From cubeloverserrors@mc.lcs.mit.edu Sun Mar 8 19:35:17 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id TAA05274; Sun, 8 Mar 1998 19:35:17 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 8 03:41:58 1998
Date: Sun, 8 Mar 1998 09:41:21 +0100 (MET)
MessageId: <1.5.4.16.19980308094102.437739b8@mailsvr.pt.lu>
To: rshep@simplex.nl
From: Georges Helm
Cc: geohelm@pt.lu, schubart@best.com, CubeLovers@ai.mit.edu
Hi,
You once asked a question about early rubik's cube solutions
(on Schubart's web page)
I have solution from 1979 by
ANGEVINE James
BEASLEY J.D.
CAIRNS Colin / GRIFFITHS Dave
CLAXTON Mike
DAUPHIN Michel (Mathematique et Pedadogie 24/79)
EASTER Bob
HOWLETT G.S.
JACKSON William Bradley
JOHNSON K.C.
MADDISON Richard
NELSON Roy
RODDEWIG Ulrich
SWEENEN John
TAYLOR Don (1978)
TRURAN Trevor (Computer Talk 7.11.1979)
Regards
Georges
Georges Helm
geohelm@pt.lu
http://ourworld.compuserve.com/homepages/Georges_Helm/
http://www.geocities.com/Athens/2715
From cubeloverserrors@mc.lcs.mit.edu Mon Mar 9 10:21:54 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA06825; Mon, 9 Mar 1998 10:21:54 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 8 18:58:07 1998
MessageId: <9803082359.AA00210@jrdmax.jrd.dec.com>
Date: Mon, 9 Mar 98 08:59:07 +0900
From: Norman Diamond 09Mar1998 0859
To: cubelovers@ai.mit.edu
Subject: Re: Taiwanese Invention of the Cube?
ReplyTo: diamond@jrdv04.enet.decj.co.jp, whuang@ugcs.caltech.edu
WeiHwa Huang replied to me:
>>As for patenting, somehow the mixture of "patent" and "Taiwan" in the
>>same sentence strikes me as an oxymoron.
>>Somehow the mixture of "trademark" and "Taiwan" strikes me as an
>>oxymoron too, even though they're not in the same sentence.
>>Want to try "copyright" next? :)
>Is it possible to copyright the Cube? That's why I didn't try it.
Some puzzle designers do copyright their designs.
When one compares patents with copyrights, copyright makes sense.
Patents are intended for inventions that improve the quality of life
and will become important in industry after the patents expire, so
that the inventors starve. Copyrights are for frivolous entertainment
like puzzles and photos, so they bring royalties for the lifetime of
the creator plus 50 years to the heirs. One can only wonder why
patents were ever granted for puzzles.
>In any case, stop sneering  Taiwan has local copyright, trademark,
>and patent laws, and has had them for decades. Sure, they haven't
>honored international copyright laws,
Guess which part of that I was sneering at.
>but then again, most other countries don't think Taiwan exists as an
>independent country.
The Republic of China also thinks Taiwan doesn't exist as an independent
country.
>When it became economically viable to honor international
>copyright, they did so  such legislation was passed in 1994.
>Perhaps you are getting a biased view from living in Japan?
No, my unbiased view was based on observations that I had made for decades.
=====
Mr. Huang and I had this discussion in private email already. I didn't
know that he was going public with it too. Anyway if I understand
correctly, Mr. Huang agreed with my point after that, so there's no
need to repeat the rest of the discussion unless I misunderstood.
[Moderator's note: In any event, further discussion on this topic
should be sent to WeiHwa Huang and Norman Diamond, rather than to
cubelovers. I somewhat regret passing _any_ of it on. The topic of
intellectual property and its legal status is vast, and has eaten
bigger lists than this. ]
=====
 Norman Diamond diamond@jrdv04.enet.decj.co.jp
[Speaking for Norman Diamond not for Digital.]
From cubeloverserrors@mc.lcs.mit.edu Mon Mar 9 11:40:13 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA07026; Mon, 9 Mar 1998 11:40:13 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 8 20:10:06 1998
MessageId:
Date: Sun, 8 Mar 1998 20:09:26 0500
To: tomkeane@mail.del.net, cubelovers
From: Charlie Dickman
Subject: Rubik's Tesseract Solution
Tom and other cubelovers,
I have completed a solution to the Rubik Tesseract and have included it in
the program and it's associated documentation but neither is ready for
prime time just yet.
I was wondering if there was anyone who would be kind enough to review the
documentation and see if the writeup of the solution is reasonably
intelligible and provide me some feedback before I make it and the program
generally available. It is an HTML document (332K selfextractingarchive)
that you can read with your browser.
Thanks,
Charlie Dickman
charlied@erols.com
From cubeloverserrors@mc.lcs.mit.edu Wed Mar 11 13:07:59 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id NAA14544; Wed, 11 Mar 1998 13:07:59 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 11 07:40:13 1998
To: cubelovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (WeiHwa Huang)
Subject: Blindfold Cubesolving
Date: 11 Mar 1998 12:39:04 GMT
Organization: California Institute of Technology, Pasadena
MessageId: <6e60l8$2bc@gap.cco.caltech.edu>
Is there anyone who knows some good techniques for blindfold cubesolving?
I can solve the cube in about 7 "peeks" or so, but that's still quite
a ways from looking at the cube once and solving it behind one's back.

WeiHwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/

Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.
From cubeloverserrors@mc.lcs.mit.edu Wed Mar 11 14:44:29 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA14952; Wed, 11 Mar 1998 14:44:29 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 11 13:58:59 1998
Date: Wed, 11 Mar 1998 13:58:48 0500 (EST)
From: Jiri Fridrich
To: WeiHwa Huang
Cc: cubelovers@ai.mit.edu
Subject: Re: Blindfold Cubesolving
InReplyTo: <6e60l8$2bc@gap.cco.caltech.edu>
MessageId:
I believe that solving the cube blindfolded in one shot is very difficult
if not impossible. One could memorize the orientation of all cubies and
their permutation. Then use algorithms for turning the cubes without
moving them, and then algorithms for permuting them. One would need to
define orintation of cubies on the cube and then the permutation
algorithms would have to preserve that orientation. This system would
presume one really long "peek" and excellent memory, of course :)
Using my system (http://ssie.binghamton.edu/~jirif), I could probably
bring down the number of peeks to four with some practice ... Of course,
seven is no sweat.
Jiri
*********************************************
Jiri FRIDRICH, Research Scientist
Center for Intelligent Systems
SUNY Binghamton
Binghamton, NY 139026000
Ph/Fax: (607) 7772577
Email: fridrich@binghamton.edu
http://ssie.binghamton.edu/~jirif/jiri.html
*********************************************
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 13 12:20:32 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA24252; Fri, 13 Mar 1998 12:20:31 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Mar 12 13:47:19 1998
Sender: mark@ampersand.com
To: whuang@ugcs.caltech.edu (WeiHwa Huang)
Cc: cubelovers@ai.mit.edu
Subject: Re: Blindfold Cubesolving
References: <6e60l8$2bc@gap.cco.caltech.edu>
From: Mark Atwood
Date: 12 Mar 1998 13:47:11 0500
InReplyTo: whuang@ugcs.caltech.edu's message of 11 Mar 1998 12:39:04 GMT
MessageId:
whuang@ugcs.caltech.edu (WeiHwa Huang) writes:
>
> Is there anyone who knows some good techniques for blindfold cubesolving?
>
> I can solve the cube in about 7 "peeks" or so, but that's still quite
> a ways from looking at the cube once and solving it behind one's back.
I have heard of something like "cubes for the blind". Probably either
have a different textured material attached to each cubie face, or a
Braille glyph embossed into each cubie face.
(Never tried to solve one blind, but I could probably solve on in about
a dozen or so glances. But for a while I worked on solving them with
my feet, after seeing someone do it on TV.)

Mark Atwood  Thank you gentlemen, you are everything we have come to
zot@ampersand.com  expect from years of government training.  MIB Zed
[ Moderator's note: You'll notice this is a different topic. Perhaps
WeiHwa Huang should consider his problem "memory solving" rather
than "blindfold solving". I've heard that John Conway has a good
memory method, I think requiring five peeks (cf Roger Frye, 20 Oct
1981). There are also several mentions of tactile cubes in the
archives. ]
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 13 15:11:08 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA24858; Fri, 13 Mar 1998 15:11:07 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Mar 12 17:13:30 1998
MessageId:
InReplyTo: <6e60l8$2bc@gap.cco.caltech.edu>
Date: Wed, 11 Mar 1998 17:33:58 0500
To: cubelovers@ai.mit.edu
From: Kristin Looney
Subject: Re: Blindfold Cubesolving
WeiHwa Huang wrote:
> Is there anyone who knows some good techniques for blindfold
> cubesolving?
>
> I can solve the cube in about 7 "peeks" or so, but that's still quite
> a ways from looking at the cube once and solving it behind one's back.
This brings back fond memories of the trip to CA for the first National
Cube contest back in '81... us nine finalists were taken on a day trip to
Disney Land and we had a race to see who could solve the cube the fastest
in the line to space mountain. As the line winds inside the building,
it is really quite dark, and we were on our hands and knees trying to
get whatever light we could from the running lights on the