
> (1152=24*48).  Herbert's approach for the third coordinate
> yields a 2304-fold reduction in the search space
> (2304=48*48).  However, the reductions in the search space
> in the two cases are not really dealing with quite the same
> issue. In the case of 1152-fold symmetry for the 2x2x2,
> there are (up to) 1152 equivalent positions which are the
> same distance from Start. If I am understanding Herbert's
> technique correctly, when positions are equivalent in the
> third coordinate, there are (up to) 2304 positions of
> the edges for which the distance from Start has the same
> lower bound.  (Maybe I should say "the same non-trivial
> lower bound", because (for example) zero would be a lower
> bound for all positions.)

I do not use the equivalence in the third coordinate as an index in a
pruning table. On the contrary, I have to "expand" the coordinate again
by a factor of 48 to get equivalence classes, which have the same
distance from start and from which I built the pruning table. But due to
the large size (12!) of edge permutations, it seems a good way (and I
see no other way) to keep track of the edge-permutation-coordinate with
only a few table-lookups.

I now have enough RAM (128MB) to implement a pruning table for all
possible coordinates of the first phase of my Two-Phase-Algorithm, which
brings the cube into the subgroup H=<U,D,R2,L2,F2,B2>. This is what Mike
Reid already did about one year ago and which seems powerful enough even
to be used as an Optimal Solver (omitting phase 2, in which the edge-
and cornerpermutations are restored). Due to this power I think of
implementing a "static" phase 2 only with a table which stores the edge-
and corner permutations of all positions up to maybe 5 face-turns in H
from start.

Using the approach for the edge permutation from above,the computation
of the starting position of phase 2 should be very fast. In the
implementation currently used, I have to switch back from the
coordinate-representation of the cube in phase 1 to a more "physical"
representation using edges and corners, apply the maneuver generated in
phase 1 and then compute the starting coordinates of phase 2. In the new
approach only coordinates could be uses.

Herbert

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec 18 11:33:50 1998
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Message-Id: <002c01be2a01$f6ba7020$7ac4b0c2@home>
From: roger.broadie@iclweb.com (Roger Broadie)
To: <Cube-Lovers@ai.mit.edu>
Cc: "Nicholas Bodley" <nbodley@tiac.net>,
        "Charlie Dickman" <charlied@erols.com>
Subject: Re: Newer mechanism? (Was: Re: re-assembling a 2x2x2?)
Date: Thu, 17 Dec 1998 21:11:22 -0000

Nicholas Bodley <nbodley@tiac.net> wrote (17 December 1998)


>
> Would really *love* to know whether there is a newer and different
>mechanism. As a somewhat casual student of these mechanisms, I've
>come to realize that for all "sizes", more than one mechanism is
>possible.
>


According to the reports on the patent case brought against Ideal for
infringement  of the Nichols patent (Moleculon Research Corp v. CBS,
Inc) there were two Ideal 2x2x2 cubes, both sold as the Rubik's Pocket
Cube, but one from Taiwan and one from Japan.  The Japanese version
used an internal sphere, which could well be the version with the
Philips screw referred to by Charlie Dickman, since it sounds like the
inside of a 4x4x4.  The Taiwanese version is less clearly described -
the internal faces are said to form a tongue and groove mechanism -
but probably also had an internal spider like the conventional 3x3x3 -
is this Nicholas Bodley's version?

Incidentally, by the time the case had been up and down to the Appeals
court a couple of times, the final decision, in 1989, was that just
these two forms infringed the patent.  The 3x3x3 and 4x4x4 were held
not to infringe.

Roger Broadie

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec 18 14:53:20 1998
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Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Dec 17 22:07:25 1998
Date: Thu, 17 Dec 1998 22:06:01 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812180306.WAA21451@cauchy.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Optimal Cube Solver

herbert writes

> I do not use the equivalence in the third coordinate as an index in a
> pruning table. On the contrary, I have to "expand" the coordinate again
> by a factor of 48 to get equivalence classes, which have the same
> distance from start and from which I built the pruning table. But due to
> the large size (12!) of edge permutations, it seems a good way (and I
> see no other way) to keep track of the edge-permutation-coordinate with
> only a few table-lookups.

> I now have enough RAM (128MB) to implement a pruning table for all
> possible coordinates of the first phase of my Two-Phase-Algorithm, which
> brings the cube into the subgroup H=<U,D,R2,L2,F2,B2>. This is what Mike
> Reid already did about one year ago and which seems powerful enough even
> to be used as an Optimal Solver (omitting phase 2, in which the edge-
> and cornerpermutations are restored). Due to this power I think of
> implementing a "static" phase 2 only with a table which stores the edge-
> and corner permutations of all positions up to maybe 5 face-turns in H
> from start.

> Using the approach for the edge permutation from above,the computation
> of the starting position of phase 2 should be very fast. In the
> implementation currently used, I have to switch back from the
> coordinate-representation of the cube in phase 1 to a more "physical"
> representation using edges and corners, apply the maneuver generated in
> phase 1 and then compute the starting coordinates of phase 2. In the new
> approach only coordinates could be uses.


herbert, you might be interested in what my sub-optimal program
(the one based on your two-stage algorithm) does about edge
permutations.  i have this extra coordinate i call "sliceedge",
(really this is just another coset space) which considers the
locations of four distinguishable edges.  there are 12*11*10*9 = 11880
possibilities for this coordinate.  when the cube is entered, it
calculates the corresponding coordinate for edges in the U-D slice,
also for edges in the F-B slice, and also for the R-L slice.
then i have lookup tables to tell me how this coordinate transforms
under turns.  this lookup table is 18 * 11880 shorts = 427680 bytes.

when stage 2 is reached, i have a lookup table that maps this
coordinate into permutations of the four U-D slice edges.  actually,
only 24 of the entries are valid, but only these occur, since we've
reached stage 2.  this lookup table is 11880 chars.
there's also a lookup table to transform the "sliceedge" coordinate
into another coordinate, which gives the locations of four
distinguishable edges among the eight U and D edges.  this coordinate
has 8*7*6*5 = 1680 possibilities, and the lookup table is 11880 shorts.

the big lookup table is the one that takes two of these last coordinates
and transforms it into a permutation of the eight U and D edges.
this table has 1680 * 1680 shorts = about 5.5 megabytes.  most of
the entries are garbage, only 40320 = 8! actually occur, since we've
reached stage 2.

so for about 6 megabytes of space, all the edge permutations are done
with lookup tables.  i haven't actually calculated how much of a speed
up this is, but it's probably good.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Fri Dec 18 15:30:11 1998
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From: "CRAWFORD, SCOTT" <CRAWFORDS@evangel.edu>
To: Cube-Lovers@ai.mit.edu
Subject: Snake
Date: Thu, 17 Dec 1998 19:50:21 -0600

This may be a little off topic, but I've recently fell in love with the
snake, making many shapes I'd never even thought of.  Are there any websites
or archives of different snake patterns?

Thanks
Scotte

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 21 14:00:52 1998
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Message-Id: <367BDD1D.DE6@hrz1.hrz.tu-darmstadt.de>
Date: Sat, 19 Dec 1998 18:06:37 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: cube-lovers@ai.mit.edu
Cc: michael reid <reid@math.brown.edu>
Subject: Re: Optimal Cube Solver
References: <199812180306.WAA21451@cauchy.math.brown.edu>

michael reid wrote:

> herbert, you might be interested in what my sub-optimal program
> (the one based on your two-stage algorithm) does about edge
> permutations.  i have this extra coordinate i call "sliceedge",
> (really this is just another coset space) which considers the
> locations of four distinguishable edges.  there are 12*11*10*9 = 11880
> possibilities for this coordinate.  when the cube is entered, it
> calculates the corresponding coordinate for edges in the U-D slice,
> also for edges in the F-B slice, and also for the R-L slice.
> then i have lookup tables to tell me how this coordinate transforms
> under turns.  this lookup table is 18 * 11880 shorts = 427680 bytes.
>
> when stage 2 is reached, i have a lookup table that maps this
> coordinate into permutations of the four U-D slice edges.  actually,
> only 24 of the entries are valid, but only these occur, since we've
> reached stage 2.  this lookup table is 11880 chars.

I already made some experience with the "sliceedge"-coordinate before. I
built it in the way:
24*position of the 4 edges + permutation of the 4 edges,
where the position range is from 0 to 494 and permutation ranges from 0
to 23.
In this way when reaching stage 2, the "sliceedge"-coordinate
automatically is in the range from 0 to 23 and you need no lookup table
at all.

> there's also a lookup table to transform the "sliceedge" coordinate
> into another coordinate, which gives the locations of four
> distinguishable edges among the eight U and D edges.  this coordinate
> has 8*7*6*5 = 1680 possibilities, and the lookup table is 11880 shorts.
>
> the big lookup table is the one that takes two of these last coordinates
> and transforms it into a permutation of the eight U and D edges.
> this table has 1680 * 1680 shorts = about 5.5 megabytes.  most of
> the entries are garbage, only 40320 = 8! actually occur, since we've
> reached stage 2.

This seems an interesting approach. Using the
edge-permutation-coordinate in the way I described it before, I need
about 20MB for the lookup-table which tells the coordinate-tranformation
under turns, which is quite a lot. Maybe I also try your method.

Herbert

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 21 14:52:35 1998
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Date: Sun, 20 Dec 1998 16:41:43 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Roger Broadie <roger.broadie@iclweb.com>
Cc: Cube Mailing List <Cube-Lovers@ai.mit.edu>,
        Charlie Dickman <charlied@erols.com>, Mark Glusker <glusk@sgi.com>
Subject: Re: Newer mechanism? (Was: Re: re-assembling a 2x2x2?)
In-Reply-To: <002c01be2a01$f6ba7020$7ac4b0c2@home>
Message-Id: <Pine.BSF.3.96.981220161825.27653I-100000@shell2.tiac.net>


On Thu, 17 Dec 1998, Roger Broadie wrote:

(Interesting to read about the lawsuit...)

}Nicholas Bodley <nbodley@tiac.net> wrote (17 December 1998)

}inside of a 4x4x4.  The Taiwanese version is less clearly described -
}the internal faces are said to form a tongue and groove mechanism -
}but probably also had an internal spider like the conventional 3x3x3 -
}is this Nicholas Bodley's version?

In the mechanism I know for a 2X2X2 Cube, at its center is a piece like
a jack, that is, one of the pieces in the traditional game, but without
the knobs at the ends. You could also think of it as three rods
intersecting at a common point, and mutually orthogonal; it's as if you
had plus and minus x, y, and z axes defined by the directions of the
rods. These create the axes of revolution for one half relative to the
other.

The cubies are hollow, and their mating faces have curved cutaways. To
keep the cubies from moving too far from each other, 12 "clips" extend
from the center outward. If you think of a deeply-grooved pulley, cut
pie-style into quarters, you have a general idea. The curved-cutout
edges of the cubies fit between two curved, parallel sides of the
"clips".

Finally, the "clips" are kept engaged with the cubies either directly by
square cross-section extensions of the center "jack", or by hollow
square rods that pivot on (smaller) cylindrical extensions. Just as the
ball in a 4^3 is locked to one half, the "jack" is, also.

The big problem with this mechanism is that unless the parts can deform
sufficiently without breaking (the actual case; they can do so), it's
impossible to assemble or to disassemble as molded. If it were made of
metal, it couldn't be assembled without design changes.

Illustrations are really needed here; this mechanism is a challenge to
describe understandably!

I'd think that the internal ball uses a variant of the tongue and groove
scheme, if it's like the 4^3.

|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer indusztry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 22 12:31:20 1998
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Date: Sun, 20 Dec 1998 21:18:09 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: "CRAWFORD, SCOTT" <CRAWFORDS@evangel.edu>
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Snake
In-Reply-To: <5B9619E72C59D211B22100A0C99CC4632EEE9E@www.evangel.edu>
Message-Id: <Pine.BSF.3.96.981220211300.2706L-100000@shell2.tiac.net>


 The Snake is delightful; you can do some interesting investigations by
starting with a straight config., and twisting each consecutive joint
according to a pattern. Just as long as you don't get physical
interferencies, you see some modestly-interesting shapes.

 It's also worth a bit of casual effort to create a "ball". All of these
things are, however, trivially easy to many of the subscribers to this
list. The snake is for mental relaxation!


|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 22 15:28:37 1998
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Date: Tue, 22 Dec 1998 16:40:58 +0900 (JST)
Message-Id: <199812220740.QAA00692@soda3.bekkoame.ne.jp>
To: cube-lovers@ai.mit.edu
From: Ishihama Yoshiaki <ishmnn@cap.bekkoame.ne.jp>
Subject: 5DRubikCube

I have made simulation of 5D RubikCube(2x2x2x2x2).
This is consisted of 2x2x2x2x2=32  5DCubes.
This is for Macintosh only.
I will not convert this program to java because it is too troublesome.
It is on my HomePage.

//----------------------------------------//

Ishihama Yoshiaki (Tokyo Japan)
E-mail: ishmnn@cap.bekkoame.ne.jp
HomePage : http://www.asahi-net.or.jp/~hq8y-ishm/

//----------------------------------------//

[Moderator's note: This is the third announcement of this website this
 month.  For all further developments, check the website. ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 22 18:44:01 1998
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Date: Tue, 22 Dec 1998 18:37:57 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812222337.SAA26516@adams.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Optimal Cube Solver

i'm glad that herbert brought up this issue of edge transformations.
because now that i think about this again, i realize that my tables
can be reduced dramatically.  i described my tables:

> there's also a lookup table to transform the "sliceedge" coordinate
> into another coordinate, which gives the locations of four
> distinguishable edges among the eight U and D edges.  this coordinate
> has 8*7*6*5 = 1680 possibilities, and the lookup table is 11880 shorts.
>
> the big lookup table is the one that takes two of these last coordinates
> and transforms it into a permutation of the eight U and D edges.
> this table has 1680 * 1680 shorts = about 5.5 megabytes.  most of
> the entries are garbage, only 40320 = 8! actually occur, since we've
> reached stage 2.

since this big table is sparse, we don't need most of it.  what i
should do is have another table (11880 char's) to transform "sliceedges"
into permutations of four edges.  the location of the four R-L slice
edges determines the location of the four F-B slice edges, so we only
need to know how they're permuted.  thus the big table can be replaced
by one with 1680 * 24 shorts that gives the permutation of the eight
U and D edges.  it no longer would have error-checking (i.e. making
sure we don't get an invalid entry in the big table), but that could
be installed with another simple table lookup, if desired.

with this new mechanism, only about 543K of tables would be needed,
the largest being a lookup table which tells how "sliceedges"
transform under face turns.  this is much better than the 6 megabytes
of tables i'm currently using.  i don't know why i didn't think of
this earlier!

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec 23 17:40:01 1998
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Date: Wed, 23 Dec 1998 23:19:05 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: cube-lovers@ai.mit.edu
Cc: michael reid <reid@math.brown.edu>
Subject: Re: Optimal Cube Solver
References: <199812222337.SAA26516@adams.math.brown.edu>

michael reid wrote:

> since this big table is sparse, we don't need most of it.  what i
> should do is have another table (11880 char's) to transform "sliceedges"
> into permutations of four edges.  the location of the four R-L slice
> edges determines the location of the four F-B slice edges, so we only
> need to know how they're permuted.  thus the big table can be replaced
> by one with 1680 * 24 shorts that gives the permutation of the eight
> U and D edges.  it no longer would have error-checking (i.e. making
> sure we don't get an invalid entry in the big table), but that could
> be installed with another simple table lookup, if desired.
>
> with this new mechanism, only about 543K of tables would be needed,
> the largest being a lookup table which tells how "sliceedges"
> transform under face turns.  this is much better than the 6 megabytes
> of tables i'm currently using.  i don't know why i didn't think of
> this earlier!
>
> mike

Is it necessary to use the table for the map from the "sliceedges" to
the 1680 "4 out of 8"-coordinate at all? I think you constructed this
"helper"-coordinate, because a 11880*11880 table-size was too big and
1680*1680 was reasonable. But 11880*24 also is small (just twice as much
as the lookup table which tells how "sliceedges" transform under face
turns).

In the way I construct the sliceedge-coordinate x, the x mod 24 gives
the permutation part, x/24 the location part. So I could compute the
edge-coordinate  at the start of phase 2 with M[x][y mod 24], where x
and y are the RL- and FB-sliceedge coordinates and M is a table with
11880*24 shorts. So I need only one tablelookup to get the coordinate.

Herbert

From cube-lovers-errors@mc.lcs.mit.edu  Thu Dec 24 11:35:18 1998
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Date: Wed, 23 Dec 1998 19:07:26 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199812240007.TAA28326@adams.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Optimal Cube Solver

herbert writes

> Is it necessary to use the table for the map from the "sliceedges" to
> the 1680 "4 out of 8"-coordinate at all?

no, i guess the "4 out of 8" coordinate is not really needed.
good point.  the tradeoff would be one extra table lookup versus
having tables that are 455K larger.  i don't know if there's a clear
choice between these two options, but either is much better than the
6 megabytes i'm using now!

mike

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 28 12:21:48 1998
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Dec 27 13:14:59 1998
Message-Id: <003301be31c4$60a798e0$335755ca@Uwe.ue.net>
From: uwe@ue.net (Uwe Meffert)
To: "Cube-Lovers" <Cube-Lovers@ai.mit.edu>
Subject: ADDITIONAL FEATURES FOR OUR PUZZLE SITE
Date: Mon, 28 Dec 1998 01:47:54 +0800


Requesting Help:

For 1999 I will be adding a lot of additional interactive puzzles &
games to the "Meffert's World of Puzzles" site in several puzzle
categories enabling a person to challenge a friend or send an
interactive  puzzle greeting.

Each Puzzle Challenge will be made available for 30 days, with actual
time taken to solve it being relayed back to the challenger.  After the
Puzzle Challenge is received, the challenge can be accepted or rejected
or a counter challenge made.

Puzzle Games will also be available online.  I am hoping to have a very
large range of categories from very simple games & puzzles such as
electronic tik tak toe (from single layer to triple layer) with some
additional new features, the traditional 8 and 15 piece sliding puzzles
with talking help function, the Orbix & Orbix Junior (12 & 6 lights in 3
colors) Electronic Reversy etc. etc. etc. to more complex puzzles &
games.

Whilst I have developed some of these already I will need a lot of help
from puzzlers worldwide to make this the best FREE interactive really
Great Puzzle and Games site for 1999.

Please spread the word to anyone you know that can contribute and other
puzzle site Web Editors that may let me use some of their existing
puzzles.

I hope to have very unique graphics and concepts to really popularize
puzzles again, this time as a Free service over the Internet.   To
appeal to the majority of the people the puzzles must not be too hard,
yet still be challenging.

Also, I am presently looking for a new Web Editor for our site, our
present Webmaster Andrew Southern is unfortunately fully tied up until
July with his studies,  please pass the word along to anyone you think
my be suitable.

Many Thanks


Warm regards and a Very Happy New Year to All.
Uwe

Uwe Meffert
P.O. Box 24455, Aberdeen, Hong Kong.
Tel. 852-2518-3080, Fax. 852-2518-3282
Email:- uwe@ue.net
Sites: www.bloodpressure.org,  www.cmd-diagnostics.com
           www.ue.edu www.ue.net www.ue.net/mefferts-puzzles

From cube-lovers-errors@mc.lcs.mit.edu  Mon Dec 28 19:00:54 1998
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Message-Id: <3685267B.D1459EC7@geocities.com>
Date: Sat, 26 Dec 1998 10:10:03 -0800
From: Jono <bagelboyj@geocities.com>
Reply-To: BagelBoyJ@geocities.com
Organization: Fine Finger Design
To: cube-lovers@ai.mit.edu
Subject: Other Cubes

Hi, cube lovers. I have a few questions.
Does anyone know if Erno Rubik is still alive?
About 4 years I vaguely remember seeing a large star-shaped
rubiks puzzle. Does anyone know where I can find one?
I am also looking for a 4x4x4 and a 5x5x5 cube. Where can I
find one?

Thanks to all.


-Jono

[ Moderator's note: Cube-lovers-request gets a lot of requests for
  information on finding 4^3 and 5^3 puzzles.  I'm pretty sure there
  is no source of 4^3 puzzles, except for the occasional auction.
  Last I heard Uwe Meffert sells 5^3 puzzles at
  http://www.ue.net/mefferts-puzzles/ --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 29 15:00:41 1998
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Date: Tue, 29 Dec 1998 02:22:08 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
Reply-To: Nicholas Bodley <nbodley@tiac.net>
To: Jono <bagelboyj@geocities.com>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Other Cubes
In-Reply-To: <3685267B.D1459EC7@geocities.com>
Message-Id: <Pine.BSF.3.96.981229015425.28800A-100000@shell2.tiac.net>


On Sat, 26 Dec 1998, Jono wrote:

}Hi, cube lovers. I have a few questions.
}Does anyone know if Erno Rubik is still alive?

Quite likely. There's a Website that might help you find out:

http://www.rubiks.com

Try a Web search (I like the meta-search engines; Metasearch and
Metafind are a couple).

}About 4 years I vaguely remember seeing a large star-shaped
}rubiks puzzle.

It might well have bees Alexander's Star. As to finding one, sorry to
say, I can't help. Rather sure it wasn't a Rubik design, though. It was
harder to manipulate mechanically than one might like. Not sure, but I
think I've seen it it a store. You might try a Web puzzle dealer.
 (...Puzzletts.com ?)

} Does anyone know where I can find one? I am also looking for a 4x4x4
}and a 5x5x5 cube. Where can I find one?

For the 5^3, in addition to Meffert, as Dan said, Dr. Christoph
Bandelow, in Germany, was selling them, as well, I'm almost certain. The
store [The Games People Play] on Mass. Ave. in Cambridge, Mass., might
have them in stock. Sorry, but I've lost track of Dr. Bandelow's
address.

My general impression is that there's enough interest in the 5^3 to have
revived production, although it could have been just one big run. Fwiw,
(honestly, not much!), my experience regarding the 4^3 agrees with Dan's
comments.


}[ Moderator's note: Cube-lovers-request gets a lot of requests for
}  information on finding 4^3 and 5^3 puzzles.  I'm pretty sure there
}  is no source of 4^3 puzzles, except for the occasional auction.
}  Last I heard Uwe Meffert sells 5^3 puzzles at
}  http://www.ue.net/mefferts-puzzles/ --Dan ]
}


|*  Nicholas Bodley   *|*  Electronic Technician {*} Autodidact & Polymath
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  The personal computer industry will have become
|*  Amateur musician  *|*  mature when crashes become unacceptable.
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 29 20:15:59 1998
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Message-Id: <199812292316.SAA06321@life.ai.mit.edu>
From: "Christoph Bandelow" <bandecbv@mailhost.rz.ruhr-uni-bochum.de>
To: cube-lovers@ai.mit.edu
Date: Wed, 30 Dec 1998 00:14:50 +0000
Subject: Availability of 5x5x5 cubes and other Rubik type puzzles

On Tue, 29 Dec 1998, Nicholas Bodley wrote about the availability of
5x5x5 cubes  and other puzzles:

> Sorry, but I've lost track of Dr. Bandelow's address.

May I help without being too much vituperated for making an unseemly
advertisement? Christoph Bandelow's email address is

Christoph.Bandelow@ruhr-uni-bochum.de

He does not only sell excellent 5x5x5 Magic Cubes (the good old ones
made in Hong Kong, not in mainland China), but also Magic
Dodecahedra, Skewbs, Pyraminxes, Impossiballs, various Puzzle Balls,
Mach Balls and other Rubik type puzzles and books about those
puzzles. It is a pleasure to deal with him for he is quick,
absolutely reliable, fair and competent. Free mail order catalog.

Christoph
--
Christoph Bandelow
mailto: Christoph.Bandelow@ruhr-uni-bochum.de
http://www.ruhr-uni-bochum.de/mathematik3/bandelow.htm

[ Objective testimonials written in the third person are always
  appreciated. --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Dec 29 21:21:03 1998
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From: Douglas Zander <dzander@solaria.sol.net>
Message-Id: <199812282147.PAA06517@solaria.sol.net>
Subject: ADDITIONAL FEATURES FOR OUR PUZZLE SITE (fwd)
To: cube-lovers@ai.mit.edu (cube)
Date: Mon, 28 Dec 98 15:47:11 CST

 I would like to voice my concern over this site.  The thing is that not
 everyone has the latest version of Netscape or IE installed.  Many
 libraries (from which I myself access the WWW) do not allow Java to be
 installed on their machines nor do they allow sound.  The highest browser
 versions I have access to in any library here in Milwaukee is 3.0
 I had gone to this site and I could not play the puzzle that was supposed
 to be there; nothing showed up on the screen. 

 I just wished to point this out to the creator of the site; in fact, to
 all creators of puzzle sites.  There are, I believe, a significant number
 of people who use public browsers without Java or sound.  Please keep this
 in mind when designing your sites.  Thank you.

--
 Douglas Zander                |
 dzander@solaria.sol.net       |
 Shorewood, Wisconsin, USA     |

[Moderator's note: Cube-lovers is not really the place to debate the
 philosophy of web design.  I must agree that for maximum usability,
 the text on a page should be displayed without requiring images, and
 that the static images should be displayed with requiring Java.
 Still, setting up a puzzle simulator without Java is more problematic
 than we can reasonably ask of a free server, and having a simulation
 that requires Java is certainly better than none. --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Dec 30 10:43:10 1998
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Date: Tue, 29 Dec 1998 23:48:14 -0500 (EST)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199812300448.XAA02148@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Re: Availability of 5x5x5 cubes and other Rubik type puzzles

>> Sorry, but I've lost track of Dr. Bandelow's address.

> Christoph Bandelow's email address is
> Christoph.Bandelow@ruhr-uni-bochum.de
> He does not only sell [...] but also [...].  It is a pleasure to deal
> with him for he is quick, absolutely reliable, fair and competent.

> Christoph Bandelow
> mailto: Christoph.Bandelow@ruhr-uni-bochum.de
> http://www.ruhr-uni-bochum.de/mathematik3/bandelow.htm

Heehee!  Had me going there until I saw the signature. :-)

I have had only one experience dealing with Mr. Bandelow.  In that one,
I did indeed find him quick, reliable, fair, and competent.  (I bought
a 5x5x5 and the Cube-family puzzle whose name I forget based on the
dodecahedron, the one that turns based on slices made parallel to the
faces, passing through edge centres.  Interestingly, my experience with
the dodecahedral puzzle taught me enough that I can now solve a
two-face scramble on the 3x3x3 without leaving the two-face subgroup,
which I previously couldn't.)

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

From cube-lovers-errors@mc.lcs.mit.edu  Tue Jan  5 17:17:11 1999
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Dec 30 20:17:39 1998
Message-Id: <v04011700b2b074d73b09@[207.172.215.201]>
In-Reply-To: <3685267B.D1459EC7@geocities.com>
Date: Wed, 30 Dec 1998 20:26:43 -0400
To: cube-lovers@ai.mit.edu
From: Charlie Dickman <charlied@erols.com>
Subject: Re: Other Cubes

>[...]I am also looking for a 4x4x4 and a 5x5x5 cube. Where can I
>find one?

Jono,

I recently saw a 4x4x4 for sale at auction at www.ebay.com. Check it out.

Charlie Dickman
charlied@erols.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Jan  5 17:47:44 1999
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From: "Chris and Kori Pelley" <ck1@home.com>
To: <cube-lovers@ai.mit.edu>
Subject: Rubik's old-timer
Date: Fri, 1 Jan 1999 22:44:07 -0500
Message-Id: <001b01be3602$2685f3e0$da460318@CC623255-A.srst1.fl.home.com>

Puducky@aol.com asked me to forward the following to this list:

I have a 96 year old friend that can do the rubiks cube in less than five
minutes, and near the end he doesn't even look at it! I thought this pretty
amazing for a man of his age. Do you know anyone I can contact about him?
Was wondering if there is a record for his age? Thank you.

Puducky@aol.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Jan  5 19:10:03 1999
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Message-Id: <19990102200254.8149.rocketmail@send104.yahoomail.com>
Date: Sat, 2 Jan 1999 12:02:54 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Original Rubik's Cube Query
To: Cube Lovers <Cube-Lovers@ai.mit.edu>

Hi,

This may be inappropriate for this list, but I thought a few cube
lovers may be interested in this.

I was talking to my college buddy the other day, and he was telling me
about a curious habit of his mom's of buying x-mas gifts and not
giving them away.  Instead, the gifts were stowed away in one of their
closets like museum pieces.  One of these "pieces" is a Rubik's cube.

Hmm... Now I became interested. I asked him about it.. he thought it
wasn't worth much, the packaging was pristine, looked new.  I asked
him to look carefully at the packaging.  The manufacturer was Ideal,
and it has a "1980" and a "Made in Hungary" stamped on it.  (i.e. an
Original Rubik's Cube, untouched in it's original packaging for almost
two decades). I told him it was probably worth more than he thinks.

Just curious.  Does anyone know how much this little gem is worth?

==
_________________________________________________________
Han Wen
Applied Materials
3050 Bowers Ave, MS 1145
Santa Clara, CA  95054
e-mail: Han_Wen@amat.com / hansker@yahoo.com

[Moderator's note: For the past few years there have been rumors
 flying around the net, and perhaps the news services, about thousands
 of dollars being paid for vintage unopened cubes, either the Ideal
 "Rubik's cubes", or the earlier "Hungarian Magic Cube".  Some of
 these rumors may even be true.  However, I feel I should warn people
 against careless speculation, since the Internet is prone to fraud of
 various forms. ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Jan  5 19:59:00 1999
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Message-Id: <3692A126.3FA0@hrz1.hrz.tu-darmstadt.de>
Date: Wed, 06 Jan 1999 00:32:54 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: cube-lovers@ai.mit.edu
Cc: michael reid <reid@math.brown.edu>
Subject: Your Optimal Solver

Mike,

I now have a Linux-partition on my PC and I compiled your optimal cube 
solver on it. It really runs fast, about 30%-40% faster than my own
optimal solver which uses the same coordinates. I then compiled your
source code with the Microsoft Visual C++ compiler with similar results.
(By the way, if there other users of the Wintel platform who are
interested in Mikes program I could send the program code to the Cube
Lovers Archives, its only 50KB).

The main reason for the different performance is the fact, that during
the tree search I only hold one cube in memory and I do not use an array
for the cube-coordinates.
But then I had another idea, which was not implemented in my program and
which does not seem to be implemented in yours and which significantly
increased the performance of my program (about 20%) with a few lines of
code (but I think it only works in face-turn-metric):

You use the lines similar to

if (p_node[1].remain_depth<DIST) continue;

which means tree pruning and we apply the next move in this depth.

An analysis shows, that the case (p_node[1].remain_depth<DIST-1) also
happens quite often (while p_node[1].remain_depth<DIST-n can never occur
for n>1 except at the very beginning of the search). In this case, if we
for example had applied the move R, we need not to check R2 and R' any
more but we can continue with another axis.
In the case of the quarter-turn-metric, if we had applied R, we still
had to check R' because the distance of the two resulting states from
start can differ by 2.

Herbert

From cube-lovers-errors@mc.lcs.mit.edu  Thu Jan  7 13:53:48 1999
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Date: Wed, 6 Jan 1999 23:02:32 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199901070402.XAA18140@cauchy.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Your Optimal Solver

herbert writes

> You use the lines similar to
>
> if (p_node[1].remain_depth<DIST) continue;
>
> which means tree pruning and we apply the next move in this depth.
>
> An analysis shows, that the case (p_node[1].remain_depth<DIST-1) also
> happens quite often (while p_node[1].remain_depth<DIST-n can never occur
> for n>1 except at the very beginning of the search). In this case, if we
> for example had applied the move R, we need not to check R2 and R' any
> more but we can continue with another axis.

interesting idea.  when i get a chance i'll see if i can also get a
performance boost using this idea.

for quarter turns, there is something similar i can do, but this is
only because of the method i used for quarter turns.  namely, i
don't ever do  R R , instead, i do  R2  and count it as two moves.
if applying the move  R  results in a branch of the tree that gets
pruned, then we do not have to try  R2.

however, if i used a different method for quarter turns, where i only
make one move at a time, then the  R2  branch would be a sub-branch
of the  R  branch.  thus it would be pruned automatically.
this suggests that it might be better to use this latter method for
searching the tree.  (the only reason i didn't do this is that i
wanted to use one function for both quarter turns and face turns.)

another idea, suggested to me by rich korf, is to use the line

     if ((node.remain_depth < ELEVEN) && (node.remain_depth < DIST))
        continue;  /*  prune this branch  */

where  ELEVEN  is just the constant 11, and  DIST  is the macro to
look into the big table for the distance of the current coset.
if the first part of the expression is  TRUE , then we evaluate
the second part.  in this case we did a tiny bit of extra work
to evaluate the first part.  but if the first part is  FALSE ,
then we save some work by not looking into the table.  we lose
a little bit of pruning (there are some cosets at distance 12)
but this is very small.  rich explained (if i understood correctly)
that every look into the big table is expensive, because it will
pull a small piece of the table into cache.  but this piece
is unlikely to be used again soon, so it probably displaced some
more useful stuff from cache.  the DIST macro is also a complicated
expression, so it is also expensive in that way.  when i tried this,
i didn't measure any significant performance boost (< 1%).  but the
cache benefit would be more noticeable for longer searches, so
perhaps my test was just too short.  it also depends upon your DIST
macro (or corresponding code); i think rich had more processing to
do besides looking into the table.  and it may also depend on the
size of your secondary cache.

i do have this in my huge optimal solver, so it must have given
some improvement there, but i don't remember how much.  i had to
do lots of tweaking for performance issues on this program.

herbert, if you have a program that uses the exact same coordinates
as mine, you will find it amusing to try the positions

*  position created by  R2 F' R2 F2 D2 F' R2 F2 R2 D2 F' D2 F'
*  inverse of the above position.

and noticing the huge difference.  because of this, i thought
about maybe solving either the input position or its inverse,
depending upon which should be faster.  my experiments showed
that it wasn't easy to predict which would be easier by looking
only at the distances of the 3 initial cosets.  but perhaps
doing a mini-search on each, and looking at how many nodes
they spawn would give a better guess.

of course, we can't expect to get the kind of performance boost
suggested by the extreme example above, but we might get something.

then i had a more ambitious idea:  maybe we could prune the search
tree by having the program realize "the inverse of this position is
too far from start".  the conclusion i eventually arrived at was that
it wouldn't be possible to keep track of the coset of the inverses
by using transformation tables.  so this idea probably won't work.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Jan 13 18:08:03 1999
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Date: Tue, 12 Jan 1999 23:18:11 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: cube-lovers@ai.mit.edu
Message-Id: <009D220A.48E392F4.2@ice.sbu.ac.uk>
Subject: Clinton and Rubik?

Date: Sun, 10 Jan 1999 09:28:19 -1000
From: drogers@math.hawaii.edu (Douglas Rogers)
Message-ID: <199901101928.JAA01661@knuth.hawaii.edu>
To: wpr3@tutor.open.ac.uk
Subject: From Monica Lewinsky to the Roubik Cube


Bill,

   Here is a quotation that might be appropriate for The
Mathematical Gazette.

   Senator Trent Lott (R-Missouri), Senate Majority Leader,
speaking on 6th January, 1999, to reporters about arrangements
for the impeachment of the President, declared, ``All sides
of this Rubik's cube have been talked about. We hope to have
this all resolved tomorrow''. [As quoted in The New York Times
for 7th January, p. A1.]

   The natural inference here is that Senators have a version of
the Rubik cube on which the US President and friends are depicted,
the object of the exercise being to twist things so as to get
the US President into or out of compromising positions - it was
thoughtful of the Senate Majority Leader to spare us the details,
and to leave this to our imagination.

   DGR.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Thu Jan 14 13:12:53 1999
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Clinton and Rubik?
Date: 14 Jan 1999 14:43:30 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <77kvqi$o6k@gap.cco.caltech.edu>
References: <cube-lovers.009D220A.48E392F4.2@ice.sbu.ac.uk>

David Singmaster <david.singmaster@sbu.ac.uk> writes:
>   Senator Trent Lott (R-Missouri), Senate Majority Leader,
>speaking on 6th January, 1999, to reporters about arrangements
>for the impeachment of the President, declared, ``All sides
>of this Rubik's cube have been talked about. We hope to have
>this all resolved tomorrow''. [As quoted in The New York Times
>for 7th January, p. A1.]

Of course, claiming that they can REsolve it implies that
it was already SOLVED at some point in the past ... :-)

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"... I guess that explains why you're automatically dogmatic!"

From cube-lovers-errors@mc.lcs.mit.edu  Mon Jan 25 14:16:47 1999
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Jan 24 00:08:55 1999
Date: Sun, 24 Jan 1999 00:07:11 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re: Corners Only, Ignoring Twist
In-Reply-To: <Pine.PMDF.3.95.980927193742.73560A-100000@PSTCC6.PSTCC.CC.TN.US>
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Reply-To: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Message-Id: <Pine.PMDF.3.95.990123235414.196055A-100000@PSTCC6.PSTCC.CC.TN.US>

On Sun, 27 Sep 1998, Jerry Bryan wrote:

> In developing a no-twist, no-flip version of the program, I decided to try
> it out on the corners only case.  Here are the results.

Here is one more tidbit on this subject.  The program which performed the
God's Algorithm search for the no-twist corners only case produced a
summary by symmetry class.  I was surprised to note that there were two
positions for which Symm(x)=M.

Normally, there is only one position for the corners where Symm(x)=M,
namely the position which fixes all the corners (i.e., Start).  However,
if twists are ignored, then the central inversion of the corners has the
property that Symm(x)=M.  This is analogous to the Pons Asinorum position
where the edges are centrally inverted.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Mon Feb  1 15:08:06 1999
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Jan 24 13:57:32 1999
Message-Id: <19990124185842.5788.rocketmail@send103.yahoomail.com>
Date: Sun, 24 Jan 1999 10:58:42 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Query on Octagon Cube Edge Parity Problem
To: Cube Lovers <Cube-Lovers@ai.mit.edu>

Hi,

I ran into an unusual scenario with the Octagon cube recently where
only ONE edge piece was flipped and all the other pieces were
positioned and oriented properly. This is bizarre of course, because
with a Rubik's cube, this is an impossible scenario; there must be a
minimum of TWO edge pieces flipped.

Does anyone understand the redundancy that allows this strange edge
parity problem?  And I guess, how to solve it.  I lamely mixed the
cube up again, resolved until the problem "went away".

For those who may not be familiar, the Octagon cube is a variant of
the Rubik's Cube. The cube is organized by color into 8 columns of
three cublets: corner, edge, corner. If you look at the top face you
see that the half of the corner cublets have been cut away so the the
face forms an octagon instead of a square. This octagon shape is
extended down through the middle and bottom layer, so that the puzzle
looks like an octagon "tube". There are a total of 10 colors, two for
the top and bottom faces, and 8 for the eight columns of
(top-middle-bottom) cublets.

==
_________________________________________________________
Han Wen
Applied Materials
3050 Bowers Ave, MS 1145
Santa Clara, CA  95054
e-mail: Han_Wen@amat.com / hansker@yahoo.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  2 17:09:46 1999
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Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jan 26 09:45:56 1999
Date: Tue, 26 Jan 1999 14:45:54 +0000
From: "Collins, Lindon" <lindon.collins@lmco.com>
Subject: Middle layer last?
To: "'Cube Lovers'" <cube-lovers@ai.mit.edu>
Message-Id: <B1F50FC23BE7D1119DA700805FBB672C870B@swnd01>

Sorry if I'm retreading old ground but I'm new to this.

1.
	I remember reading somewhere an article that advocated solving
the middle layer of the cube last.  I have got some fairly short moves
to place cubes from the middle layer onto the bottom layer, but when it
comes to solving the middle layer, I seem to be trusting to luck that I
have reached a favourable position.  I cannot see how I am going to
reduce my average number of moves to solve the cube using this method.
I think there are two possibilities:-

	1.  There are some cool moves for solving the middle layer last
that I have missed.
	2.  I should forget about solving the middle layer last, and
stick to my usual method
	(ie. top,middle,bottom)

2.
	A more general question is: What is the shortest (practical)
method for solving the cube that anyone knows of?  (keyword: "practical"
- don't say 22 moves)

Thanks,
Lindon Collins
Swindon, UK

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  2 17:56:32 1999
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Date: Mon, 1 Feb 1999 17:16:54 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199902012216.RAA07304@chern.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Query on Octagon Cube Edge Parity Problem
Cc: hansker@yahoo.com

> I ran into an unusual scenario with the Octagon cube recently where
> only ONE edge piece was flipped and all the other pieces were
> positioned and oriented properly. This is bizarre of course, because
> with a Rubik's cube, this is an impossible scenario; there must be a
> minimum of TWO edge pieces flipped.

the "octagon" puzzle has some "edges" with only one visible face.
namely, these are the  U-D layer edges on a cube, which were shaved
when the cube was modified in shape.  these edges have no visible
orientation, so flip one of these along with the edge that's
definitely in the wrong orientation.

in a similar way, it's possible to get positions that appear to have
the wrong permutation parity.  there are four vertical columns of
two corners and an edge each.  these do not have any fixed "home"
location, so that any permutation of these also constitutes a "solved"
state.  (well, at least most people would consider it to be solved.)
but swapping two of these columns creates an odd permutation parity.
thus you can swap two columns, and also swap a pair of edges or
corners, which gives the impression of incorrect parity.

for a simple example, do  R2 F2 R2  from the solved position.
the edges  UF  and  DF  have been swapped, and it looks like nothing
else has happened.  in fact, the  FL  column has been swapped with
the  BR  column as well.

mike

[Moderator's note: I hadn't noticed that this had such an obvious
 answer.  Thanks also to Jon Ferro, Steve LoBasso, der Mouse,
 Guy N. Hurst, Michael Ehrt, and Christ van Willegen for also
 providing the answer.  I've selected Mike Reid's, since he points out
 the other notable ambiguity of the Octagon.

 What wasn't noted is that the Spratt wrench can be used to flip the
 noted edge along with three of the ambiguous edges. --Dan]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  2 19:27:15 1999
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Message-Id: <36B77B8B.2A223AA5@sgi.com>
Date: Tue, 02 Feb 1999 14:26:19 -0800
From: Derek Bosch <bosch@sgi.com>
To: "Collins, Lindon" <lindon.collins@lmco.com>
Cc: "'Cube Lovers'" <cube-lovers@ai.mit.edu>
Subject: Re: Middle layer last?
References: <B1F50FC23BE7D1119DA700805FBB672C870B@swnd01>

well, if you want "simple", the only two moves you really need
are:

 the edge 3-cycle...  U2^U2v  (UB->DF->UF)
 and the 2-edge flip...  ^U^U^U2vUvUvU2  (flips UF and UB)

assuming you are holding the cube so that the left and right faces are solved
(where ^ is moving the middle slice up, and v is moving the middle slice down)

note, once you get used to those moves, you can create variations that move
and flip, like:

U'^U'^U2vU'vU

these are also REALLY ergonomic - very easy to do rapidly...
-- 
Derek Bosch        "A little nonsense now and then
(650) 933-2115      is relished by the wisest men"... W.Wonka
bosch@sgi.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  2 21:16:59 1999
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Date: Tue, 2 Feb 1999 20:27:26 -0500
Message-Id: <00131725.C22092@scudder.com>
From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject: Re: Middle layer last?
To: "'Cube Lovers'" <cube-lovers@ai.mit.edu>

1.  I learned the solution of top, middle, and then bottom many years ago.
I forgot most of it, so when I asked Kristin Looney to remind me how to do
the cube, she showed me her solution, which is top corners, bottom cFrom cube-lovers-errors@mc.lcs.mit.edu  Sat Feb  6 01:46:02 1999
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Date: Tue, 2 Feb 1999 20:27:26 -0500
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From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject: Re: Middle layer last?
To: "'Cube Lovers'" <cube-lovers@ai.mit.edu>

1.  I learned the solution of top, middle, and then bottom many years ago.
I forgot most of it, so when I asked Kristin Looney to remind me how to do
the cube, she showed me her solution, which is top corners, bottom corners,
top and bottom edges at the same time, then middle edges.  It is a much
easier solution to do and to teach.  A variant of it can be found at
http://www.unc.edu/~monroem/rubik.html, although in this variant he is more
rigid about how to solve the top and bottom edges.  So, I'd forget your
original method of top/middle/bottom, because it does not revel in the
corner/edge dichotomy of the cube.  I only use top/middle/bottom when
solving cubes with complicated patterns, such as those at
http://www.wunderland.com/WTS/Kristin/CustomCubes.html.

2.  The shortest, pratical, method for solving the cube, in my opinion, is
the one discussed above.  Kristin, who does not have incredibly fast hands,
did very well in speed cube competitions when pitted against people who had
hot hands but the top/middle/bottom solution.  Also, I can teach someone
this solution in an hour.

-Jacob

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  9 15:30:14 1999
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Date: Sun, 7 Feb 1999 21:18:58 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: reid@math.brown.edu
Cc: CUBE-LOVERS@ai.mit.edu
Message-Id: <009D3667.F01254DB.8@ice.sbu.ac.uk>
Subject: Re: Query on Octagon Cube Edge Parity Problem

	Similar parity problems can be produced by recolouring a cube.  I once
sold a cube to someone who came back a few minutes later with two centers
exchanged.  I accused him of taking it apart, but then I fiddled with it and
got it back right, which amazed me even more.  Then I discovered that two
opposite faces of the cube were colored red!  Some time later, I had an example
with two adjacent sides having the same color.
	In 1980, Tamas Varga showed me some cubes with just two colors and I
then made up numerous such color variants.  E.g. using just two colors, have
the three faces of one color meet at an corner, or not meet at a corner (these
are the only two ways of coloring the faces with two colors, three faces of
each color).  Also fun is a three color cube - two opposite sides red, two
opposite sides white and two opposite sides blue.  Then every corner is  red,
white, blue  - except half of them are red, blue, white.  All these are
difficult to solve for people who have only done ordinary cubes.
	In the mid 1980s, Edward Hordern showed me a cube which I recall he
said Nob Yoshigahara had invented, but my example was made by Marcel Gillen.
This appear to be a  4^3,  but when turned, it moves eccentrically.
Examination shows that it is a  3^3 with three layers of pieces glued to three
adjacent faces.  Edward's original example had no colors, so it took some time
to solve as one didn't know where pieces went.  Further, the eccentric movement
causes parts to protrude, making it hard to hold and to move.  All in all, a
most enjoyable variant.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  9 20:27:47 1999
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Message-Id: <19990208030401.14755.rocketmail@send1e.yahoomail.com>
Date: Sun, 7 Feb 1999 19:04:01 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Moves for Solving the Pyramorphix puzzle
To: Cube Lovers <Cube-Lovers@ai.mit.edu>


Hi

For those cube masters out there getting bored, you may want to play
with Meffert's latest puzzle, the Pyramorphix.  It only takes about
1-2 hours to solve, but it provides some Square-1-like entertainment.
For those who may not be familiar, the Pyramorphix looks like the
Pyraminx's little brother with 4 pieces instead of 9 pieces for each
face.  However, unlike its big brother, the Pyramorphix turns a lot
differently (90 deg rotations), morphing into different shapes as you
twist and turn.  As you know, with the Pyraminx, you're always
rotating little pyramids about one of the 4 tips. However, with the
Pyramorphix, when starting out with the pyramid shape, you rotate an
edge and its two adjacent corners (i.e. the whole edge of a given
face).  The rotation axes lie along the intersection of the 4 center
triangle pieces.

There appears to be 6 general shapes you can "morph" the Pyramorphix
into I'll affectionately call: the Pyramid, the Butterfly, the Crown,
the Rocket, the Airplane and the truncated Star of David.  (There are
actually two truncated Star of Davids, mirror images of one another).
You'll know what I mean if you actually have the little creature in
your hands. The hardest part of this puzzle is figuring out how to
morph between all these different shapes.

I solve the Pyramorphix by first solving the 4 corner pieces and then
orienting the 4 center pieces.

First, a few notation definitions. Hold the Pyramorphix so that you
have one of the faces facing you.  I'll call that face Front (F).  The
face on the bottom will be called Down (D), and the faces left and
right of the F face will be called Left (L) and Right (R) respectively.

An edge will refer to a center piece and its two adjacent corner
pieces. I'll specify which edge by indicating the two faces the edge
intersects (e.g. R-D edge is the edge formed by the interesection of
the Right and Down faces).
Now, unfortunately, I can't just refer to rotations of edges, because
as you'll see, sometimes you need to rotate strange shapes that look
nothing like an edge.  Instead, it's better to refer to slices.
Namely the plane about which one of the edges rotates on.  So, if you
look at the F face, you'll see three slices: the R-D slice (60 deg),
the L-D slice (120 deg) and the horizontal F-slice (0 deg).  When the
Pyramorphix is in its pyramid shape, you can rotate the two edges on
the R-D and L-D slices, but you cannot rotate the corner piece sitting
on the F-slice.  Don't worry, when we morph the pyramid into the
Rocket, you'll see that you'll be able to rotate the tip of the rocket
about the F-slice.

Now, that I've thoroughly confused you, here are my notations for
actual moves:
R - 90 deg clockwise rotation about the R-D slice
L - 90 deg clockwise rotation about the L-D slice
F - 90 deg clockwise rotation about the F-slice
R2 - 180 degree rotation about the R-D slice
R' - 90 degree counterclockwise rotation about the R-D slice

Finally, hold the Pyramorphix in your hands, so the the F-face is
facing towards you with the tip of the triangle pointing up.  If you
now rotate the puzzle so the the D-face is facing towards you, you
should see an upside-down triangle with the tip pointing down.  I will
refer to the three corner pieces that you see as:

DL - left corner piece
DR - right corner piece
DM - middle corner piece
_______________________________________________________
To solve the four corner pieces, first get them in their proper
positions by performing 180 rotations of the edges. Now, you need to
orient the corners by making appropriate clockwise or counterclockwise
twists.  Here are some moves to do this:

Name: Single corner twister
Move: (R L' R' L) ^2
Shapes: Butterfly - Star of David - Airplane - Airplane - Star of
David - Star of David - Butterfly - Pyramid
Action: Clockwise (CW) twist of DM corner

Name: Left-side double corner twister
Move: (R L R' L' ) ^2
Shapes: Butterfly - Star of David - Crown - Star of David - Crown -
Star of David - Butterfly - Pyramid
Action: CCW twist of DL corner, CW twist of the DM corner

Name: Right-side double corner twister
Move: (R' L' R L) ^2
Shapes: Butterfly - Star of David - Crown - Star of David - Crown -
Star of David - Butterfly - Pyramid
Action: CCW twist of DR corner, CW twist of the DM corner

Name: Triple corner twister
Move: (R' L R L') ^2
Shapes: Butterfly - Star of David - Star of David - Airplane -
Airplane - Star of David - Butterfly - Pyramid
Action: CW twist of DL corner, CW twist of DR corner, CCW twist of DM
corner
_______________________________________________________
Now, to orient the center pieces in place, here are some moves to do
this:

Name: Double edge swapper
Move: R' L2 F2 L2 R
Shapes: Butterfly - Rocket - Rocket - Butterfly - Pyramid
Action: Swap F <--> L and R<-->D center pieces

Name: Tricycle swap
Move: (L R2 F' R2 L') (R' L2 F L2 R)
Shapes: (Butterfly - Rocket - Rocket - Butterfly - Pyramid ) ^2
Action: Permute F --> R --> L --> F center pieces

Name: Edge swapper
Move: (R' L2 F L2 R) (L R2 F R2 L') (R' L2 F L2 R)
Shapes:  (Butterfly - Rocket - Rocket - Butterfly - Pyramid ) ^3
Action: Swap F <--> D center pieces

With these short collection of moves, you should be able to readily
solve the Pyramorphix...
_______________________________________________________
Epilogue:

So, you've solved the Pyramorphix, bored silly and you want to know
what's next?!  Check out Meffert's Puzzle Ball.  It's actually a bit
more challenging.  Happy puzzling.... :)



==
_________________________________________________________
Han Wen
Applied Materials
3050 Bowers Ave, MS 1145
Santa Clara, CA  95054
e-mail: Han_Wen@amat.com / hansker@yahoo.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb  9 22:42:43 1999
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Message-Id: <714F77ADF9C1D111B8B60000F863155102DD6CBA@tbjexc2.tbj.dec.com>
From: Norman Diamond <Norman.Diamond@dec-j.co.jp>
To: <cube-lovers@ai.mit.edu>
Subject: Re: Moves for Solving the Pyramorphix puzzle
Date: Wed, 10 Feb 1999 11:34:47 +0900

Hey!  Everyone on this list _already_ knows moves to solve
Pyramorphix, although it often requires careful staring at the
thing in order to recognize the exact configuration each time.

You see, everyone on this list knows how to solve 3x3x3
Rubik's cube.  And everyone knows how to solve 2x2x2
Rubik's cube because it's a subset of the 3x3x3, without
edges or centers.  And everyone knows how to solve
Pyramorphix because it's a subset of the 2x2x2, where
some of the corners don't need orienting.

-- Norman.Diamond@dec-j.co.jp
[Not speaking for Compaq]

From cube-lovers-errors@mc.lcs.mit.edu  Thu Feb 11 21:17:42 1999
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Message-Id: <3.0.3.32.19990210083221.0092a700@cscan02.caddscan.com>
Date: Wed, 10 Feb 1999 08:32:21 -0500
To: Cube-Lovers@ai.mit.edu
Reply-To: <bmain@caddscan.com>
From: Bryan Main <bmain@caddscan.com>
Subject: Puzzle Stores in London area

I'll be going to the london/cambrige area in a few days and was wondering
if anyone knew where there are any puzzle stores that have cube puzzles.
I'm not really looking for 3x3x3 cubes since I have enough, mainly I'm
looking for the other ones like pyrimix, megamix etc.  thanx in advance.

bryan

[Please respond directly to Bryan; if there are particularly
worthwhile and non-repettive suggestions he can send them to
cube-lovers --Moderator ]

From cube-lovers-errors@mc.lcs.mit.edu  Fri Feb 19 12:26:12 1999
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Date: Thu, 18 Feb 1999 23:45:18 -0400 (EDT)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Edges only, Ignoring Flips, Face Turn Metric
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Message-Id: <Pine.PMDF.3.95.990218232951.21673A-100000@PSTCC6.PSTCC.CC.TN.US>

I have completed a God's Algorithm run in the face turn metric for the
group consisting of edges only ignoring flips.  The size of the group is
therefore 12!  The results are as follows:

  Distance    Patterns   Positions    Branching
    From                               Factor
   Start

     0f            1              1
     1f            2             18     18.000
     2f            9            243     13.500
     3f           75           3240     13.333
     4f          920          42535     13.128
     5f        11406         542234     12.748
     6f       136423        6529891     12.043
     7f      1386164       66478628     10.181
     8f      6481303      310957078      4.678
     9f      1969536       94443600      0.304
    10f          129           4132      0.000

  Total      9985968      479001600

I should mention that Herbert Kociemba was the first to calculate the
"patterns" column.  He did it as a part of an investigation into
developing an IDA* optimal solver, with the patterns column being part of
a patterns data base used by the IDA* algorithm.

I used my usual program where (for example) I calculate the set of all
positions which are 10f from Start by forming the products of all
positions which are 5f from Start in lexicographic order, and throwing
away the duplicates and the ones that are shorter than 10f.

This technique is reasonably efficient when the branching factor is fairly
constant, as it is at this distance from Start for larger problems such as
the whole cube.  However, it is very inefficient for this particular
problem.  I have to calculate about (542234^2)/48 products just to get the
129 products I keep at 10f from Start.  (The "divide by 48" takes symmetry
into account.)  In a "one level at a time" search by contrast, the tail of
the distribution usually goes very quickly because there are so few
positions in the tail.

I have come to believe that any corners only (with or without twist) or
edges only (with or without flip) group, or the group which keeps both
corners and edges but without twists and flips, will be a fairly poor
pattern data base for IDA*.  The problem is that any such search space
will have a diameter which is too small, and more importantly will have an
average distance from Start which is too small.

Here is why I think the diameter and average distance from Start will be
too small for these groups.  Consider the quarter turn metric.  We know
immediately that the maximum branching factor is 12 because there are 12
quarter turns.  We know almost as immediately that the maximum branching
factor beyond one move from Start is 11 because there is always at least
one quarter turn that goes closer to Start.  Finally, readers of
Cube-lovers know that the maximum branching factor is asymptotic to about
9.3 because of Dan Hoey's syllable analysis.  Syllable analysis takes into
account moves which commute because they are on opposite faces such as
RL=LR.

(Similar analysis for the face turn metric yields an asymptotic maximum
branching factor of about 13.3)

I have come to think of syllable analysis not just as an upper limit for
the branching factor but as a predictor for the branching factor. Indeed,
the actual branching factor differs from the branching factor "predicted"
by syllable analysis only because of duplicate positions which arise from
processes which are not accounted for by syllable analysis.

Such duplicate positions must exist by the finiteness of the problem, else
a God's algorithm search would be infinite.  But such duplicate positions
are non-trivial and are generally not very close to Start. With the full
cube, they are quite rare as close to Start as has been searched so far
(10f and 12q, respectively).

What happens with a typical search is that the branching factor stays
relatively constant until within a couple of levels of the mode of the
frequency distribution of the distances from Start.  The branching factor
then declines rapidly due to duplicate positions, and there is a short
tail in the frequency distribution just past the mode.

The key point is that syllable analysis is identical for all groups
involving corners only, edges only, corners and edges, and/or with or
without twists and flips.  Hence, the basic branching factor is the same
for all such groups.  Therefore, the mode of the distribution is reached
much sooner when the group is smaller and the average distance from Start
is much smaller.

What would be desirable for a pattern data base for IDA* would be a
subgroup of G whose branching factor was smaller so that the mode, the
diameter, and the average distance from Start would be larger.  That is,
what would be desirable would be a subgroup which was not constrained by
standard syllable analysis.  Failing that, it seems to me that the only
way to improve the pattern data base is to make it larger.  In this light,
I interpret what Mike Reid (and more recently) Herbert Kociemba have done
with their IDA* programs is to find clever ways to make their pattern data
bases as large as possible, but they do it in such a way (using symmetry
and other equivalence classes) that their large data bases are small
enough to store in memory'

As I think has been mentioned before, a group such as <U,R> has a small
branching factor but is not suitable as a pattern data base for IDA*
because the distance from Start for a particular position in <U,R> may be
larger than the distance from Start for the same position in G.

Jerry Bryan

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From: Douglas Zander <dzander@solaria.sol.net>
Message-Id: <199902192002.OAA04117@solaria.sol.net>
Subject: 2by alternative mechanism
To: cube-lovers@ai.mit.edu
Date: Fri, 19 Feb 99 14:02:03 CST

Hello,  I remember someone asked if there were two different mechanisms 
to the 2x2x2 cube but I can't remember if it was answered.  Last night
while doing a patent search I came across a mechanism totally different
than what my own cube looks like.  It is U.S. Patent # 5826871  Oct '98
Basically, it is a standard Rubik's Cube with the central layers hidden
underneath and one quadrant is fixed in place.  I am wondering if there 
exists a web site with all the diagrams of all different puzzles from
their patent pages.  As you may know, patents and patent diagrams are 
*not* covered by copyright issues, you may photocopy them for personal
viewing and even display them on a web page.  I think such a web page
would be nice to have; a web page of all the diagrams of the working 
mechanisms of all the puzzles discussed on this list.  (The thing is 
though that soon the U.S. Patent office will be adding the diagrams to
their web pages. www.uspto.gov)   Anyone wish to start such a page?
I have access to a patent depository.  :-)

-- 
 Douglas Zander                | 
 dzander@solaria.sol.net       |
 Shorewood, Wisconsin, USA     |

From cube-lovers-errors@mc.lcs.mit.edu  Mon Feb 22 13:23:12 1999
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Date: Mon, 22 Feb 1999 17:23:57 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: cube-lovers@ai.mit.edu
Message-Id: <009D4210.9784799A.12@ice.sbu.ac.uk>
Subject: Fwd: Request for spectacular cube-solving  -  Can anyone help?

From:	mytv-film@t-online.de
To:	MX%"david.singmaster@sbu.ac.uk"
Subj:	rubik's cube
Date: Wed, 17 Feb 1999 17:49:19 +0100
To: david.singmaster@sbu.ac.uk
Subject: Rubik's cube

Hello Mr. Singmaster,

my name is Gvksen and I'm working for the new German TV show "Guinness -Show
of Records" on behalf of the German Broadcast Station ARD.

In one of our next shows we would like to present a person who is able to solve
a classical Rubik's Cube in a spectacular way. For example by studying the cube
first for a couple of minutes and then solving it without looking at it. On an
internet homepage with different links to Rubik's cube fans we've read that there
is a young man in U.K. called John White who is able to do that: Studying the
cube for 10 minutes and than solving it behind his back in 136 seconds. Now,
we're looking for this young man, who is or was a mathematics student at the
University of Warwick.

Our investigation up to now brought us to you, so you are considered to be one
of the leading "Rubik's cube experts" in the world. In addition you were a judge
during the "Rubik's Cube World Championship". So you may know a lot of persons
and beyond that a lot of different variations of solving a Rubik's cube in an
impressive way.

We would be very happy if you could give us a hint or some advice concerning our
idea of presenting this category of record in our show.

One characteristic feature of our show is the fact that we present a record as a
competition. That means: We would invite the record holder and a challenger who
believes himself capable of being able to solve a cube in a similar way. In the
best case this challenger should be a person who speaks German. Do you know
anyone who could be suitable?

I hope that these are not too much questions and I would like to thank you even
now for your help and kind co-operation.

Our e-mail address: mytv-film@t-online.de

kind Regards

Gvksen
MyTV Film and Tv Production

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Tue Feb 23 11:10:32 1999
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Message-Id: <19990222194015.5998.rocketmail@send105.yahoomail.com>
Date: Mon, 22 Feb 1999 11:40:15 -0800 (PST)
From: pete beck <just_puzzles@yahoo.com>
Subject: Cubes in the news (was: Request for spectacular cube-solving...)
To: David Singmaster <david.singmaster@sbu.ac.uk>, cube-lovers@ai.mit.edu

It must be the age of the CUBE.

Pete Beck
----------------
from Yahoo! News Technology Headlines Monday February 22 1:19 PM ET

Corrected: Rubik Cube Whiz Offers Millennium Bug Solution

LONDON (Reuters) - A man who solved the riddle of Rubik's cube has
invented a test kit to detect where the millennium computer bug will
strike.

At the age of 12, Patrick Bossert shot to fame when he worked out his
own solution to the mystifying cube and wrote a bestseller about it
that sold 1.5 million copies.

Now 30, he and a team of software experts at London-based WSP Business
Technology have developed Delta-T Probe, a program that can work out
whether microchips embedded in electronic equipment are likely to fail
on January 1, 2000.

Delta-T works by electronically detecting equipment to identify chips
that process date and time, making it likely to malfunction when 1999
becomes 2000.

``Only a small percentage of systems will fail to recognize the next
millennium, but finding out which ones might go wrong is a huge and
costly process,'' said Bossert, technical director at WSP Business
Technology, a unit of consulting engineering group WSP Group Plc.

Bossert estimates hundreds of millions of chips are buried deep inside
equipment in Britain. The chips control devices such as security
systems, fire alarms, production lines, medical equipment and
telecommunications.

Bossert expects one in 500 embedded systems will take equipment back
in time to Jan. 1, 1900, causing equipment to fail.

British supermarket chain Sainsbury's Plc is among major companies
that have tested Delta-T. Sainsbury's said a trial run at one of its
stores in Devon, southwestern England, had been a success.



Hi, this is Pete Beck's personal e-mail site, a.k.a. Just Puzzles is a
hobby mail order seller of "Mechanical puzzles" specializing in
Rubik's Cube type puzzles. HOME is at
<www.freeyellow.com/members4/justpuzzles/> - post POB 267, Wharton NJ
07885, answering machine is 973-625-4191. Current as of 10 Nov 1998.

From cube-lovers-errors@mc.lcs.mit.edu  Wed Feb 24 12:52:23 1999
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Date: Mon, 22 Feb 1999 12:13:16 -0800 (PST)
From: Tim Smith <tzs@tzs.net>
To: cube-lovers@ai.mit.edu
Subject: Re: Fwd: Request for spectacular cube-solving  -  Can anyone help?
In-Reply-To: <009D4210.9784799A.12@ice.sbu.ac.uk>
Message-Id: <Pine.LNX.4.04.9902221210120.6439-100000@52-a-usw.rb1.blv.nwnexus.net>

> In one of our next shows we would like to present a person who is able to
> solve a classical Rubik's Cube in a spectacular way.

A spectacular way I'd like to see someone do the cube is while juggling.
Instead of that old routine where the juggler juggles some fruit and
eats the fruit while juggling it, juggle three (or more!) cubes, and
solve them at the same time.

--Tim Smith

From cube-lovers-errors@mc.lcs.mit.edu  Wed Feb 24 13:57:12 1999
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Date: Tue, 23 Feb 1999 15:41:05 -0800 (PST)
From: "Jorge E. Jaramillo" <kingeorge@rocketmail.com>
Subject: Number of moves
To: cube <Cube-Lovers@ai.mit.edu>

A lot has been said about world records solving the
cube when it comes to time but I don't recall seeing
anywhere how many moves the person who holds the
record did.
Does anyone know?


===
Jorge E Jaramillo

From cube-lovers-errors@mc.lcs.mit.edu  Wed Feb 24 15:56:12 1999
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Message-Id: <36D3B818.E20EFCD6@ibm.net>
Date: Wed, 24 Feb 1999 00:28:08 -0800
From: "Jin 'Time Traveler' Kim" <chrono@ibm.net>
Reply-To: chrono@ibm.net
To: cube-lovers@ai.mit.edu
Subject: Acquiring rare puzzles
References: <199902192002.OAA04117@solaria.sol.net>

This may be common knowledge to many of you on this list, but I thought
I'd mention these and give everyone a fair shot (as well as maybe
angering some who used it as a resource to locate some harder to find
pieces).

1) If you are looking for some rare pieces like the Revenge, Megaminx,
Alexander's Star, these items often come up for auction at

http://www.ebay.com

In the three weeks or so that I've been actively participating in
auctions, I've seen several of each of the following: Rubik's Revenge,
Alexander's Star, Missing Link, Megaminx, and Whippit.  I've also seen
Skewb, 5x5x5, Rubik's Magic, Pyraminx, Mickey's Challenge, and any
number of others come up for bidding.  Haven't seen a Rubik's Domino
(2x3x3), though.  For a general idea of what's there, try doing a search
for the key word "rubik" at the site.  That's just for starters.  I even
saw a Cuboctahedron (yes, just a shaved down 3x3x3 cube) go for over 60
dollars.  Incredible.  Using eBay I have personally added two more
Revenges to my collection.



2) There is also someone on the net who seems to be contacting people
directly and trying to sell a stock of puzzles he ran across.  His
email is jo_schumacher@[see moderator's note].  Try emailing him for
availability and pricing.  For example, he was last selling mint
Revenges for $109 and Alexander's Stars for $70.  I haven't bought
anything from him yet so consider it an "at your own risk" venture.
  
-- 
Jin "Time Traveler" Kim
chrono@ibm.net
http://www.slamsite.com/chrono
'95 PGT - SCPOC

[ Moderator's note: At least one person on the list has complained
  about getting spam from schumacher, and it sure looked like spam to
  me.  I certainly can't countenance sending e-mail to offer goods for
  sale to list members who haven't indicated a willingness to receive
  such an offer.  When it comes to spammers, "at your own risk"
  includes "at the risk of helping to make this medium unusable.

  I usually steer the numerous requests from people seeking Rubik's
  Revenge to ebay (and to yard sales, and to simulations).  But note
  that the FTC says that internet auctions are the growth zone for
  Internet fraud.  This can be somewhat mitigated by sending your
  money through an escrow agent--see ebay for information on that. ]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Feb 24 17:03:02 1999
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From: "Schmidt, Greg" <Greg.Schmidt@gonzo.cle.ab.com>
To: "'Tim Smith'" <tzs@tzs.net>, cube-lovers@ai.mit.edu
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone help ?
Date: Wed, 24 Feb 1999 13:53:02 -0500

While we're at it, I might as well add that I think the juggler should
be blindfolded!

-- Greg

[ And holding her breath underwater! And counting to one hundred
  backwards with her toes!

  Greg also notes that you can't trust ebay's auction ending dates.
  Caveat browsor.   -- Moderator ]

From cube-lovers-errors@mc.lcs.mit.edu  Thu Feb 25 11:25:26 1999
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Message-Id: <4.1.19990224195654.0092e1c0@mail.vt.edu>
Date: Wed, 24 Feb 1999 20:38:44 -0500
To: cube-lovers@ai.mit.edu
From: Kevin Young <keyoung3@vt.edu>
Subject: Oddzon version of the cube
In-Reply-To: <36D3B818.E20EFCD6@ibm.net>
References: <199902192002.OAA04117@solaria.sol.net>

I am curious if anyone has had the same problems with the current official
Rubik's cube that is in production.  I bought a Rubik's cube last year in a
toy store.  This was the newest version made by Oddzon.  I noticed that
there was a clear sticker overtop of each of the colored stickers.  In less
than a month this clear sticker peeled up.  I just don't remember the
original cubes by Ideal wearing down that fast.  In fact I still have a
couple cubes from Ideal with stickers still intact.

I contacted Oddzon by email and have received several responses from them,
forwarding my email to the appropriate department.  Currently my concern is
with under review by the Quality Control and Marketing department.  They
have claimed that this concern about the stickers is not common.  Maybe I
just got a bad cube, I don't know.  But, in order for them to make a better
product, I'm sure it's going to take a strong voice from the people
actually buying the products.  I just encourage everyone if they have
experienced the same thing with the stickers with the Oddzon product, to
contact them.

Most of the time, I still play with my cube I got from Ideal from 1982.  It
was called "Rubik's Cube Deluxe".  It was the official Rubik's cube that had
colored tiles instead of stickers.  That cube is holding up like new.  I
also made a motion to Oddzon to put the "Rubik's Cube Deluxe" back into
production.  That motion is now with the Senior Marketing Manager at
Oddzon.  It's his voice to decides which products go into production.  I
also encourage anyone that wants to see that version back into production,
to contact Oddzon.

Oddzon has been incredibly helpful and incredibly nice.  They have informed
me that two new Rubik's products will be coming out this year from Oddzon.
They should be out by the middle of the summer.  They werent at liberty to
say what products they were.  I can think of some that I'd like to see back
into production.  I'll have to wait and see.

On another note.  I do not endorse ebay, however, I have had a bit of
success with ebay.  I won two different auctions on ebay.  They were both
for Cubes by Ideal, never opened or played with, cellophane still intake.
I was lucky enough to get them for under 20 bucks apiece.  But, use
caution.  I wouldn't recommend ever auctioning on an item from a seller
with negative or low feedback.  Definantly be careful.  I've been really
lucky with the products I've won.

Cheers,
Kevin

From cube-lovers-errors@mc.lcs.mit.edu  Thu Feb 25 17:28:33 1999
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Date: Thu, 25 Feb 1999 08:58:31 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone  help ?

At 13:53 24-2-1999 -0500, you wrote:
>While we're at it, I might as well add that I think the juggler should
>be blindfolded!

Hey! _I'm_ already teaching my blind friend how to solve a cube!
We marked the colors of the cube with braille letters spelling
1 - 6 dots. We're having a terribly hard time to teach him to solve
it. It's fun, though...


Would a team of 3-4 blind people competing to solve the cube be
considered 'spectacular'? I think juggling is too hard. I'll ask
my gf, though (she knows how to juggle a bit, _and_ how to solve
the cube).

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar  8 21:24:30 1999
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Date: Thu, 25 Feb 1999 09:02:15 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: Megaminx solving times?

Hi,

I've been practising the Megaminx, and I can now solve it
without resorting to formulas written down on paper. I can
do it in about 10 minutes. How does this compare to other
people's times?

And, what method do you use? The method I developed relies
heavily upon the standard cube moves, and I solve the Mega-
minx going down from one flat top in rings. I needed to
adapt 1 (one!) standard cube formula to get it to work on
the Megaminx.

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar  8 22:00:14 1999
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Date: Thu, 25 Feb 1999 10:56:09 -0700 (MST)
From: Paul Hart <hart@iserver.com>
To: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
In-Reply-To: <4.1.19990224195654.0092e1c0@mail.vt.edu>
Message-Id: <Pine.BSF.3.96.990225103436.94998B-100000@anchovy.orem.iserver.com>

On Wed, 24 Feb 1999, Kevin Young wrote:

> I noticed that there was a clear sticker overtop of each of the colored
> stickers.  In less than a month this clear sticker peeled up.  I just
> don't remember the original cubes by Ideal wearing down that fast.  In
> fact I still have a couple cubes from Ideal with stickers still intact.

[...]

> They have claimed that this concern about the stickers is not common.
> Maybe I just got a bad cube, I don't know.

No, it's not just you.  I had this exact problem with an Oddzon cube.
Eventually all Oddzon cubes will develop this problem after any extended
use, I believe.  The "stickers" appear to be actually nothing more than
some sort of colored paper glued to the cube surface, with a thin
transparent plastic film covering the paper.

Like Kevin mentions, this plastic film begins to peel up at the corners
and edges of the stickers after extended usage of the cube, perhaps due to
fingerprint oils.  Eventually the film will fall off (or it gets to the
point where it must be deliberately removed), and the underlying colored
paper is completely ruined not long after.  I noticed this very same
phenomenon with one of the "C4" cubes, those cubes with Rubik's profile in
the center of the one of the sides.

It seems that these newly manufactured cubes are not up the level of
quality of cubes from "back in the day".  All of the original Ideal cubes
that I have seen have stickers that appear to be made of straight colored
plastic.  Even the clone knock-offs from that era used these stickers.
These solid plastic stickers seem to hold up much better than the
paper-based ones used on the Oddzon products.  After my disappointing
results with my Oddzon cube, I pledged to never again buy one of their
products until they change or improve their sticker design.

Paul Hart

--
Paul Robert Hart        ><8>  ><8>  ><8>        Verio Web Hosting, Inc.
hart@iserver.com        ><8>  ><8>  ><8>        http://www.iserver.com/

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar  9 12:30:38 1999
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References: <36D3B818.E20EFCD6@ibm.net> <199902192002.OAA04117@solaria.sol.net>
Date: Thu, 25 Feb 1999 10:06:33 -0800
To: Kevin Young <keyoung3@vt.edu>
From: Patrick Weidhaas <weidhaas1@llnl.gov>
Subject: Re: Oddzon version of the cube
Cc: RUBIK cube group <cube-lovers@ai.mit.edu>

Kevin Young wrote:

>...  I bought a Rubik's cube last year in a
>toy store.  This was the newest version made by Oddzon.  I noticed that
>there was a clear sticker overtop of each of the colored stickers.  In less
>than a month this clear sticker peeled up....

Kevin,

I do not have an answer for you, but your email made me wonder why stickers
are being used at all? As far as I know, nobody has produced a cube (or
variation) where the plastic "cubies" are colored appropriately without
relying on stickers. Is that process so much more expensive, or do the
toy-makers want to give their customers a chance to cheat by switching the
stickers in case they can't get the puzzle solved?

Patrick

------------------------------------------------------------------
Patrick P. Weidhaas                     e-mail: weidhaas1@llnl.gov
Parallel I/O Project                    voice: 925-422-7704
Lawrence Livermore National Laboratory  fax:   925-422-6287
P.O. Box 808,  L-560
Livermore, CA 94551-0808


From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar  9 13:06:35 1999
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Message-Id: <36D5AE30.23D2@zeta.org.au>
Date: Fri, 26 Feb 1999 07:10:24 +1100
From: Wayne Johnson <sausage@zeta.org.au>
Reply-To: sausage@zeta.org.au
To: Kevin Young <keyoung3@vt.edu>
Cc: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
References: <199902192002.OAA04117@solaria.sol.net> <4.1.19990224195654.0092e1c0@mail.vt.edu>

Kevin Young wrote:

> I am curious if anyone has had the same problems with the current official
> Rubik's cube that is in production...

However, it must be said that they are not suddenly aware of this. This
has been a problem for some time. Frankly, I have a Rubik's original
from 5 years ago with the same propblem. Oddzon don't seem to built the
items or make them in colours that Rubik himself even approves of (take
the new colours of the magic for instance - terrible).

I am currently on the lookout for an asian fake cube because they have
always been (dare I say it) better quality in the sticker dept. My Fake
is 15 years old with all stickers nicely attached.

Come on Oddzon... a cube that wears out in a month?.... I think this is
very clever and deliberate marketing.

Regards,
Wayne

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar  9 13:31:19 1999
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Date: Thu, 25 Feb 1999 18:12:51 -0500
To: cube-lovers@ai.mit.edu
From: Kevin Young <keyoung3@vt.edu>
Subject: Re: Request for spectacular cube-solving  -  Can anyone help ?
In-Reply-To: <3.0.32.19990225085830.00963b40@mail.spc.nl>

I am currently trying to learn how to do it without looking.  Have a long
ways to go.  I use to be able to do it, watching the cube of course,  in
less than a minute.  After 17 years and not being a school age boy anymore,
and forgetting some of my tricks, I can still do it everytime in approx. 90
seconds.  Long ways away from being a world champion.  But, currently I'm
working on the no looking thing.  Right now, I can put two pieces at a time
in their appropriate position, behind my back, however, after those pieces
are set, I have to look.  Good luck to all that are trying to learn how to
do it blindfolded.

Regards,

Kevin Young

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar  9 14:01:38 1999
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From: Norman Diamond <Norman.Diamond@dec-j.co.jp>
To: cube-lovers@ai.mit.edu
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone  help ?
Date: Fri, 26 Feb 1999 09:24:41 +0900

Christ van Willegen [c.v.willegen@spcgroup.nl] wrote:

>We marked the colors of the cube with braille letters spelling
>1 - 6 dots.

I have a magic domino which was actually manufactured that way.
Bought it from Christoph Bandelow about 13 years ago.

Dr. Bandelow, you're on this list, right?  Were there ordinary Rubik's
cubes with the same feature?

(Question to self:  How can the words "ordinary" and "Rubik's"
be placed next to each other in a sentence?)

-- Norman.Diamond@dec-j.co.jp
[Not speaking for Compaq]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 14:54:38 1999
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Message-Id: <199902261047.CAA19201@f15.hotmail.com>
From: "Philip Knudsen" <philipknudsen@hotmail.com>
To: cube-lovers@ai.mit.edu
Subject: RE: Request for spectacular cube-solving
Date: Fri, 26 Feb 1999 02:47:40 PST

What about solving the cube while performing a full-length rap?
I even know someone who can do that: MYSELF ;-)
Actually I am a singer, but I _do_ rap from time to time _and_ I also 
happen to speak german. But I can't joggle :-(

____________________________________
Philip K
Vendersgade 15, 3th
DK - 1363 Copenhagen K
Denmark
Phone:  +45 33932787
Mobile: +45 21706731
E-mail: philipk@bassandtrouble.com
E-mail: philipknudsen@hotmail.com

[Moderator's note: Philip also mentions his good experiences with ebay.
 The cube-lovers list isn't running a poll on them, so no one else needs to
 send reports of their performance.  Pay your money, take your chances, seek
 advice from appropriate sources. ]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 15:49:18 1999
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Date: Fri, 26 Feb 1999 09:06:55 -0500 (Eastern Standard Time)
From: Dale Newfield <din5w@cs.virginia.edu>
Reply-To: DNewfield@cs.virginia.edu
To: cube-lovers@ai.mit.edu
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone  help ?
In-Reply-To: <3.0.32.19990225085830.00963b40@mail.spc.nl>
Message-Id: <Pine.WNT.4.03.9902260905060.-424545@biff.cs.virginia.edu>

On Thu, 25 Feb 1999, Christ van Willegen wrote:
> Hey! _I'm_ already teaching my blind friend how to solve a cube!
> We marked the colors of the cube with braille letters spelling
> 1 - 6 dots. We're having a terribly hard time to teach him to solve
> it. It's fun, though...

How difficult is it to read braille when the characters are in arbitrary
orientations?  Have you thought of using some other coding technique that
might prove more easy to distinguish in any orientation?  (Any idea what
that might be?)

-Dale

[Moderator's note: I'm inordinately proud of my own invention, which I
thought I mentioned years ago but can't find in the archives: Wire
symbols glued to a destickered cube, polished to a high gloss.  The
symbols are blank opposite dot, square opposite circle, and plus
opposite X.  The supergroup is marked by a cutout at a corner of each
face center and the adjacent cubies.  I can solve it behind my back,
but when I lent it to a blind computer scientist, he gave up. --Dan]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 19:40:41 1999
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Fwd: Request for spectacular cube-solving - Can anyone help ?
Date: 26 Feb 1999 20:52:13 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <7b71ht$3pp@gap.cco.caltech.edu>
References: <cube-lovers.3.0.32.19990225085830.00963b40@mail.spc.nl>

Christ van Willegen <c.v.willegen@spcgroup.nl> writes:
>Would a team of 3-4 blind people competing to solve the cube be
>considered 'spectacular'? I think juggling is too hard. I'll ask
>my gf, though (she knows how to juggle a bit, _and_ how to solve
>the cube).

How about team solving?  n people, n cubes, everyone makes one move
and passes the cube to the left.  Repeat.  :-)

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"Pop", "Soda", or "Coke"?  http://www.ugcs.caltech.edu/~almccon/pop_soda/

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 20:15:40 1999
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Date: Sat, 27 Feb 1999 05:00:50 +0100 (MET)
From: Martin Moller Pedersen <tusk@daimi.au.dk>
Message-Id: <199902270400.FAA482087@bonestell.daimi.au.dk>
To: cube-lovers@ai.mit.edu
Subject: help on 5x5x5 wings

I am trying to solve my new cube the 5x5x5 cube.

I have managed to solve all of it except the wings.

The wings are the y's in the following diagram:


ZyZyZ
yZZZy
ZZZZZ
yZZZy
ZyZyZ

I have many moves for the 3x3x3 but I can't figure out how to apply these moves
to the wings.

Thanks for all help.

/Martin

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 20:47:02 1999
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Date: Thu, 4 Mar 1999 23:09:00 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199903050409.XAA24597@cauchy.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re: Edges only, Ignoring Flips, Face Turn Metric

jerry writes

> I have completed a God's Algorithm run in the face turn metric for the
> group consisting of edges only ignoring flips.  The size of the group is
> therefore 12!  The results are as follows:
[ ... ]

very interesting.  i hope that you'll also do the quarter turn metric.

> I have come to believe that any corners only (with or without twist) or
> edges only (with or without flip) group, or the group which keeps both
> corners and edges but without twists and flips, will be a fairly poor
> pattern data base for IDA*.  The problem is that any such search space
> will have a diameter which is too small, and more importantly will have an
> average distance from Start which is too small.

another shortcoming of this coset space for ida* is that transformations
aren't easy to compute.  for the cosets spaces i've used, they always
split up as a product of smaller coset spaces.  then i use transformation
tables for everything.

ida* spend a lot of time moving from a position to its neighbors.
instead of keeping the cube position, i just keep track of which
coset i'm in.  then i need to find out what coset i'll be in if i
apply the turn  F  (for example).  i always do this by using
transformation tables.  to simplify things, suppose that my coset
space had 1000000 cosets.  i could use a table with 18 * 1000000
entries that tells me which coset i go to by applying a given turn.
if my coset space is a product of two spaces, each with 1000 cosets,
then i only need a tranformation table with 18 * 1000 entries for
the first coordinate and one of the same size for the second coordinate.

this is really addressing implementation issues of ida*, not so much
the effectiveness of it.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 21:22:07 1999
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Message-Id: <36E4A993.5AB5@zeta.org.au>
Date: Tue, 09 Mar 1999 15:54:43 +1100
From: Wayne Johnson <sausage@zeta.org.au>
Reply-To: sausage@zeta.org.au
To: cube-lovers@ai.mit.edu
Subject: Speed cube times

Hello All,

There doesn't seem to be any records these days kept of people's current
solving times for the cube. Perhaps they should be shared here?

My current time for solving the 3x3x3 Rubik's cube is 47 seconds using
the Petrus method and involved:

Building the 2x2x2,
finishing the bottom and mid layer, 
1 edge alignment, 
1 corner swap, 
2 sunes, 
and 1 clockwise edge rotation.

How are other people's speeds fairing?

Wayne Johnson
http://www.zeta.org.au/~sausage/rubikscube.html

[Moderator's note: Since this may result in a large number of very
 small messages, please send your answers to Wayne, who I hope will
 summarize the results for the list. ]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 22:03:13 1999
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To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Megaminx solving times?
Date: 9 Mar 1999 15:46:55 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <7c3fpf$mte@gap.cco.caltech.edu>
References: <cube-lovers.3.0.32.19990225090214.009654a0@mail.spc.nl>

Christ van Willegen <c.v.willegen@spcgroup.nl> writes:
>I've been practising the Megaminx, and I can now solve it
>without resorting to formulas written down on paper. I can
>do it in about 10 minutes. How does this compare to other
>people's times?

I've never solved mine for speed, because I'm afraid of more
stickers falling off (already 3 are missing and I have to "deduce"
what they are).  10 minutes sounds reasonable -- I'm not sure I've
ever resorted to formulas written on paper.   (For one thing, I'm
not sure I know of any notation!)

>And, what method do you use? The method I developed relies
>heavily upon the standard cube moves, and I solve the Mega-
>minx going down from one flat top in rings. I needed to
>adapt 1 (one!) standard cube formula to get it to work on
>the Megaminx.

This brings up, actually, a rather embarrasing point for me as a puzzle
solver.  At first I had no idea how to generalize the standard cube moves
I used to the Megaminx.  So, eventually I figured out a new method,
which was:
  1.  Solve a large chunk of it by normal moves, perhaps leaving
    only three faces unsolved;
  2.  Solve the edges of the remaining faces (if you can solve an
    Alexander's Star, you can do this);
  3.  Solve the corners.

I found this quite effective.

About a year later, when my Megaminx was in storage and I was
playing with the Cube, I suddenly realized that my method for the
Megaminx would work perfectly well for the Cube!  (O, for that
matter, anything with a similar structure of "corners" with three
faces and "edges" with two faces.) I chastized myself
heavily for not realizing this "obvious generalization", and
with it was able to work out more moves for the Cube.

--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"Pop", "Soda", or "Coke"?  http://www.ugcs.caltech.edu/~almccon/pop_soda/

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 22:50:04 1999
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Date: Tue, 9 Mar 1999 19:08:55 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: weidhaas1@llnl.gov
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D4DE8.BDBAD646.12@ice.sbu.ac.uk>
Subject: Re: Oddzon version of the cube

	Re: coloured cubies.
	I only ever saw one version of the cube where the black plastic ccubies
had painted faces.  The colours were rather paler than on the ordinary cubes
and one colour was violet.  I can't find it immediately, but I should find it
if I stop an look.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 10 23:23:35 1999
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Date: Tue, 9 Mar 1999 19:24:29 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: sausage@zeta.org.au
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D4DEA.EA4E5AAE.11@ice.sbu.ac.uk>
Subject: Re: Oddzon version of the cube

	It's depressing that manufacturers can't provide a decent cube.  When
the C4 cube was introduced about 10(?) years, I found the mechanism very poor
and poeple told me that their examples broke within an hour of buying it.  As
you say, the Asian pirates had become very good in 1982 or so and I believe
Ideal was actually buying production from some of the same companies.
	Regarding cubes with printed colours, I have located mine.  It has
violet replacing orange, but is otherwise the usual colours and arrangement.
The colours are pretty good, but because the plastic surface is not perfectly
smooth, it gives the effect of a matte finish, rather than a glossy finish,
which is why I remember the colours as a bit paler.  My records indicate this
was bought in a regular Rubik Cube packaging, but I don't recall when and I've
never seen the technique used again.  I suppose one has to place squares of
coloured material against the cube and then fuse the colour into the surface of
the plastic and this seems likely to be more expensive than the use of
stickers.  The elimination of orange may be due to the fact that many orange
colours are based on cadmium which is toxic and not permitted on objects for
children.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 18:02:02 1999
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From: Norman Diamond <Norman.Diamond@dec-j.co.jp>
To: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
Date: Wed, 10 Mar 1999 09:36:17 +0900

Patrick Weidhaas [weidhaas1@llnl.gov] wrote:

>As far as I know, nobody has produced a cube (or variation) where the
>plastic "cubies" are colored appropriately without relying on stickers.

I think I have mentioned on this list before that I bought one in India
in 1996.

>Is that process so much more expensive,

I think it is not.  I am not an expert on manufacturing so can't really
say if it's more expensive to make a multicolored plastic piece than
it is to make (or buy) adhesive tapes and punch stickers out of them
for attachment to unicolored pieces.  But I do think, when the version
with multicolored plastic pieces could be retailed for 35 rupees, the
cost of manufacture must be less than 35 rupees, the difference between
this method of manufacture and the more common method must be less than
35 rupees, and I'd lay odds on a distributor not even noticing the
difference in costs if they went that way.

(35 rupees was about 120 yen then.  35 rupees would be about 100
yen now, though the product's price in rupees might have risen.)

>or do the toy-makers want to give their customers a chance to cheat
>by switching the stickers in case they can't get the puzzle solved?

Interesting.
Is this the reason why the stickers come off by themselves :-)
Maybe they don't know that some people used to disassemble cubes and
rearrange the cubies :-)  Of course early Rubik's Revenges used to
disassemble themselves that way.

-- Norman.Diamond@dec-j.co.jp
[Not speaking for Compaq]

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 18:27:53 1999
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Date: Tue, 9 Mar 1999 22:40:45 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Patrick Weidhaas <weidhaas1@llnl.gov>
Cc: Kevin Young <keyoung3@vt.edu>, RUBIK cube group <cube-lovers@ai.mit.edu>
Subject: Re: Oddzon version of the cube
In-Reply-To: <v04020a07b2fb4082cb5b@[134.9.13.42]>


On Thu, 25 Feb 1999, Patrick Weidhaas wrote:

{snips}

}Kevin,
}
}I do not have an answer for you, but your email made me wonder why stickers
}are being used at all? As far as I know, nobody has produced a cube (or
}variation) where the plastic "cubies" are colored appropriately without
}relying on stickers.

Ideal once made a deluxe Cube that had individual colored plastic tiles
attached to the cubies. It also had a mechanism (same general principle,
just different details) that would self-align as you began a maneuver
with a slight misalignment, instead of jamming. In other words, it was a
good bit more tolerant of misalignment. It would be delightful if they'd
reissue it!

}Is that process so much more expensive, or do the
}toy-makers want to give their customers a chance to cheat by switching the
}stickers in case they can't get the puzzle solved?

It was maybe almost twice the price of the standard Cube, iirc.
Swapping stickers is silly! Learn how to disassemble (but reassemble as
a solved Cube; iirc, there are 11 wrong ways to do it, essentially).

Most of the movable-piece puzzles can be disassembled, but not all.

}Patrick


|*  Nicholas Bodley   *|*  Autodidact & Polymath * Electronic Tech. (ret.)
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  Frequent crashes are unacceptable in a mature
|*  Amateur musician  *|*  computer industry.
--------------------------------------------------------------------------

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Message-Id: <3.0.32.19990310090119.00960d80@mail.spc.nl>
Date: Wed, 10 Mar 1999 09:01:20 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: Stickers Re: Oddzon version of the cube

>I do not have an answer for you, but your email made me wonder why stickers
>are being used at all? As far as I know, nobody has produced a cube (or
>variation) where the plastic "cubies" are colored appropriately without
>relying on stickers. Is that process so much more expensive, or do the
>toy-makers want to give their customers a chance to cheat by switching the
>stickers in case they can't get the puzzle solved?

Switching stickers is a no-no. You can so this a couple of times, after
that they'll just fall off...

I've seen cubes that have real plastic colors! They are 3by's in black,
with colored square bricks on the faces that form the colors. They
are about $5 (I think). I haven't bought one to check quality (yet?).
Perhaps if the mechanism is alright, these might be better suited for
the cube-addict. But I'm afraid that the mechanism won't be able to stand
lots of use.

Perhaps I'll just go ahead and try the experiment. After all, it's only
$5...

Christ van Willegen

[ Moderator's note: I take it you mean they are available in the
  Netherlands?  Anywhere else? ]

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 20:18:19 1999
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From: "Philip Knudsen" <philipknudsen@hotmail.com>
To: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
Date: Wed, 10 Mar 1999 00:51:10 PST

Patrick Weidhaas wrote:

>As far as I know, nobody has produced a cube
>(or variation)where the plastic "cubies" are
>colored appropriately without relying on stickers....

Cubes like that are probably more expensive to manufacture. However, a
"Deluxe" version, using plastic tiles, was produced by Ideal in the
early 80's. Pretty hard to get now, i'm afraid.
I also have one that seems newer, this also has plastic tiles. It came
on a card, that had the name "Old Brand Magic Cube" on it. The back of
the card has a (poor) solution printed, and there is also a picture of
an Octagon, supposedly by same manufacturer. Apart from the plastic
tiles, there is nothing deluxe about this last one - the turning is o.k.
but not VERY smooth.

____________________________________
Philip K
Vendersgade 15, 3th
DK - 1363 Copenhagen K
Denmark
Phone:  +45 33932787
Mobile: +45 21706731
E-mail: philipk@bassandtrouble.com
E-mail: philipknudsen@hotmail.com

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 21:43:05 1999
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From: "Philip Knudsen" <philipknudsen@hotmail.com>
To: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
Date: Wed, 10 Mar 1999 01:02:17 PST

Paul Hart wrote:

>After my disappointing results with my Oddzon cube,
>I pledged to never again buy one of their products
>until they change or improve their sticker design.

You're right about their Cube, but the other Rubik products by OddzOn
are fine, or at least acceptable. For instance, I think the Eclipse is
MUCH better in design than the Magic Strategy Game by Matchbox, 10 years
earlier. Same game, but much more attractive and somehow also better
gameplay.
Another example is Rubik's Bricks, which is Rubik's version of the Soma
Cube. There is a short mention of it in "Cubic Compendium", but OddzOn
were the first to market it - and their version is excellent!

BTW has anyone seen/tried any of the other new stuff by OddzOn, like
"Rubik's Infinity" or Rubik's Double Tangram" ???

__________________________________
Philip K
Vendersgade 15, 3th
DK - 1363 Copenhagen K
Denmark
Phone:  +45 33932787
Mobile: +45 21706731
E-mail: philipk@bassandtrouble.com
E-mail: philipknudsen@hotmail.com

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 22:12:57 1999
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Date: Wed, 10 Mar 1999 10:43:27 -0500 (EST)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199903101543.KAA28632@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube

> [...] made me wonder why stickers are being used at all?  As far as I
> know, nobody has produced a cube (or variation) where the plastic
> "cubies" are colored appropriately without relying on stickers.

A while ago, I took the smoothest-acting Cube I have, peeled off all
the stickers, took the thing apart, and painted all the facicles.
Et voila! no more sticker problems!

Now, I just need to do that with one of the 5-Cubes I have, the one
that's suffering from the Dread Orange Sticker Disease; it's already
lost one orange sticker completely, and about four more are so loose
that only a piece of masking tape is keeping them with the Cube.
(Assuming I can figure out how to get it apart non-destructively.)

I agree, it would be much more pleasant if the plastic itself were
coloured.  But that would require at least six different plastics,
instead of one, which is probably why it's not done commercially.
Low volume already makes the things expensive....

On the other hand, I wonder how much more it really would cost to do
coloured plastics.  Anyone with enough experience in the industry to
say?

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 11 23:48:09 1999
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Date: Wed, 10 Mar 1999 22:04:21 +0000
From: David Singmaster Computing <david.singmaster@sbu.ac.uk>
To: Norman.Diamond@dec-j.co.jp
Cc: CUBE-LOVERS@ai.mit.edu
Message-Id: <009D4ECA.6A22DEF3.10@ice.sbu.ac.uk>
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone  help ?

	Yes there were ordinary cubes made with markings fo the blind.  Rainier
Seitz, product manager fro Arxon, the German distributor, made the first
examples by heating a needle and making dice-like marking of one to six spots.
Several versions were made commercially or by specialist firms.  I have
examples with zero to five spots in this style, alos with brass studs of five
sizes and then Ideal (perhaps only Arxon) produced a version with moulded
plastic facelets having symbols on them: +, -, hollow circle, square, triangle
and solid circle.  Meffert made pyramids with four different textures of
surface  I asked a blind friend if they were easyily distinguished and he said
yes.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 00:15:29 1999
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Date: Wed, 10 Mar 1999 22:19:21 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re : Re: Edges only, Ignoring Flips, Face Turn Metric
In-Reply-To: <199903050409.XAA24597@cauchy.math.brown.edu>
To: michael reid <reid@math.brown.edu>
Cc: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9903102221.A@GN209A.PSTCC.CC.TN.US>

On Thu, 04 Mar 1999 23:09:00 -0500 michael reid
<reid@math.brown.edu> wrote:

> jerry writes
>
> > I have completed a God's Algorithm run in the face turn metric for the
> > group consisting of edges only ignoring flips.  The size of the group is
> > therefore 12!  The results are as follows:
> [ ... ]
>
> very interesting.  i hope that you'll also do the quarter turn metric.
>

I have completed a run out to 11q (took a long time).
Regrettably, the diameter proved to be greater than 11q.  I
now have a run in progress out to 12q.  It's going *very*
slowly.  The problem I described where my method is
inefficient calculating the tail of the distribution is
even worse for the quarter turn metric than for the face
turn metric for this particular problem.  Also, to
calculate to 11q or 12q I have to store all the positions
out to 6q, which I can do.  I don't think I can store out
to 7q on the computer I have.  Even if I could, a run to
13q or 14q would be too slow, I think.

I know from parity considerations that the diameter is
greater than 12q, so in some ways my run to 12q is a fool's
errand.  That is, there are less than 12!/2 odd positions
through 11q, so there must be at least a few at 13q.  My
only hope is that all the even positions will show up by
the time I get to 12q.  If so, I would know that the rest
of the odd ones must be at 13q.  Otherwise, I am doomed for
now.

I have an idea of how to approach the inefficiency in the
tail.  Since I am calculating ends-with for each position I
calculate, I know for sure for each position I calculate
which quarter-turns go further from Start and which go
closer.  The idea is that once I get to the tail of the
distribution, I once again begin storing calculated
positions in memory (those which are at the maximal
distance which I am able to calculate).  From there, I
continue further from Start in a more traditional fashion,
leaping one level at a time rather than many levels at a
time.  This works because I have knowledge of which quarter
turns go closer to Start, and hence I don't have to worry
about comparing against those positions closer to
Start which I am not able to store.  If I had time to put
this plan into action, the run time for the tail of the
distribution should be only a few minutes or a few seconds.

----------------------------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 14:16:07 1999
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Date: Wed, 10 Mar 1999 23:47:56 -0500
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From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject: Re: Request for spectacular cube-solving
To: cube-lovers@ai.mit.edu

Wei-Hwa Huang suggested:
>How about team solving?  n people, n cubes, everyone makes one move
>and passes the cube to the left.  Repeat.  :-)

Well, I'm left handed and Kristin Looney is right handed, and her solution 
is the one I use as well.  So we have solved it together, each contributing 
one hand.  Of course it helps a little that both of us can solve it one 
handed, but hey....

-Jacob Davenport

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 14:57:16 1999
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Message-Id: <36E7D039.265049BA@marlboro.edu>
Date: Thu, 11 Mar 1999 09:16:25 -0500
From: Jim Mahoney <mahoney@marlboro.edu>
Organization: Marlboro College
To: Martin Moller Pedersen <tusk@daimi.au.dk>
Cc: cube-lovers@ai.mit.edu
Subject: Re: help on 5x5x5 wings
References: <199902270400.FAA482087@bonestell.daimi.au.dk>

Martin Moller Pedersen wrote:
>
> I am trying to solve my new cube the 5x5x5 cube.
> I have managed to solve all of it except the wings.
> The wings are the y's in the following diagram:
>
> ZyZyZ
> yZZZy
> ZZZZZ
> yZZZy
> ZyZyZ
>
> I have many moves for the 3x3x3 but I can't figure out how to apply these moves
> to the wings.

Hi Martin.

Here's an excerpt from a longer disucussion of the NxNxN cube
which I posted to cube-lovers some time ago.  Other folks have
done similar work, and published consistent results.

In what follows a "cubie" is one of the small, colored cubes
that make up the NxNxN, a "slice" is an NxN plane of the cube
(even if the inside cubies don't exist), and an "orbit" is
a set of cubies which can be moved into each other's places,
like the corners or edges.  The method below can be made
to work for any kind of orbit, including the "wings" you ask about.

Good luck,

 Jim Mahoney

-- excerpt from http://www.marlboro.edu/~mahoney/cube/NxN.txt --

=====================================================================
(VI) How to Cycle Three Cubies  =====================================
=====================================================================

The basic idea is to find a move sequence that will (1) take a chosen
cubie off from its "hot seat" on a chosen slice *without* (here's
the trick) disturbing any other cubie on that slice.  The rest
of the cube can be completely scrambled by this operation.  Then (2)
rotate the chosen slice, (3) undo step (1), putting the original
cubie back into its original slice and undo whatever changes were
made to the other cubies, and (4) undo step 2.

The sequence always of the form
   A R A' R'
where "A" is step 1, "R" is a rotation of a single slice, and
the ' mark means, as usual, the inverse operation.

Here's a detailed example, using the Corner orbit of a 3x3x3 cube,
with the top layer as the "chosen slice" and the cubie marked "1" in
the unfolded sketch of a cube below as the focus of attention. In
eight moves the cubies in locations 1, 2, and 3 will trade places.

The starting position:

             U
    a  -  1  - 2 - d -        (a,1,2,d,e,3,g,h) are a Corner orbit.
    |  L  |  F | R | B
    e  -  3  - g - h -        (U, D, L, R, F, B) are the possible
             D                clockwise rotations.


 (1) Get "b" off the chosen slice, without disturbing any other
     cubie on that slice.  Replace it with the cubie that you
     want to put in its place.

                 e  -  a  - 2 - d -
 ->   L  ->      |     |    |   |
                 3  -  1  - g - h -


                 e  -  a  - 2 - d -
 ->   D  ->      |     |    |   |
                 h  -  3  - 1 - g -


                 a  -  3  - 2 - d -
 ->   L'  ->     |     |    |   |           After L D L'
                 e  -  h  - 1 - g -

      The top layer was (a,b,c,d); now it is (a,f,c,d).
      "b" has been taken off the top slice, and "f" is in its place.

 (2) Rotate the chosen slice to place a new cubie in the hot seat.

                 3  -  2  - d - a -
 ->   U  ->      |     |    |   |           After (L D L') U
                 e  -  h  - 1 - g -


 (3) Undo step 1, which pops the chosen cubie "b" back to its
      original slice, *and* (here's the key part), restore (nearly) all
      other cubies to their original locations, since none of the
      disturbed ones were on the slice that rotated in step (2).

                    3  -  1  - d - a -
 -> L D' L' ->      |     |    |   |        After (L D L') U (L D' L')
                    e  -  2  - g - h -


 (4) Undo step 2, restoring the chosen slice back to its original
position.

                    a  -  3  - 1 - d -
   ->  U'  ->       |     |    |   |        After (L D L') U (L D' L') U'
                    e  -  2  - g - h -

So the move sequence to cycle corners (1,2,3) is simply
(L D L') U (L D' L') U'   (reading left to right).

With a few extra moves before this sequence (which should be undone
afterwards) to arrange the cubies which should be moved into the
places which are actually modified by this operation (or a similar
one), this trick and its variations can be used to put back all 8
corners into their proper places.

And with a bit of exploration, this same idea can be used to cycle
three cubies of any type, in any orbit, on any layer, without
disturbing anything else.  For the Edge-Singles on the 3x3x3, for
example, to bring an edge off the top slice without disturbing
anything else on top, step (1) can be S D S', where "S" vertical is a
rotation of a center slice.

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 15:58:21 1999
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Message-Id: <36E7D3A7.1796@ameritech.net>
Date: Thu, 11 Mar 1999 08:31:03 -0600
From: Hana Bizek <hbizek@ameritech.net>
Reply-To: hbizek@ameritech.net
To: cube-lovers@ai.mit.edu
Subject: color

Hello, cube-lovers,

May I address an issue of cube colors, brought here by Dr. Singmaster?

Color problem is crucial to those of us who engage in creating
multi-cube designs, particularly if those designs are 3-dimensional. In
such designs it is possible to suppress colors and create designs of,
e.g., three colors only on its faces. No, I have not gone insane, I know
what I am talking about.

It is most unpleasant for a designer to have stickers falling off his or
her cubes! The cubes should be well-made and their colors distinct. I
find the orange and red colors to be nearly identical in hue. The red is
light and the orange is dark, which is bad. I try to solve the problemn
by suppressing orange in my 3-, 4- and 5-color designs, but it is
irritating. Why  can't the cube makers replace red-orange colors by
pink-dark red for better contrast?

Another issue is parity pairs. In solution algoirithms you  don't kave
to know them, but they are crucial in 3-dimensional-design algorithms.
They make the above-mentioned color suppression possible. Now why can't
the manufacturers sell such pairs? If they don't know what parity pairs
are, I will tell them.

Hana Bizek
physicist, and
3-d Rubik's cube designer

[Moderator's note: By parity pairs, I rather suspect he means mirror-image
pairs.]

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 16:53:45 1999
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From: Jacob_Davenport@scudder.com (Jacob Davenport)
Subject: Re: help on 5x5x5 wings
To: cube-lovers@ai.mit.edu, Martin Moller Pedersen <tusk@daimi.au.dk>

I have a decent solution to the 5x5x5 on my web page at
www.wunderland.com/WTS/Jake.  I call those cubies "wings"
also, and I foolishly assumed that I invented the term.
Either that, or you have looked at my solution and are
still having trouble.  Please let me know if this page
is helpful or not.  If not, I would be happy to explain
further how I solve the wings.

Indeed, 3x3x3 moves will not help you with the wings.

-Jacob

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 17:23:13 1999
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Message-Id: <3.0.32.19990311084938.00953240@mail.spc.nl>
Date: Thu, 11 Mar 1999 08:49:39 +0100
To: DNewfield@cs.virginia.edu, cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone help ?

At 09:06 26-2-1999 -0500, Dale Newfield wrote:
>On Thu, 25 Feb 1999, Christ van Willegen wrote:
>> Hey! _I'm_ already teaching my blind friend how to solve a cube!
>> We marked the colors of the cube with braille letters spelling
>> 1 - 6 dots. We're having a terribly hard time to teach him to solve
>> it. It's fun, though...
>
>How difficult is it to read braille when the characters are in arbitrary
>orientations?  Have you thought of using some other coding technique that
>might prove more easy to distinguish in any orientation?  (Any idea what
>that might be?)

It's not the characters that count, just the numbers of dots. And it
seems to be reasonably easy to read the number of dots in any
position.

Besides, it was the easiest and quickest thing we could think of.

The problem with (most) blind people is lack of 3D concepts (I'd
almost put a Duh! here). I know I solve the first layer of the
cube with insight in the movement of cubelets. I've had lots of
trouble describing what I do to my friend! Diagrams are impossible
to draw, so you'd have to describe in words _exactly_ where cubelets
are to be placed w.r.t. the rest of the cube before a formula can
be applied.

Too bad we didn't have any time to practise, yet. But we will!

>
>-Dale
>
>[Moderator's note: I'm inordinately proud of my own invention, which I
>thought I mentioned years ago but can't find in the archives: Wire
>symbols glued to a destickered cube, polished to a high gloss.  The
>symbols are blank opposite dot, square opposite circle, and plus
>opposite X.  The supergroup is marked by a cutout at a corner of each
>face center and the adjacent cubies.  I can solve it behind my back,
>but when I lent it to a blind computer scientist, he gave up. --Dan]

That's also a nice idea. I might try that if the braille dots thing
doesn't work. Do you have pictures of this thing on-line?

[Moderator's reply: No pics, but the above description should get you
 pretty close.  The only symbols you might want help with are
    Dot: Made with a short tight spiral of wire,
    Plus and X: Double-outlined so that they can be made with a
         continuous strand of wire, end to end.
 After gluing them down, I put an extra cover of cement to protect the
 wire and blunt the sharp ends.  Silicone protectant ("Armor All" TM in
 the US) gives it a good feel.  I used steel wire about .2mm thick,
 like a paperclip, but perhaps I should stay vague so you will invent
 your own variation. ]

Dr. Bandelow! Please adjust your machines to make cubes like this :-)

[But building it is half the fun! --Dan]

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 18:03:02 1999
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In-Reply-To: <009D4DE8.BDBAD646.12@ice.sbu.ac.uk>
Date: Thu, 11 Mar 1999 10:14:14 -0800
To: David Singmaster <david.singmaster@sbu.ac.uk>
From: Patrick Weidhaas <weidhaas1@llnl.gov>
Subject: Re: Oddzon version of the cube
Cc: cube-lovers@ai.mit.edu

Thanks for your info re coloured cubies. I also got some info from Kevin
Young, see below.
Patrick

Date: Thu, 25 Feb 1999 15:58:16 -0500
To: Patrick Weidhaas <weidhaas1@llnl.gov>
From: Kevin Young <keyoung3@vt.edu>
Subject: Re: Oddzon version of the cube

Patrick,

As a long time cubist, I can assure you that at one time Ideal did make a
cube called "Rubik's Cube Deluxe".  They used tiles instead of stickers.
They were colored appropriately.  In fact, that is still the cube
that I use almost all of the time.  The tiles are colored with blue on
one side opposite of white, red opposite of orange, and yellow opposite of
green.  On the center red tile, was written in Gold Lettering "Rubik's
Cube Deluxe".  This placement of the Rubik's Cube logo differs from the
Ideal sticker version, where that version has the logo on a sticker on top
of the white center cube.  The gold lettering has faded over the past 17
years, but the cube is like new other than that.  If I remember correctly
this was a limited release by Ideal.  If you can find another one out there
that someone is willing to sell to you, I recommend getting it.

Sincerely,

Kevin Young

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 12 18:39:11 1999
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From: WaVeReBeL@webtv.net
Date: Thu, 11 Mar 1999 10:34:35 -0800 (PST)
To: Cube-Lovers@ai.mit.edu
Subject: Re:  Speed cube times
Message-Id: <21178-36E80CBB-632@mailtod-121.bryant.webtv.net>

My records:

LAYER BY LAYER - 33 sec.

I can get it under a minute almost every time.  My average is about 55
sec.

I use a very eclectic approach in order to get as few moves as possible.
I'll start off w/ Jiri's or Lars Petrus' method for the top, depending
on which might get fewer moves.  For the bottom, I use a whole bunch of
different methods that I have learned on the web to finish it w/ as few
series as possible.  Usually it takes 3 series of moves to finish off
the last layer.  Often only 2 series, at the most 4.  I average about 65
turns.

My hands aren't super fast, so I try to make up for it by looking ahead
and limiting the number of moves.  If I take the time to practice the
hand movements and fingering, I think I can increase my time
dramatically.  My goal is to get it down to average under 30 seconds
(yeah right). =)


CORNERS FIRST - 47 sec.

My average is about 1 min.  I basically use Matthew Monroe's method w/ a
few tricks of my own.  I just wish there is more info on this method on
the web.  The fingering is much easier and faster to do than the layer
method, but it takes me longer because I have to use so many more turns.
I'm guessing there is lots of info I don't know about in the books that
have been written on this method, but I don't know how to get my hands
on 'em.  Could somebody point me to a web page or a book store that
could help?  I'm looking for more series of moves on this method.


ONE HANDED! - 1 min 12 sec.

I have been so addicted to cubing recently that I would spend hours at a
time.  So much so that my arms and hands would get tierd, and I would
sometimes get pains in my wrists & forearms.  (But as an addict I still
keep on going)  One day my right hand gave up, so I thought...what the
hell, why not try w/ my left by itself!  At first it took me about 5
min, then I got it down to 3 min.  After a little bit more practice, I
now average about 1 min, 45 sec!  Again, I'm not super fast, I just look
ahead and use as few moves as possible.  On the web I saw that the
record for one hand is 53 seconds.  I'm hoping to beat that someday.
Does anybody else on this list specialize one handed?  Got any good one
handed records?


MY FEET!!! - 7 min 19 sec.

I didn't think I could do it, but after my success w/ one hand, I had to
try it.  I also remember seeing somebody do it on that's incredible.

At first it took me about 20-30 minutes...I didn't bother timing myself,
because I didn't think I could do it.  The hardest part is doing long
sequences of moves in the last layer.  I've messed up a lot at that
point.  Sometime later I tried it again, and did it in 10 min, then
finally my record.  I would practice this a lot more, but my legs get
tierd lifting and manipulating the cube after just a couple of tries.
With some practice, I could probably get it down to 5 minutes.

-----

This brings me to a couple of ideas for a spectacular feat. (No pun
intended)  How about solving two cubes at once...one in eace
hand...solving w/ feet and hands at the same time?

-----

I have to say I'm pretty proud of these accomplishments.  I'm new to
this.  4 months ago I couldn't even solve a cube.  I just barely learned
how in November.  Now I'm addicted.  I wore out the stickers on two
brand new cubes in my first 3 weeks. (A problem which is currently being
addressed on this mailing list)  It has gotten so bad that It's
interfered w/ my school work.  (As I'm writing this, I should be
studying for my Trig exam)

-Alex Montilla-

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 13:18:24 1999
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From: WaVeReBeL@webtv.net
Reply-To: WaVeReBeL@webtv.net
Date: Thu, 11 Mar 1999 13:47:33 -0800 (PST)
To: Cube-Lovers@ai.mit.edu
Subject: Local Cubists
Message-Id: <10765-36E839F5-696@mailtod-122.bryant.webtv.net>

I feel like I was born in the wrong era.  I'm 20 years old and in
college right now.  If I had been born 10-15 years earlier, I might be
training to be in competitions.  I was a little too young when it was in
its prime in the 80's.  I got my hands on a cube this past November &
now am hooked for life.  I went from 30 mins to 3 mins in the 1st month,
& in 2 more months got it down to under a minute.  I bring a cube
everywhere I go.  Tons of people come up to to talk to me about it.  99%
of the time they would say:  "You know what I used to do?" and then
either: 1) peel off the stickers or 2) take it apart.  I used to laugh
along at this amusing anecdote, for I myself admit to both.  But, after
the 100th person...Its just annoying.

There is no one I can share my hobby with.  Of all these people, in the
4 months I've been into this, I have only met one person who actually
knew how, but it was years ago, and has since lost interest.  I got one
of my friends into it, but he isn't at a level where he can compete w/
me yet.  Plus, he's too busy to put some serious time into it.  With the
exception of my one friend, I feel all alone when it comes to cubes.

Right now I am looking for any cube enthusiasts, beginner through
advanced, in my area who want to get together to compete, buy/sell/trade
books & cube related stuff, share secrets/techniques that can only be
taught in person (there are limits as to what text & 2D images on a web
page & in books can convey about manipulating a 3D cube), etc.

All of my knowledge on cubes is from people's web pages, but practically
all of them are made to teach the beginner.  My current goal is a 30
second average.  At a 55 sec average w/ a 33 sec record, I'm not too far
from my goal.  I achieved this using moves I've gotten from the web.  I
would like to talk to people personally, and see some of the books no
longer in print.  I read somewhere that there are over a hundred
written.  Please contact me if you have books I can look at, or know
where I can get a hold of them.

I would like also to join any local clubs if any still exist or, If
there are enough people interested, start a new one.  A lot of people
who have seen me playing with mine in public have asked me if the
Rubik's Cube has "come back" like a new craze.  Although there isn't, I
would sure like to see one.  And how about a new championship?  That
would be cool.  Not a world wide thing, but it wouldn't be hard to set
up a small local one with a small prize like a Megaminx.  Any thoughts
on this?  Has this been attempted lately?

I live in Carson, California.  So, if anyone in the LA/South Bay area is
interested, please contact me.

-Alex Montilla-
waverebel@webtv.net
ipiiika@aol.com

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 14:13:32 1999
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Reply-To: <noel@mud.ca>
From: "Noel Dillabough" <noel@mud.ca>
To: "'Christ van Willegen'" <c.v.willegen@spcgroup.nl>
Cc: <cube-lovers@ai.mit.edu>
Subject: RE: Stickers Re: Oddzon version of the cube
Date: Thu, 11 Mar 1999 20:47:24 -0500
Message-Id: <000801be6c2a$49a18160$030a0a0a@noel>
In-Reply-To: <3.0.32.19990310090119.00960d80@mail.spc.nl>

I too found a "tiled" cube that was about $5 USD with plastic tiles rather
than stickers.  At first this seemed ideal since there can be no wear and
tear on the stickers.  However quite soon afterwards the tiles start to come
off.  Perhaps if they were glued better...

> with colored square bricks on the faces that form the colors. They
> are about $5 (I think). I haven't bought one to check quality (yet?).
> Perhaps if the mechanism is alright, these might be better suited for
> the cube-addict. But I'm afraid that the mechanism won't be
> able to stand
> lots of use.
>

I would be willing to pay more for puzzles of better quality.  On a similar
note, old versions of the Rubik's Revenge puzzle are fetching prices over
$75 USD.  The puzzles are basically unusable, as they are so stiff and
brittle from age that they fall apart.  The best buy is to buy a used
Rubik's Revenge, one that was actually used during the 80s and was worn in.
These puzzles are not so stiff and are actually usable.

Anyway back to my point, I was wondering how many people on this list would
be interested in getting a manufacturer to do a good quality run of 4x4x4
cubes?  A place in the U.S. called "Puzzlets" was supposed to have a sign up
list to create a production run of these cubes, but I have heard that they
are no longer in business.  Perhaps there is enough interest in the cube
lovers' list?

[Moderator's note: Has anyone else found aged, unused 4^3s more brittle than
 the originals? Even the early ones were usually stiff; I needed to take
 them apart and apply wax or other lubricant.  And still they broke much
 too easily, due to the tiny necks on the face centers.  Perhaps the only
 advantage aged, used cubes have is that the stiff ones whose owners didn't
 lubricate them are long broken. --Dan]

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 15:58:49 1999
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Date: Fri, 12 Mar 1999 01:01:52 -0500
From: michael reid <reid@math.brown.edu>
Message-Id: <199903120601.BAA15248@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: Re : Re: Edges only, Ignoring Flips, Face Turn Metric

i guess i'm not sure what you're doing, jerry.  but i don't think
it should be *that* difficult.  the number of configurations is
12! = about 480 million.  if you divide out by symmetry, you get
about 10 million configurations.  this should be small enough to
store in memory and do a complete breadth-first search of the space.

mike

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 16:49:33 1999
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Message-Id: <36E8B44F.3A68D055@erco.com>
Date: Fri, 12 Mar 1999 07:29:35 +0100
From: "michael ehrt" <m.ehrt@erco.com>
Reply-To: m.ehrt@erco.org
To: cube-lovers@ai.mit.edu
Cc: der Mouse <mouse@rodents.montreal.qc.ca>
Subject: Re: Oddzon version of the cube
References: <199903101543.KAA28632@Twig.Rodents.Montreal.QC.CA>

> Now, I just need to do that with one of the 5-Cubes I have, the one
> that's suffering from the Dread Orange Sticker Disease; it's already
> lost one orange sticker completely, and about four more are so loose
> that only a piece of masking tape is keeping them with the Cube.
> (Assuming I can figure out how to get it apart non-destructively.)

That's not a big problem. Underneath each center piece there's a screw
(just like in some (or all?) 3x3s, although we don't need to open it to
take it apart). Just open one of them and the whole thing comes apart.
And if you've never taken it apart, enjoy the beautiful mechanism.
Putting it together again is a pretty nice puzzle itself :-)

Michael

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 17:46:20 1999
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Message-Id: <006c01be6c55$1fdab740$ca685dcb@uwe>
From: "UMroaming" <uwe@ue.net>
To: "Cube-Lovers" <cube-lovers@ai.mit.edu>
Subject: MULTI COLORED PLASTIC SURFACES
Date: Fri, 12 Mar 1999 14:27:24 +0800

der Mouse <mouse@rodents.montreal.qc.ca> wrote:

>> [...] made me wonder why stickers are being used at all?  As far as I
>> know, nobody has produced a cube (or variation) where the plastic
>> "cubies" are colored appropriately without relying on stickers.
>
>A while ago, I took the smoothest-acting Cube I have, peeled off all
>the stickers, took the thing apart, and painted all the facicles.
>Et voila! no more sticker problems!


It is not to difficult to make cubes or other puzzles without stickers and to
spray paint the surfaces, the price is about the same.  I once made a
Pyraminx test run using this method.  As David Singmaster mentioned the
plastic surface under the sticker has some flow lines which are unavoidable
and the sticker serves in part to hide these imperfections.

>>The one which I bought in India did not have spray-painted
>>surfaces, it was made out of multicolored plastic.  So I think that
>>manufacturers can afford it.

The problem with molding the cubes out of colored plastic is the corner
pieces.  as they have to be molded in 3 different colored plastic, such
tooling and molding procedure is extremely expensive and I can not imagine
that India has the technology to produce such an item.  What would make more
sense is to make the stickers out of small clip on plastic tiles such as is
used in my Impossiball this would not increase production cost, but totally
new tooling would need to be made costing around US$50,000.00.  I made a
survey for my Pyraminx many years ago.  Using either colored plastic tiles
or the none slip  fluorescent stickers and the stickers won.

>Now, I just need to do that with one of the 5-Cubes I have, the one
>that's suffering from the Dread Orange Sticker Disease; it's already
>lost one orange sticker completely, and about four more are so loose
>that only a piece of masking tape is keeping them with the Cube.
>(Assuming I can figure out how to get it apart non-destructively.)
>
>I agree, it would be much more pleasant if the plastic itself were
>colored.  But that would require at least six different plastics,
>instead of one, which is probably why it's not done commercially.
>Low volume already makes the things expensive....

No problem to use different colored plastics only the corner pieces pose a
problem.

>On the other hand, I wonder how much more it really would cost to do
>colored plastics.  Anyone with enough experience in the industry to
>say?

As mentioned above there would be no cost increase apart from new tooling
having to be made for the corner pieces and the flow marks on some of the
surfaces will be noticeable.

The best solution would be spray painting, as is being done with my Orbix
and the 4 colored rings on my 3D-Puzzle Balls.

I hope that this clears up your discussion on why Manufacturers use
stickers.


Regards
Uwe

HAPPY PUZZLING
Uwe Meffert
P.O. Box 24455, Aberdeen, Hong Kong.
Tel. 852-2518-3080, Fax. 852-2518-3282
Email:- uwe@ue.net
www.ue.edu www.ue.net www.mefferts-puzzles.com

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 18:31:44 1999
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Message-Id: <3.0.32.19990312090954.009582d0@mail.spc.nl>
Date: Fri, 12 Mar 1999 09:09:55 +0100
To: cube-lovers@ai.mit.edu
From: Christ van Willegen <c.v.willegen@spcgroup.nl>
Subject: Re: Plastic colors Re: Stickers Re: Oddzon version of the cube
Reply-To: Christ van Willegen <c.v.willegen@spcgroup.nl>

At 09:01 10-3-1999 +0100, I wrote:
>I've seen cubes that have real plastic colors! They are 3by's in black,
>with colored square bricks on the faces that form the colors. They
>are about $5 (I think). I haven't bought one to check quality (yet?).
>Perhaps if the mechanism is alright, these might be better suited for
>the cube-addict. But I'm afraid that the mechanism won't be able to stand
>lots of use.
>
>Perhaps I'll just go ahead and try the experiment. After all, it's only
>$5...

I was wrong... They are not $5, but $2.50 :-)  They're using some quite
agressive glue to put the plastic colors on the faces. The color sceme
is weird (White <-> Red, Green <-> Yellow, Blue <-> Orange, White, Green
and Yellow are clockwise). When I bought it, two cubelets were glued
together! Also, there are traces of the glue everywhere.

The machanism is not too well, but it doesn't fall apart easily. The
spring loaded mechanism is quite strong! When I took it apart, the springs
pulled the centre pieces together quite a bit.

I think that after some time, it may turn quite well. It will never be
a professional cube, however...

On a side note, my gf claims she saw these cubes at Intertoys or Toys-R-Us
for about $1.50...

>
>[ Moderator's note: I take it you mean they are available in the
>  Netherlands?  Anywhere else? ]

As far as I could tell, these are manufactured somewhere in Europe. I could
buy lots and send them to someone willing to re-ship them locally (as in:
In the States). I know the moderator usually doesn't approve of scemes like
this, but if people _really_ want one, it's the only way... Shipping is
_way_ over $1.50 apiece.

If people really want one, let me know, and also if you'd be willing to
re-ship from the USA (to someone else there).

Christ van Willegen

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 19:01:32 1999
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Date: Fri, 12 Mar 1999 11:10:18 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: whuang@ugcs.caltech.edu
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D5001.60154AD6.25@ice.sbu.ac.uk>
Subject: Re: Fwd: Request for spectacular cube-solving - Can anyone help ?

	Very hard to have a person do just one move and pass it on.  Perhaps
allow five seconds?
	A bit of spectacular solving would be to have someone make five or six
moves and let the solver work out how to unscramble it in the same number of
moves.  Kate Fried in Budapest could do four moves, perhaps five, regularly.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 15 19:37:47 1999
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Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 12 07:39:01 1999
Date: Fri, 12 Mar 1999 12:34:43 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: c.v.willegen@spcgroup.nl
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D500D.2B3620CA.27@ice.sbu.ac.uk>
Subject: RE: Stickers Re: Oddzon version of the cube

	Cubes with tiles instead of stickers came out very early on.  I recall
one called Gyro Cube, which I think was Korean, about 1981?.  Then Ideal took
on the idea as the Deluxe Cube.  I've got an example made in China (real PRC,
not Taiwan, and for the Chinese market).  Recently, I've bought an example at a
newsagent's in London for somewhere in the #1 to #2 range and it worked
moderately well (the previous two which I bought recently were virtually
immovable!).  A large version of this (about 90mm or 3 1/2 in on an edge) was
on sale from a street vendor in December, but I don't know if he has any left
and I haven't seen it elsewhere.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

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From: WaVeReBeL@webtv.net
Reply-To: WaVeReBeL@webtv.net
Date: Fri, 12 Mar 1999 10:31:40 -0800 (PST)
To: Cube-Lovers@ai.mit.edu
Subject: Re: Oddzon version of the cube
Message-Id: <1095-36E95D8C-2128@mailtod-121.bryant.webtv.net>
In-Reply-To: "Philip Knudsen" <philipknudsen@hotmail.com>'s message of Wed, 10 Mar 1999 00:51:10 PST

I too have seen many of  these cubes w/ plastic tiles.  I have many
different brands including the "Old Brand Magic Cube".  I find them at
swap meets.  Most sell for $2, but I recently found a vendor who sells
'em for ONLY $1!  I immediately bought 5 on the spot.  The next time I
go down there, I think I'll buy out the shop's stock of cubes.  Either
that, or ask how to get them myself.

The quality is pretty bad.  The turning is pretty sticky, but w/ some
lube and a little wear and tear, it's alright.  Plus, they break apart
pretty easily.  When they're really worn out, cubies start popping out
all the time.  If you use WD-40, it'll eat away at the plastic resulting
in really smooth turning.  It'll be great for about 3-4 weeks of daily
cubing, but the WD-40 will take its toll, and the cube will start
falling apart.  But for a few bucks every 3-4 weeks is worth it to me.

The good thing about these is that you can make your own "deluxe" cube.
First, with a little bit of effort, and a good razor blade, you can pry
off the plastic tiles.  Next, strip off all the stickers (If they
haven't already fallen off by themselves) and wipe off the sticky residue
on your smoothest, sturdiest cube.  Then, super glue the tiles in any
color arrangement you want!  (the tiles come in the standard colors)
The tiles may have a slightly rough surface after you pry them off, so
you might have to strip off any scarring/hardened glue w/ a razor to
make sure it's nice and flat when glued down.

This process isn't easy.  It took me hours.  But in the end, it's all
worth it.  I've only done one so far, but it works so well, I don't need
to do another one yet.  I've been cubing daily, hours at a time for
about two months with the same cube.

-Alex Montilla-

P.S.
I live in Carson, CA.  E-mail me if you want to know where I get em.

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 13:34:31 1999
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Date: Fri, 12 Mar 1999 15:17:53 -0400
To: cube-lovers@ai.mit.edu
From: Kristin Looney <kristin@wunderland.com>
Subject: Re: Request for spectacular cube-solving
Cc: alison@wunderland.com

Jake wrote:

>Well, I'm left handed and Kristin Looney is right handed, and her solution
>is the one I use as well.  So we have solved it together, each contributing
>one hand.  Of course it helps a little that both of us can solve it one
>handed, but hey....

wow!  that explains it!   I had forgotten that you are left handed!
I started teaching Alison to solve the cube last night, and we did
a few trials at the solving-it-together-with-on-hand-each thing,
and had a really hard time at it.

-Kristin Looney
kristin@wunderland.com
http://wunderland.com/Home/Rubik.html

To all the fishies in the deap blue sea, Joy.

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 14:07:10 1999
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Message-Id: <00e001be6cdf$a0cabaa0$75c4b0c2@home>
From: roger.broadie@iclweb.com (Roger Broadie)
To: <cube-lovers@ai.mit.edu>
Cc: "Martin Moller Pedersen" <tusk@daimi.au.dk>
Subject: Re: help on 5x5x5 wings
Date: Fri, 12 Mar 1999 23:25:23 -0000

Martin Moller Pedersen <tusk@daimi.au.dk> wrote

>I am trying to solve my new cube the 5x5x5 cube.

>I have managed to solve all of it except the wings.

>The wings are the y's in the following diagram:


>ZyZyZ
>yZZZy
>ZZZZZ
>yZZZy
>ZyZyZ

Here's a set of explicit processes:  a lower-case letter means a turn
of the layer containing the wing piece next to the outer layer denoted
by the corresponding capital letter, and in the same sense.  The
brackets show the movement of the pieces in the upper layer.

l F' L F l' F' L' F          (Bl, lF, Lf)

F2 r2 D R2 D' r2 D R2 D' F2  (Br, Fl, Rf)

r' U b U' F2 U b' U' F2 r    (Bl, lF, Fr)

R2 U2 l D' l' U2 l D l' R2   (Br, Fl, bR)

b L2 D l D' L2 D l' D' b'    (Lb, Fl, fL)

l2 U2 r' l U2 l' U2 l U2 r l U2 r' U2 l U2 r l2 U2  (Fl, rF)

The final sequence swaps a pair of pieces in the front face.  There's
been a lot of discussion of this move in Cube-lovers over the years.
The process I've quoted changes pieces in the central nine on the back
face, but nothing will show if all the cube except the top layer is
solved.  The other processes change no other pieces.

Roger Broadie

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 14:43:02 1999
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Message-Id: <36EAE6E9.6B1F0608@okanagan.net>
Date: Sat, 13 Mar 1999 14:30:04 -0800
From: Karen Loewen <loewens@okanagan.net>
Reply-To: Karen Loewen <loewens@okanagan.net>
To: cube-lovers@ai.mit.edu
Subject: Speed Cubing

I have I question for any one willing to answer. I was wondering if the
people who can get the rubik's cube under 45 seconds did you actually
figure it all out by yourself. Or did you find out how through books,
email, websites etc? My best time is 90 seconds but I can't seem to beat
it the way I do it. I don't want any one telling me ways to do it faster
because I want to find out for myself. But I am wondering are there
certain ways to achieve faster times. Please just answer yes or no. Also
I have just ordered the 5x5x5 and I was wondering how much harder is it
than the 4x4x4. Thanks.

[Moderator's note: Send responses to Karen; I hope she will send
 cube-lovers a summary of the results of her survey. ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 15:39:51 1999
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Message-Id: <36EB0B28.1169@ameritech.net>
Date: Sat, 13 Mar 1999 19:04:40 -0600
From: Hana Bizek <hbizek@ameritech.net>
Reply-To: hbizek@ameritech.net
To: cube-lovers@ai.mit.edu
Subject: parity pairs
References: <36E7D3A7.1796@ameritech.net>

> [Moderator's note: By parity pairs, I rather suspect he means mirror-image
> pairs.]

Let me tell you what I mean by parity pairs, why very few have probably
heard about this concept and why they are crucial in 3-dimensional (3-d)
cube art.

Suppose one has two cubes of identical color scheme such that the color
on both cubes' up, down, front and back faces are exactly the same. If
the color of the left face of one cube is identical to the color of the
right face of the other cube, such a pair of cubes is said to form a
parity pair. The color scheme is still identical, but the ORIENTATION of
the faces is reversed for one of the members of the pair. One cannot
obtain parity pairs by conventional cube manipulations, but must obtain
them either from the manufacturer, or switch the faces themselves
manually. I would prefer to buy such pairs from the toymaker, for it
pains me to tamper illegally with  those stickers. I have devised a
simple algorithm to do it as painlessly as possible, but it still is a
pain. But will a manufacturer sell me parity pairs?

The reason so few people know about parity pairs is that such pairs are
moot in solution algorithms. You do not need to concern yourself at all
with parity pairs, you just have one cube and painlessly solve it. Ditto
for 2-dimensional (2-d) designs (unless you treat them as lxmx1)
designs. However, they are essential in 3-d cube art. They are
responsible for reflection-equivalent designs, designs of fewer than six
colors and ultimately fractal design prototypes. They also determine
special symmetries in a 3-d design. They are the cornerstone of 3-d
design theory. Without their presence all of the 3-d designs I have
constructed would not be possible.

Why all this  self-serving fuss about parity pairs and 3-d designs? The
point is this: given four parity pairs, one can construct a 2x2x2 larger
clean design, that has three colors only on its six faces. The internal
faces that touch are colored the same. Those colors are hidden inside
the design or suppressed. Such an array of cubes, when used as corners,
produce, e.g., reflection-equivalence in a design. Go to your cube
collection, extract four parity pairs and see for yourselves.

So I think you got the idea, Mr. Moderator. Just one slight correction;
I am a "she," not a "he." You will find this almost incredible, but
women too, love the cube.

Hana Bizek
(female) physicist and
3-d Rubik's cube designer

[Moderator's note: On the contrary, there are several women on
 cube-lovers, and Dame Kathleen Ollerenshaw is well-known as one of
 the earliest writers about Rubik's cube and one of the first victims
 of Cubist's Thumb.  I just didn't know that "Hana" was a woman's
 name, and I had forgotten that this information was presumed by a
 mention of you in the archives.  I apologize for the oversight.

 As for nomenclature, the reason no one knows about "parity pairs" is
 that the term is ambiguous--"parity" could refer to representatives
 of any even division of a set into two parts.  If you wish to enable
 people to know what you mean without going through your somewhat
 confusing description, then you should use the term "enantiomorphic
 pairs", "chiral pairs", or "mirror-image pairs".  I believe these are
 the standard terms used by chemists, physicists, and everyone else,
 respectively.

 There is an interesting question, though, which your hobby may give
 you a particular ability to answer.  According to _Rubik's Cubic
 Compendium_, the most common color scheme has red opposite orange,
 blue opposite green, and white opposite yellow.  This permits two
 mirror-image color schemes, distinguished by whether red, white, and
 blue go clockwise or anticlockwise around a corner.  The question is
 whether there is a tendency for one of these schemes to predominate,
 and if so, which and by how much?  For instance, one enantiomorph
 predominates extremely strongly in the manufacture of dice, though I
 don't know why. --Dan ]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 16:11:07 1999
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Message-Id: <19990314063522.18949.rocketmail@send103.yahoomail.com>
Date: Sat, 13 Mar 1999 22:35:22 -0800 (PST)
From: Han Wen <hansker@yahoo.com>
Subject: Temporary Fix for OddzOn Sticker Peel
To: Cube Lovers <Cube-Lovers@ai.mit.edu>

Hi,

For those folks out there, like myself, frustrated with OddzOn sticker
peel problem, there is an effective method to prolonging the life of
these pitiful stickers. I got this technique from one of the posts on
the Rubik's website.

Currently, the plastic laminate that makes up the surface of the
stickers for OddzOn Rubik's Cubes starts to peel at the edges after
only a few weeks of intensive playing.

Well, get yourself some long-lasting acrylic nail polish and paint
over all the stickers. (Make sure you do this when the cube is brand
new.) I know, it's a pain in the ass, but it's worth it. The first
coat lasts for about 2-3 weeks before the edges start peeling again.
What I do then is take a razor blade and cut off the peeled away
sections and then nail polish over the stickers again.  You'll
probably get another 2-3 weeks of intensive play again before the cut
laminate starts peeling again.  However, after I repeated this process
once more (razor blade cut/nail polish over), the stickers have now
lasted over several months without additional peel. Hope this helps...

==
_________________________________________________________
Han Wen
Applied Materials
3050 Bowers Ave, MS 1145
Santa Clara, CA  95054
e-mail: Han_Wen@amat.com / hansker@yahoo.com

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 16:52:10 1999
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Date: Mon, 15 Mar 1999 13:27:05 -0700 (MST)
From: Paul Hart <hart@iserver.com>
To: Noel Dillabough <noel@mud.ca>
Cc: cube-lovers@ai.mit.edu
Subject: RE: Stickers Re: Oddzon version of the cube
In-Reply-To: <000801be6c2a$49a18160$030a0a0a@noel>
Message-Id: <Pine.BSF.3.96.990315131452.7422L-100000@anchovy.orem.iserver.com>

On Thu, 11 Mar 1999, Noel Dillabough wrote:

> On a similar note, old versions of the Rubik's Revenge puzzle are
> fetching prices over $75 USD.  The puzzles are basically unusable, as
> they are so stiff and brittle from age that they fall apart.

Really?  I haven't noticed this in my own collection as far as I can see.

> Anyway back to my point, I was wondering how many people on this list would
> be interested in getting a manufacturer to do a good quality run of 4x4x4
> cubes?

I think this would be an excellent idea.

> A place in the U.S. called "Puzzlets" was supposed to have a sign up
> list to create a production run of these cubes, but I have heard that
> they are no longer in business.

Is Puzzletts out of business?  Their web site is still up at least.  Check
it out at:

    http://www.puzzletts.com/

[Moderator's note: If you get a response from puzzletts, please contact
 cube-lovers-request@ai.mit.edu.  I've had several people say they don't
 answer.]

> Has anyone else found aged, unused 4^3s more brittle than the originals?
> Even the early ones were usually stiff; I needed to take them apart and
> apply wax or other lubricant.  And still they broke much too easily, due
> to the tiny necks on the face centers.

A year or two ago I had the very good fortune of stumbling across a number
of 4x4x4 cubes that were brand new in their unopened original boxes from
1982.  Of the two cubes that I personally opened and used extensively, I
did not notice any unusual stiffness.  The 4x4x4 does suffer from the
known weak neck in the center pieces that is prone to snapping, but aside
from that the cubes were in excellent condition and have held up very
well.  I'm not sure if lubricating the cube will remedy the tendency for
the neck to break, but perhaps it would since it seems to happen when the
cube jams slightly.  The first of the two cubes to suffer a broken center
piece became my spare parts cube that I use to keep the other in good
running condition.

Paul Hart

--
Paul Robert Hart        ><8>  ><8>  ><8>        Verio Web Hosting, Inc.
hart@iserver.com        ><8>  ><8>  ><8>        http://www.iserver.com/

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 17:24:14 1999
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Message-Id: <01BE6F31.56407CA0.Jean.LEBLANC@wanadoo.fr>
From: Jean Leblanc <Jean.LEBLANC@wanadoo.fr>
To: "Cube-Lovers (Adresse de messagerie)" <cube-lovers@ai.mit.edu>
Subject: Re:speed cube times
Date: Mon, 15 Mar 1999 22:14:43 +0100

Bonjour les fous du cube

Honestly, I think I am an unrecognized champion. I CAN solve the cube 
within 5 minutes or more, especially when I throw it through the window 
into the garden (my dog loves cubes, too).
If somebody can do worse, please tell me !
I didn't make up a method to solve 3*3*3, nor 4*4*4.
My cube (my only 3*3*3) is a poor clone of the 80's ; it creaks and get 
jammed but it still works !
I'm a poor lonesome cubist...
My wife and my children are not interested in cubes; shall I sacrifice my 
family to a plastic coloured God ?
"Il faut savoir raison garder !"
After all, I should be very interested in a new fabrication of 4*4*4, 
because mine is broken.

Jean Leblanc
Muret
 France.

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 17:55:07 1999
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Message-Id: <714F77ADF9C1D111B8B60000F863155102DD6DAD@tbjexc2.tbj.dec.com>
From: Norman Diamond <Norman.Diamond@dec-j.co.jp>
To: cube-lovers@ai.mit.edu
Subject: Old 4^3s
Date: Tue, 16 Mar 1999 09:45:38 +0900

Noel Dillabough [noel@mud.ca] didn't write, but his message contained:

>[Moderator's note: Has anyone else found aged, unused 4^3s more brittle than
> the originals? Even the early ones were usually stiff; I needed to take
> them apart and apply wax or other lubricant.  And still they broke much
> too easily, due to the tiny necks on the face centers.  Perhaps the only
> advantage aged, used cubes have is that the stiff ones whose owners didn't
> lubricate them are long broken. --Dan]

Nob Yoshigahara told me that he had designed a correction for the original
design of the 4^3 cubes so that they would not fall apart.

In my experience, early 4^3s easily fell apart, and then when a cubie hit the
floor it easily broke.  In my experience, later 4^3s don't easily fall apart.
I would guess that the manufacturer accepted Nob-sensei's advice.

-- Norman.Diamond@dec-j.co.jp
[Speaking for Norman Diamond not for Compaq]

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 18:24:27 1999
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From: "Jorge E. Jaramillo" <jejarami@usa.net>
To: "cube" <Cube-Lovers@ai.mit.edu>
Subject: Easy to find tile cubes
Date: Mon, 15 Mar 1999 21:38:27 -0500

I live in Colombia South America and here it is very easy to find cubes with
tiles instead of colored stickers. They are also very cheap (less than U$ 2).
They are (I guess) those Asian cubes that are not that durable. They have
this mechanism of one screw under the center tile. If you twist them many
times they unscrew and come apart but since they are so cheap when I break
one I just buy another one

Jorge E. Jaramillo
kingeorge@hotmail.com
Cut the chain and chase the dream
Savatage 1984

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 19:02:58 1999
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Date: Tue, 16 Mar 1999 08:59:41 +0000 (GMT)
From: the terminal sloth <alex@slothdom.demon.co.uk>
To: Cube-Lovers <cube-lovers@ai.mit.edu>
Subject: Re: MULTI COLORED PLASTIC SURFACES
In-Reply-To: <006c01be6c55$1fdab740$ca685dcb@uwe>
Message-Id: <Pine.LNX.4.05.9903160855180.3329-100000@slothdom.demon.co.uk>

On Fri, 12 Mar 1999, Uwe Meffert wrote:
> ...
> It is not to difficult to make cubes or other puzzles without stickers and to
> spray paint the surfaces, the price is about the same.  I once made a
> Pyraminx test run using this method.  As David Singmaster mentioned the
> plastic surface under the sticker has some flow lines which are unavoidable
> and the sticker serves in part to hide these imperfections.

Why can't you take a modelling knife or a file and remove these lines? And
use a bit more paint where necessary (easier with acrylic than with spray
paints).
Obviously this isn't practical for large-scale manufacture.

<snip>
> I hope that this clears up your discussion on why Manufacturers use
> stickers.


Alex
-- 
Alexander Lewis Jones - the terminal sloth
sometimes I sits and thinks, sometimes I just sits

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 19:32:55 1999
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Message-Id: <199903160937.KAA24418@bednorz.get2net.dk>
From: "Klodshans" <klodshans@get2net.dk>
To: cube-lovers@ai.mit.edu
Date: Tue, 16 Mar 1999 10:36:42 +0000
Subject: Re: OddzOn version of the cube
Reply-To: klodshans@get2net.dk

Wayne Johnson <sausage@zeta.org.au> wrote:

> I've got the new magic.
> Works exactly the same except that the colours are pretty ugly.
> Matchbox did a nice one that was rainbow on black.
> The new ones are yellow on red.

I agree. I have all the different Magics and the OddzOn version is
not an improvement. The weird thing is that their website
www.rubiks.com
shows a picture of the Magic thats looks like one of the Matchbox
ones (BTW on the same site I read the other day that OddzOn is
planning to make their own deluxe cube).

Anyway, in the message that Wayne responded to, I was not
talking about Magic, but the "Magic Strategy Game" by Matchbox
which is a completely different thing. As I wrote, this has been re-
launched by OddzOn under the new name "Eclipse", and in this
case the OddzOn guys have actually improved the original design,
in my opinion.


______________________________________
Philip K
E-mail: philipk@bassandtrouble.com
E-mail: klodshans@get2net.dk
web:    http://hjem.get2net.dk/philip-k

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 20:02:41 1999
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Date: Tue, 16 Mar 1999 09:41:52 +0000 (GMT)
From: "M. P. Baker" <mpb2@mcs.le.ac.uk>
Reply-To: "M. P. Baker" <mpb2@mcs.le.ac.uk>
Subject: Re: Stickers vs. tiles
To: cube-lovers@ai.mit.edu

David Singmaster <david.singmaster@sbu.ac.uk> wrote:

> Recently, I've bought an example [of a tiled cube] at a
> newsagent's in London for somewhere in the #1 to #2 range and it worked
> moderately well (the previous two which I bought recently were virtually
> immovable!).  A large version of this (about 90mm or 3 1/2 in on an edge) was
> on sale from a street vendor in December, but I don't know if he has any left
> and I haven't seen it elsewhere.

These cubes appear to be available from street vendors all over the UK. I've
recently bought some in Leicester and Plymouth. They also sell them in a
"gadget" shop at the end of my street, the sort of place that also sells blow-up
aliens etc. The large ones are really nice for displaying patterns on, and came
with a solution entertainingly translated from some Chinese type script :-)
--------------------------------------------
Matthew Baker
Dept. of Mathematics and Computer Science
University of Leicester

mpb2@mcs.le.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Tue Mar 16 20:35:46 1999
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Date: Tue, 16 Mar 1999 05:14:55 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Cube Mailing List <Cube-lovers@ai.mit.edu>
Subject: Taking apart the 5^3
Message-Id: <Pine.SUN.3.96.990316050216.6308M-100000@sunspot.tiac.net>


As nearly as I can remember, you can begin dismantling one of these by
rotating the top slice by maybe 30 degrees or so, then prying upward on
one of the "wing" cubies (between the center and the corner cubies). Use
your thumb, nail side down, and lift.

Experiment with different amounts of rotation; you'll find a position
where the "wing" cubie's "foot" will push aside many others. Once it's
disengaged, life becomes easier.

This method, if you choose the proper cubie to pry, and align it
properly with the "loosest" neighbor below it, is harmless.

Just possible that I'm suggesting the wrong cubie to pry, but iirc, the
center cubies are more directly held than the "wings".

Believe me, the insides of a 5^3 are utterly amazing. The scheme used
for the 3^3 can't hold a 5^3 together unaided; the mechanism is an
extension of that in the 5^3, but has an additional set of retaining
surfaces, generally spherical in their geometry.  The shape of a "foot"
on a corner cubie is something to behold; it could be a bit of a
challenge to define it in a CAD program.

As I've said before, don't even think of allowing your cat to watch
the process!

Sorting the pieces for reassembly is part of the fun.

|*  Nicholas Bodley   *|*  Autodidact & Polymath * Electronic Tech. (ret.)
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*
|*  Amateur musician  *|*
--------------------------------------------------------------------------

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 13:35:41 1999
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Message-Id: <199903170235.VAA11730@garnet.sover.net>
Date: Tue, 16 Mar 1999 21:37:49 -0500
To: Nicholas Bodley <nbodley@tiac.net>
From: Nichael Lynn Cramer <nichael@sover.net>
Subject: Re: Taking apart the 5^3
Cc: Cube Mailing List <Cube-lovers@ai.mit.edu>
In-Reply-To: <Pine.SUN.3.96.990316050216.6308M-100000@sunspot.tiac.net>

Nicholas Bodley wrote:
>As nearly as I can remember, you can begin dismantling one of these by
>rotating the top slice by maybe 30 degrees or so, then prying upward on
>one of the "wing" cubies (between the center and the corner cubies). Use
>your thumb, nail side down, and lift.

Oh course, if you want to wimp out, the center face cubie is held on by a
screw (at least his is true on my 5Xs).

Just take off the sticker (this is probably easier on the orange side...)
and the rest becomes pretty easy.
---
Nichael Cramer                 loose shoes,
nichael@sover.net                 a tight schedule,
http://www.sover.net/~nichael/       and a warm place to write Lisp...

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 14:07:04 1999
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Message-Id: <015c01be7033$e646df80$8ccfaf8b@uwe>
From: "UMroaming" <uwe@ue.net>
To: "Christ van Willegen" <c.v.willegen@spcgroup.nl>
Cc: "Cube-Lovers" <cube-lovers@ai.mit.edu>, "Jing Meffert" <jing@ue.net>
Subject: Re: MULTI COLORED PLASTIC SURFACES
Date: Wed, 17 Mar 1999 13:05:23 +0800

Dear Christ

Thanks for your email, I will answer / comment in text

> From: Christ van Willegen <c.v.willegen@spcgroup.nl>
> Date: Wednesday, March 17, 1999 12:29 AM
> Subject: Re: MULTI COLORED PLASTIC SURFACES

> Uwe Meffert

> I saw one of your newer cubes in Eindhoven, The Netherlands
> last week. They are black, and have plastic tiles as the colors.
> By the way, 'newer' as in: I've never seen them before.
> Prices vary from $1.50 to $2.50 (about 3 to 5 DM, your name
> makes me assume you're German, but I may be wrong, of course).
> I saw them at 'Jolie/Promida' in Eindhoven, my gf saw them at
> Intertoys (she's not sure...) in Veldhoven.

YES I AM GERMAN LIVING IN HONG KONG

> I've modified one of those cubes (with glue and electricity wire)
> to bear symbols that can easily be discerned by hand (as you may
> be aware from previous discussions on this list, one of my friends
> is blind, and I wanted to learn him how to solve the cube).

> I assume (from glue residues found on the cube (even glueing together
> two cubelets!)) that the tiles are glued on. How 'hard'/'expensive'
> would it be to make these tiles not square, but pre-formed? It
> would make the cubes nicer, and even more useful for blind people
> (and seeing people who would like to learn to solve the cube
> behind their backs). The forms would be:

THE CUBES THAT YOU REFER TO ARE NOT MINE BUT A CHEAP COPY OUT OF CHINA.

Different textured labels including tile labels should be easily purchasable
from a good stationary store and you can easily up a few samples by hand.

About 12 years ago I made 50k pieces of my Pyraminx with 4 different texture
material for the Blind which I donated to several Blind Intitutes around the
world and I understand that some of the players where able to solve the
pyraminx by themselves without any help.

> - Filled square
> - Open square
> - Filled circle
> - Open circle
> - Plus sign
> - Star (6-points)

> Do you like the idea? Is it marketable? Is it produceable? It's

Produceable yes marketable NO.

> probably a bit harder to produce. I don't know if the tiles are
> glued on by hand, or if they are glued on by machines. It would
> make a big difference...

All the self adhesive labels are glued on by hand one side at a
time using a sort of special scotch tape with its adhesive
properties being lower then the adhesive on the labels.

> The cubes I have (2 of them) suggest hand production. A few tiles
> are glued on a bit skew, and the colors are not in the same order.
> I'd assume a machine would glue them on in perfect alignment and
> always in the same order.

Thats because the cubes are from a small copy company that does
not care about quality

> By the way: I can solve the ImpossiBall! I'm using about the same
> sequences of rotations as I do on the cube.

CONGRATULATIONS

With warm regards
Uwe


HAPPY PUZZLING
Uwe Meffert
P.O. Box 24455, Aberdeen, Hong Kong.
Tel. 852-2518-3080, Fax. 852-2518-3282
Email:- uwe@ue.net
www.ue.edu www.ue.net www.mefferts-puzzles.com

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 14:36:56 1999
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From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
To: Cube Lovers <cube-lovers@ai.mit.edu>
Subject: Re : Re: Edges only, Ignoring Flips, Face Turn Metric
In-Reply-To: <199903120601.BAA15248@euclid.math.brown.edu>
Message-Id: <SIMEON.9903170012.A@pavilion.pstcc.cc.tn.us>
Date: Wed, 17 Mar 1999 00:34:12 -0500 (Eastern Standard Time)

On Fri, 12 Mar 1999 01:01:52 -0500 michael reid <reid@math.brown.edu>
wrote:

> i guess i'm not sure what you're doing, jerry.  but i don't think
> it should be *that* difficult.  the number of configurations is
> 12! = about 480 million.  if you divide out by symmetry, you get
> about 10 million configurations.  this should be small enough to
> store in memory and do a complete breadth-first search of the space.
>

The way you describe the search is how Herbert Kociemba did it, but
it is not how my program does it.  I think his program only took an
hour or two. I am applying my program to a problem to which it is not
well suited because I do not have time to write one more like
Herbert's.

I tend to think that the most fundamental design decision in a
program which does a Start rooted breadth first search for a cube
space is to decide whether the search space can fit in memory.  If it
can, and if there is an easy way to index the search space, then the
permutations themselves do not have to be stored.  All that has to be
stored is the distance from Start for each permutations.  These
distances are usually stored one per byte, or sometimes one per
half-byte.  There is even some discussion the Cube-lovers archives
about how the storage can be reduced to two bits per permutation.

If the search space cannot fit in memory, then it seems to me to be
the case that some representation of the permutations themselves must
be stored in addition to the distance from Start for each
permutation.  My program is designed to search as much as possible of
the 4.3*10^19 search space for the entire cube group, so it stores
permutations.  To make it into a program to search edges only without
flips, I simply fixed the corners and the flips, plus I made the
lexicographic ordering consider edges before corners.  But it still
stores the permutations.  It's sort of a quick and dirty solution
which runs very slowly for the problem at hand.

When a search space consists of the elements of a cube group, it is
easy to index the search space.  But when a cube group is reduced by
symmetry the result is generally not a group and the resultant search
space is (in my experience) not very easy to index.  The thing about
Herbert's program that I have trouble comprehending is that he is able
to reduce the search space by symmetry and still have the indexing be
well behaved.  He has posted a clear exposition of his method, so the
problem is in my understanding rather than in his explanation.

The reason reduction by symmetry results in  poorly behaved indexing
for the search space is because not all positions are equally
symmetric. There is much discussion of this phenomenon in the
archives under the general heading of "the real size of cube space".
Herbert seems to have overcome this problem for the edges problem.
But if I understand correctly, he does not believe the same solution
can be applied to the corners.

If Q[n] is the set of permutations which are n moves from Start, then
my program is calculating the product Q[6]Q[6] (all products of the
form st for s and t in Q[6]) as a way to determine Q[12].  For the
whole cube, most such products are in fact 12q from Start and most
such products are distinct.  There is very little wasted time or
energy.  But for edges only without flips, Q[12] is in the tail of
the distribution so most such products are either duplicate or are
less than 12q from Start.  Nearly all the products are a waste of
time.

My program does reduce by symmetry to certain extent.  If R[n] is the
set of representatives (patterns) which are n moves from Start, then
I only store R[n].  (R[n] is about 48 times smaller than Q[n].) Q[n]
is inferred via pointers to R[n], and is represented as Q[n]=R[n]^M,
where M is the set of 48 rotations and reflections of the cube.
Secondly, I only produce elements of R[2n] rather than elements of
Q[2n], which in theory speeds up the program by about 48 times but
which in practice only seems to speed it up by about 20 times.  But
for the edges without flips search, this kind of a speedup is utterly
dwarfed by all those wasted products from Q[6]Q[6].  My program
always runs into this problem when it gets into the tail of a
distribution.

----------------------------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us
Pellissippi State Technical Community College

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 15:30:12 1999
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Message-Id: <199903171610.RAA18270@bednorz.get2net.dk>
From: "Klodshans" <klodshans@get2net.dk>
To: cube-lovers@ai.mit.edu
Date: Wed, 17 Mar 1999 17:09:42 +0000
Subject: Re: OddzOn version of the cube
Reply-To: klodshans@get2net.dk
In-Reply-To: <4.1.19990316202534.0092a160@mail.vt.edu>

Kevin Young <keyoung3@vt.edu> wrote:

> Where at on www.rubiks.com does it say that Oddzon
> is planning on making a deluxe version?

At www.rubiks.com, go into the "news" section. The 
announcement for an OddzOn deluxe Cube was added on March 
11th.


______________________________________
Philip K
E-mail: philipk@bassandtrouble.com
E-mail: klodshans@get2net.dk
web:    http://hjem.get2net.dk/philip-k

[Moderator's note: http://www.rubiks.com/deluxe.html says in one place
 that they are planning to have "super high quality durable stickers"
 and in another that "there won't be stickers" so it's anyone's guess
 what they have in mind.  They're also considering "holographic center
 labels" vs. "the classic look" and say they will be conducting a
 poll.  Rumors that the Tartan design is being promoted by one of its
 inventors are unverified, and those were gifts, not bribes.  --Dan]

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 16:04:42 1999
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Message-Id: <36EFE0A5.3B50AF48@switchview.com>
Date: Wed, 17 Mar 1999 12:04:37 -0500
From: Michael Swart <Michael.Swart@switchview.com>
Organization: Switchview
To: Roger Broadie <roger.broadie@iclweb.com>, cube-lovers@ai.mit.edu
Cc: Martin Moller Pedersen <tusk@daimi.au.dk>
Subject: Re: help on 5x5x5 wings
References: <00e001be6cdf$a0cabaa0$75c4b0c2@home>

Regarding 5x5x5 wings
Roger Broadie said:
> l F' L F l' F' L' F          (Bl, lF, Lf)
> F2 r2 D R2 D' r2 D R2 D' F2  (Br, Fl, Rf)
> r' U b U' F2 U b' U' F2 r    (Bl, lF, Fr)
> R2 U2 l D' l' U2 l D l' R2   (Br, Fl, bR)
> b L2 D l D' L2 D l' D' b'    (Lb, Fl, fL)
> l2 U2 r' l U2 l' U2 l U2 r l U2 r' U2 l U2 r l2 U2  (Fl, rF)

Wow, that's a big help.
I discovered the following long-winded maneuvers a while ago. Changing
'wings' on D.

RT = (r' D' r D' r' D2 r) (right thingy)
LT = (l D l' D l D2 l') (left thingy)

1. RT D2 RT D2 LT D2 LT D2     (Fl, Bl)(fL, Fr)  
2. RT D LT D' RT LT D' RT D LT (fL, bL)(Fl, Br)

(Side note, RT and LT are used by me to get the last squares of the
fourth layer)

Michael Swart
Michael.Swart@switchview.com

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 16:30:30 1999
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Message-Id: <36EEEF3F.2645@zeta.org.au>
Date: Wed, 17 Mar 1999 10:54:39 +1100
From: Wayne Johnson <sausage@zeta.org.au>
Reply-To: sausage@zeta.org.au
To: cube-lovers@ai.mit.edu
Subject: Speed cubing results - March 99

Hello all,

This is the list so far. There are others to go on, but I need CURRENT
best times, and CURRENT averages:

Name		Method		Best Time  Average Time

Lindon Collins	Layer by Layer	38 sec	   47.5 sec
Jiri Fridrich	Fridrich	14 sec	   20 sec  
Ryan Heise	Fridrich	34 sec	   43 sec  
Ryan Heise	Petrus		40 sec	   56 sec  
Wayne Johnson	Petrus		47 sec	   65 sec  
Wayne Johnson	Layer by Layer	58 sec	   75 sec  
Karen Loewen	Karen Loewen	90 sec	   ***	  
Clive McCaig	Layer by Layer	38 sec	   60-75 se
Alex Montilla	Layer by Layer	33 sec	   55 sec  
Alex Montilla	Corners first	47 sec	   60 sec  
Alex Montilla	One hand only	73 sec	   01:45	  
Alex Montilla	Feet only	07:19	   N/A	  
Han Wen		Fridrich	25 sec	   40 sec  

Wayne

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 17:02:28 1999
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From: Douglas Zander <dzander@solaria.sol.net>
Message-Id: <199903120225.UAA13599@solaria.sol.net>
Subject: cube with colors attached
To: cube-lovers@ai.mit.edu (cube)
Date: Thu, 11 Mar 99 20:25:33 CST

I received as a present many years ago the Game of Rubik's Cube.  (I forget
the exact name)  but just in case anyone wonders about it, it is a hard
mechanism to turn.  The cubies each had plastic colored blocks fitted into
the cube surface with holes for pegs.  What you were supposed to do was
place a peg into a hole then play some game with an opponent.  Does anyone
else have this cube game?  How rare is this?

Also, I wish to start a new topic about the cube.  Has anyone ever thought
of making a large cube out of wood so that there is a lot of wood (or clay)
on the outside of the cube and then carve something like a human head out
of the material? (a bust)  Then the object is to scramble and reconstruct
the head.  (Has this been talked about before?)  I think a white bust of
an ancient Greek (like white marble statues) would be cool.

--
 Douglas Zander                |
 dzander@solaria.sol.net       |
 Shorewood, Wisconsin, USA     |

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 17:40:24 1999
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Date: Tue, 9 Mar 1999 22:46:57 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
Reply-To: Nicholas Bodley <nbodley@tiac.net>
To: Norman Diamond <Norman.Diamond@dec-j.co.jp>
Cc: cube-lovers@ai.mit.edu
Subject: Magic Domino: WTB; also, What's its mechanism like?
In-Reply-To: <714F77ADF9C1D111B8B60000F863155102DD6D3A@tbjexc2.tbj.dec.com>


 Highly unlikely that anyone has one that they'd like to part with, but
I'd like to buy a Magic Domino. [...]

 I'd love to know what the mechanism of a Magic Domino is like, inside.

|*  Nicholas Bodley   *|*  Autodidact & Polymath * Electronic Tech. (ret.)
|*   Waltham, Mass.   *|*  -----------------------------------------------
|*  nbodley@tiac.net  *|*  Frequent crashes are unacceptable in a mature
|*  Amateur musician  *|*  computer industry.

[Moderator's note: Anyone who has one for sale, please contact nbodley by
 e-mail. --Dan]

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From: TSWatts@aol.com
Message-Id: <2d4d92d3.36eef932@aol.com>
Date: Tue, 16 Mar 1999 19:37:06 EST
To: cube-lovers@ai.mit.edu
Subject: Puzzlets

On 3/16/99 hart@iserver.com (Paul Hart) asked if Puzzlets is still in
business.  I can verify that they are in fact still in business in the
Supermall in Auburn, Washington (near Seattle).  They recently moved from a
location in downtown Seattle which may be the source of the confusion.

I can't say anything one way or the other about their tendency to not answer
Emails, but I have spoken to the owner recently about various cube-like
puzzles and he was very knowledgeable and willing to talk to me at length
(unfortunatley I don't remember his name).  I'm surprised he isn't a part of
this Email group!

I can also tell you that he DOES still maintain a list of people who would be
interested in paying to get somebody to do another run of 4x4x4 cubes (aka
"Rubik's Revenge").  Apparently no manufacturer will do it unless they can do
a run of 30,000 of them!  Also, I know he has a list of people who would be
willing to pay a premium for a used version of the 4x4x4, if anyone's
interested.  I traded my 4x4x4 with him for some other merchandise since I
find the 5x5x5, which I only just learned existed about two months ago, to be
basically the same degree of difficulty as the 4x4x4, just bigger.  They do
have some 5x5x5's in stock.

-Tom Watts
Puyallup, WA, USA

From cube-lovers-errors@mc.lcs.mit.edu  Wed Mar 17 19:17:00 1999
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From: WaVeReBeL@webtv.net
Date: Wed, 17 Mar 1999 15:41:21 -0800 (PST)
To: Cube-Lovers@ai.mit.edu
Subject: Re: Speed cubing results - March 99
Message-Id: <18576-36F03DA1-15@mailtod-122.bryant.webtv.net>
In-Reply-To: Wayne Johnson <sausage@zeta.org.au>'s message of Wed, 17 Mar
	1999 10:54:39 +1100

As far as timing is concerened, do you include a preview before the
timer is started, or is it done "cold" (No looking at all before the
timer starts)?  If a preview is allowed, how long do you get?  Is there
a standard for this that everyone is going by?  For longer times this
shouldn't matter too much, but for record keeping and fast times such as
Jiri's 14 seconds, I can see how a few seconds in the begining can
really make a difference.  Is there a FAQ about this?

Also, are you people timing yourselves or do you have someone to do it?
I do it myself.  It doesn't affect my time too much, but I can shave off
a second or two if someone else does it.

-Alex-

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 12:34:38 1999
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From: Derrick Schneider <derrics@ibm.net>
To: "cube-lovers@ai.mit.edu" <cube-lovers@ai.mit.edu>
Subject: RE: Puzzlets
Date: Wed, 17 Mar 1999 16:27:10 -0800

I also can't comment on their non-responding trend (though a recent
move might be reason enough), but the owner's name is Mike Green, I
believe, and he's the one you e-mail when going to www.puzzletts.com.

Derrick

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 13:22:39 1999
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Date: Thu, 18 Mar 1999 00:53:22 -0500 (EST)
From: der Mouse  <mouse@rodents.montreal.qc.ca>
Message-Id: <199903180553.AAA24726@Twig.Rodents.Montreal.QC.CA>
To: Cube-lovers@ai.mit.edu
Subject: Re: Taking apart the 5^3

> As nearly as I can remember, you can begin dismantling one of these
> by rotating the top slice by maybe 30 degrees or so, then prying
> upward on one of the "wing" cubies (between the center and the corner
> cubies).  Use your thumb, nail side down, and lift.

Well, I fiddled with it and finally managed to get one of my 5-Cubes
apart (I used the one with the loose stickers).

I found it more effective to turn a "thick slice" (ie, the outer two
slices turned together) about 45 degrees, then pry with my thumb
between the corner and wing of the turned slice.  (This is perhaps
ambiguous.  Start with a solved 5-Cube, turn the U face 45 degrees
clockwise, so the URF cubie and the RF wing cubie next to it are just
above the middle of the F face.  Then stick your thumb between those
two cubies, nail towards the URF corner cubie, and lever the wing cubie
down.)  It's harder to get the last wing cubie back in than it is to
take the first wing cubie out, but by reversing the move I described
above I find it not too difficult.

Now I just need to find paints that will stick well to the plastic
these things are made of.  (The paints I used for the 3-Cube I painted
don't stick as well as I'd like.)

> Believe me, the insides of a 5^3 are utterly amazing.  [...]  The
> shape of a "foot" on a corner cubie is something to behold;

True.  Quite impressive to look at.  Indeed, once you've taken out the
off-center face cubies and the wing cubies, you're left with something
that looks like a ricketey skeletal 3-Cube (and indeed can, if you're
careful, be manipulated as such).

Amusingly, I realized that as long as you get that "ricketey 3-Cube"
put together in its solvable orbit, it's impossible to put the rest of
the 5-Cube together unsolvably!  (Unless you've marked the face cubies
so they're distinguishable, of course.)

					der Mouse

			       mouse@rodents.montreal.qc.ca
		     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 13:40:33 1999
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Message-Id: <36F09AFF.752D8854@ibm.net>
Date: Wed, 17 Mar 1999 22:19:44 -0800
From: "Jin 'Time Traveler' Kim" <chrono@ibm.net>
Reply-To: chrono@ibm.net
To: skouknudsen@get2net.dk
Cc: cube-lovers@ai.mit.edu
Subject: Some more Rubik's Revenge notes
References: <199902251022.LAA13413@bednorz.get2net.dk>

Check that, the Korea cube is definitely of poorer quality than either
Macau or Hong Kong yet at the same time has its own merits.  The reason
that the cube felt more like it was going to break was because the
middle "wing" cubelets and corner cubelets are "hollow."  The material
is also softer, but somehow it feels like the corner pieces are sturdier
than the earlier ones made in Macau and Hong Kong.  I am assuming it was
made later because seldom are puzzles made of higher quality in later
runs, especially when the tooling is probably more expensive and when
the print run is so short.  But all 3 versions wear the same "(c) I.T.C.
1982" (Ideal Toy Company?) with the different "Made in..." markings.

Anyway, the puzzle was easily disassembled without the need to remove a
screw (There IS no screw unlike the others).  Just like popping open a
standard cube.  Looser tolerances went into making this piece so this
was not a difficult task.  Like I said, the corner cubelets actually
feel sturdier than the others but the overall the plastic used is much
softer and tolerances aren't tight.  Of course, this makes for an easier
turning puzzle but tends to "stick" a lot because of the hollow pieces.
If these "hollows" were somehow smoothed in, this would be a pleasure to
work on instead of the stiffer (and more break prone) Macau and Hong
Kong cubes.  As it is, the center windows tend to bind easily against
each other and I think ultimately it will break just as easily (if not
more easily) than the Macau or Hong Kong deals.  Oh yeah, the Korea
Revenge also feels slightly lighter than the Macau or Hong Kong cubes
for obvious reasons.

Hmmm... I think I'll cc the cube list with this.  Might be interesting
to some.  I wonder how I could squeeze some of this into the Cube FAQ...

--
Jin "Time Traveler" Kim
chrono@ibm.net
http://www.slamsite.com/chrono
'95 PGT - SCPOC

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 14:09:55 1999
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From: "Johan Myrberger (EBC)" <Johan.Myrberger@ebc.ericsson.se>
Reply-To: johan.myrberger@bigfoot.com
To: "'cube-lovers@ai.mit.edu'" <cube-lovers@ai.mit.edu>
Subject: RE: cube with colors attached
Date: Thu, 18 Mar 1999 09:24:44 +0100

> ... Has anyone ever thought of making a large cube out of wood so
> that there is a lot of wood (or clay) on the outside of the cube and
> then carve something ...

Something like this has been manufactured. I believe it was by Disney corp.
They made cubes which loked like the head of Donald Duck and Mickey Mouse if
my memory serves me.

regards
/Johan Myrberger
mailto:Johan.Myrberger@bigfoot.com

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 14:41:45 1999
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From: C.McCaig@queens-belfast.ac.uk
Date: Thu, 18 Mar 1999 09:56:55 GMT
To: Cube-Lovers@ai.mit.edu
Message-Id: <009D54AE.1E20C937.3@a1.qub.ac.uk>
Subject: Re: Speed cubing results - March 99


> As far as timing is concerened, do you include a preview before the
> timer is started, or is it done "cold" (No looking at all before the
> timer starts)?  If a preview is allowed, how long do you get?  Is there
> a standard for this that everyone is going by?

i usually do it cold.  i use the layer-by-layer method, so a preview
isnt really all that advantageous.  i've tried jiri's method, but i
couldnt get used to it, and only once managed to break 60 secs with it.

> Also, are you people timing yourselves or do you have someone to do it?
> I do it myself.  It doesn't affect my time too much, but I can shave off
> a second or two if someone else does it.

i just use my watch, with the alarm set to go off, and the display reading
the seconds (so the alarm goes off at :00) and then look when i've
completed the cube.

> -Alex-

Clive

--
Clive McCaig
Dept. Applied Mathematics
Queens University Belfast
Northern Ireland

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 19:04:31 1999
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Date: Thu, 18 Mar 1999 07:35:57 -0500 (EST)
From: Jiri Fridrich <fridrich@binghamton.edu>
To: WaVeReBeL@webtv.net
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Speed cubing results - March 99
In-Reply-To: <18576-36F03DA1-15@mailtod-122.bryant.webtv.net>
Message-Id: <Pine.SOL.L3.93.990318072843.17846C-100000@bingsun1>


I recommend that we accept the same rules as during the 1st (and the last)
world championship. We had a chance to pick up the cube and look at it for
15 seconds. It was then returned to the table and the actual solving
followed. Most competitors actually needed only 5-10 sec. to figure out
the first couple of moves.

Timing? I think most of us when we practice do the timing ourselves.

I have one more point regarding the average. We should standardize
this as well. For example, one can solve the cube 12 times, remove the
worst and the best time and average the remaining 10. Or, do you want to
list all the times during a practice and average them together?

Jiri

*********************************************
 Jiri FRIDRICH, Research Scientist
 Center for Intelligent Systems
 SUNY Binghamton
 Binghamton, NY 13902-6000
 Ph/Fax: (607) 777-2577
 E-mail: fridrich@binghamton.edu
 http://ssie.binghamton.edu/~jirif/jiri.html
*********************************************

From cube-lovers-errors@mc.lcs.mit.edu  Thu Mar 18 19:37:40 1999
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Message-Id: <36F18BF0.5950@hrz1.hrz.tu-darmstadt.de>
Date: Fri, 19 Mar 1999 00:27:44 +0100
From: Herbert Kociemba <kociemba@hrz1.hrz.tu-darmstadt.de>
Reply-To: kociemba@hrz1.hrz.tu-darmstadt.de
To: Cube Lovers <cube-lovers@ai.mit.edu>
Subject: Re: Re : Re: Edges only, Ignoring Flips, Face Turn Metric
References: <SIMEON.9903170012.A@pavilion.pstcc.cc.tn.us>

Jerry Bryan wrote:
>
> On Fri, 12 Mar 1999 01:01:52 -0500 michael reid <reid@math.brown.edu>
> wrote:
>
> > i guess i'm not sure what you're doing, jerry.  but i don't think
> > it should be *that* difficult.  the number of configurations is
> > 12! = about 480 million.  if you divide out by symmetry, you get
> > about 10 million configurations.  this should be small enough to
> > store in memory and do a complete breadth-first search of the space.
> >

> When a search space consists of the elements of a cube group, it is
> easy to index the search space.  But when a cube group is reduced by
> symmetry the result is generally not a group and the resultant search
> space is (in my experience) not very easy to index.  The thing about
> Herbert's program that I have trouble comprehending is that he is able
> to reduce the search space by symmetry and still have the indexing be
> well behaved.  He has posted a clear exposition of his method, so the
> problem is in my understanding rather than in his explanation.

I think you are right to say that the indexing of a cube group reduced
by symmetries does not behave very well. For this reason I must build a
table which maps the index to a representative of the corresponding
equivalence class. I have no method to directly compute the index.

About 10 million entries would be possible but quite a lot, so I
defined two edge permutations x and y as "equivalent" if x = MyN with
two symmetries M and N. So I reduced by another factor of about 48 and
got 208816 classes. If x is a representative of such a class with index i,
Mx with an arbitrary symmetry M is a representative of a "real" symmetry
class. The "well behaved" index of the latter is computed by 48*i +
Index(M), where index(M) enumerates the symmetries from 0 to 47.
The problem with that which I did not realize first is, that Mx and M'x
could be equivalent, which led to wrong results when computing the God's
Algorithm for positions more than 3 face turns from start (I compared my
results with Jerry's, who made a quick run for positions up to 6 face
turns). With some exta computation this problem could be fixed.

Herbert Kociemba

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 10:32:05 1999
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From: WaVeReBeL@webtv.net
Date: Fri, 19 Mar 1999 02:07:36 -0800 (PST)
To: Cube-Lovers@ai.mit.edu
Cc: IBILUIE@aol.com, IPIIIKA@aol.com
Subject: Re: Speed cubing results - March 99
Message-Id: <9085-36F221E8-14@mailtod-122.bryant.webtv.net>
In-Reply-To: Jiri Fridrich <fridrich@binghamton.edu>'s message of Thu, 18
	Mar 1999 07:35:57 -0500 (EST)

I think that if we are to be keeping track records & averages, we should
ALL stick to one standard.  Using tournament rules sounds like a good
idea.  This makes comparing times more accurate.

Almost everybody responded w/ a different preview time (anywhere from no
preview to 15 seconds).  People like me who started cold had a
disadvantage to those who had a preview.  A 15 second preview sounds
good to me.  This gives enough time to familiarize oneself w/ the cube,
look for pieces, and plan out the first few moves.  I've been timing
myself cold which means much time is wasted at the beginning.  Having a
preview helps a lot.

When it comes to averages, I guess there is no standard.  I agree w/
disregarding the high & low extremes though.  They can distort the
average (arithmetic mean).  This should give a more accurate
representation.  Also, the more entries calculated into the average the
better.

I hope I'm not going overboard.  It's not like we're in a tournament.
If all that is necessary is an informal rough estimate, then you can
disregard this entire message. =)

-Alex Montilla-

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 11:06:11 1999
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Message-Id: <199903191105.MAA28298@bednorz.get2net.dk>
From: "Klodshans" <klodshans@get2net.dk>
To: cube-lovers@ai.mit.edu
Date: Fri, 19 Mar 1999 12:04:43 +0000
Subject: RE: cube with colors attached
Reply-To: klodshans@get2net.dk
In-Reply-To: <0F8184664EA9D21192F70008C75D16925CAEFF@esealnt145>

Johan Myrberger <Johan.Myrberger@bigfoot.com> wrote:

> > ... Has anyone ever thought of making a large cube out of wood so
> > that there is a lot of wood (or clay) on the outside of the cube and
> > then carve something ...
> 
> Something like this has been manufactured. I believe it was by Disney corp.
> They made cubes which looked like the head of Donald Duck and Mickey Mouse if
> my memory serves me.

These "cubes" were made by Disney in Spain. They work with a 
2x2x2 mechanism. The mechanism seems to be different to the 
one used in the Rubik's Mini Cube - you can see a screw inside 
the "cube" when turning slightly on two axis simultaniously. A bit 
similar to the Pyramorphix - maybe these were also manufactured 
by Meffert ?

I have seen them for sale at Puzzle-shop
www.puzzle-shop.de
and from Pete Beck/Just Puzzles
www.freeyellow.com/members4/justpuzzles/

They are quite fun to operate - one can turn Mickey's ears so they 
point backwards instead of upwards, or turn his eyes so they look 
like i don't know what. Pretty perverse ;-)

Philip

______________________________________
Philip K
Vendersgade 15, 3th
DK - 1363 Copenhagen K
Denmark
Phone:  +4533932787
Mobile: +4521706731
E-mail: philipk@bassandtrouble.com
E-mail: klodshans@get2net.dk
web:    http://hjem.get2net.dk/philip-k

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 12:02:52 1999
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Date: Fri, 19 Mar 1999 11:14:58 +0000
From: David Singmaster Computing <david.singmaster@sbu.ac.uk>
To: c.v.willegen@spcgroup.nl
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D5582.3042380C.305@ice.sbu.ac.uk>
Subject: RE: Fwd: Request for spectacular cube-solving  -  Can anyone help ?

	It is true that some blind people have limited 3-D perception, but a
colleague once told me he came into a graduate student room and heard the only
blind student in his class explaining subdivisions in n-dimensions to the other
students!

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 12:47:00 1999
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Date: Fri, 19 Mar 1999 12:16:04 +0000
From: David Singmaster <david.singmaster@sbu.ac.uk>
To: hbizek@ameritech.net
Cc: cube-lovers@ai.mit.edu
Message-Id: <009D558A.B96AC759.280@ice.sbu.ac.uk>
Subject: RE: parity pairs

	Conway noted the two mirror-image orientations of the standard colour
pattern (W/Y, B/G, R/O).  One of the corners has  BOY  at a corner and he called
this a  BOY,  versus the mirror-image  YOB.  I think he read the colours clockwise?
Certainly most of the production was  BOY  and one had to hunt a bit for  YOBs.
Some cubists were particularly keen to have one orientation rather than the
other.

DAVID SINGMASTER,  Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411;  fax: 0171-815 7499;
email:  zingmast  or  David.Singmaster  @sbu.ac.uk

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 13:23:41 1999
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Message-Id: <001001be7220$d2f8e740$0237a8c0@uwe>
From: uwe@ue.net (Uwe Meffert)
To: "der Mouse" <mouse@rodents.montreal.qc.ca>
Cc: "Cube-Lovers" <cube-lovers@ai.mit.edu>
Subject: Re: Taking apart the 5^3
Date: Fri, 19 Mar 1999 23:53:58 +0800


>>From: der Mouse <>
>>To: Cube-lovers@ai.mit.edu <Cube-lovers@ai.mit.edu>
>>Date: Friday, March 19, 1999 11:03 AM

>>>I found it more effective to turn a "thick slice" (ie, the outer two
>>>slices turned together) about 45 degrees, then pry with my thumb
>>>between the corner and wing of the turned slice....

That procedure is not recommended as it voids the implied warranty and has
the danger of permanently stripping the thread inside the center of the
cube.

If you must take the cube apart do so by prying off one of the center small
squares and then loosening one of the screws, which later after re-assembly
should be re-tightened.

Regards
Uwe


HAPPY PUZZLING
Uwe Meffert
P.O. Box 24455, Aberdeen, Hong Kong.
Tel. 852-2518-3080, Fax. 852-2518-3282
Email:- uwe@ue.net
www.ue.edu www.ue.net www.mefferts-puzzles.com

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 15:00:22 1999
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Date: Fri, 19 Mar 1999 11:48:07 -0500
To: Cube-Lovers@ai.mit.edu
From: Kevin Young <keyoung3@vt.edu>
Subject: Mustering Interest in the Rubik's Cube

Hi-

I've been a cubist since elementary school in 1980.  My interest increases
and decreases in waves, however, it never dies.  I'm now back in school at
Virginia Tech as a computer science major.  Does anyone have any
suggestions on how to muster serious  interest with some of my peers at the
University?

Thank you,

Kevin Young

From cube-lovers-errors@mc.lcs.mit.edu  Fri Mar 19 19:31:58 1999
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Date: Fri, 19 Mar 1999 16:21:33 -0500 (Eastern Standard Time)
From: Jerry Bryan <jbryan@pstcc.cc.tn.us>
Subject: Re : RE: parity pairs
In-Reply-To: <009D558A.B96AC759.280@ice.sbu.ac.uk>
To: Cube Lovers <cube-lovers@ai.mit.edu>
Message-Id: <SIMEON.9903191633.A@GN209A.PSTCC.CC.TN.US>

On Fri, 19 Mar 1999 12:16:04 +0000 David Singmaster
<david.singmaster@sbu.ac.uk> wrote:

> 	Conway noted the two mirror-image orientations of the standard colour
> pattern (W/Y, B/G, R/O).

W/Y, B/G, R/O is the "differ by yellow" standard, which I prefer as
"the" standard.  However, there are also references in Cube-Lovers
archives to W/B, R/O, and Y/G as a standard or as the tournament
standard.  I have no idea who gets to be the standards body to select
"the" standard.

But as one example of why I like the W/Y, B/G, R/O standard, many of
the local maxima at 12q from Start are only "somewhat symmetric", but
the eye's sense of symmetry in looking at them can be much stronger.
The reason is that the eye (or my eye, at least) can easily identify
W/Y as the "same" color, B/G as the "same" color, and R/O as the
"same" color.  And when such identifications are made, the symmetry
of many of the 12q local maxima is much stronger than it would be
otherwise.  I really haven't looked at them with any other color
scheme, but I can't imagine that the apparent symmetry would look as
strong otherwise.

Also, in all the various discussions about stickers, falling off and
otherwise, there have been comments about cubes where it is hard to
tell the colors apart, depending on the exact colors which are used,
how faded the colors are with age, etc.  I guess my experience has
been pretty positive in that my stickers have not fallen off and with
one notable exception, the colors seem easy to distinguish.  The
exception is that with my 2x2x2 Pocket Cube, it is very difficult to
distinguish the orange from the red stickers unless I have very, very
good lighting conditions.  This particular cube has always been this
way.  I can think of no reason that a 2x2x2 should be this way as
compared to a 3x3x3 or a 4x4x4, but it does seem to be the case.

----------------------------------------
Jerry Bryan

From cube-lovers-errors@mc.lcs.mit.edu  Mon Mar 22 13:38:42 1999
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Date: Sat, 20 Mar 1999 06:50:39 -0500 (EST)
From: Nicholas Bodley <nbodley@tiac.net>
To: Uwe Meffert <uwe@ue.net>
Cc: der Mouse <mouse@rodents.montreal.qc.ca>,
        Cube-Lovers <cube-lovers@ai.mit.edu>
Subject: Re: Taking apart the 5^3
In-Reply-To: <001001be7220$d2f8e740$0237a8c0@uwe>
Message-Id: <Pine.BSF.3.96.990320064432.15379L-100000@shell2.tiac.net>


On Fri, 19 Mar 1999, Uwe Meffert wrote:

}>I found it more effective to turn a "thick slice" (ie, the outer two
}>slices turned together) about 45 degrees, then pry with my thumb
}>between the corner and wing of the turned slice....

}That procedure is not recommended as it voids the implied warranty and has
}the danger of permanently stripping the thread inside the center of the
}cube.
}
}If you must take the cube apart do so by prying off one of the center small
}squares and then loosening one of the screws, which later after re-assembly
}should be re-tightened.

Since I was the first to suggest this method, I'll retract the advice.
Indeed, it would be really unfortunate to strip the threads in the
plastic that hold a screw in place.

Mr. Meffert is, and has been, a formidable inventor and manufacturer of
cube-like puzzles for quite some time, for those who don't recognize his
name. I'd follow his advice; definitely! Do note that he said "loosen",
not "remove" the screw. I posted recently about special care in
reinserting a removed screw.

My regards to all...
NB

|*  Nicholas Bodley   *|*
|*   Waltham, Mass.   *|*
|*  nbodley@tiac.net  *|*
|*  Amateur musician  *|*

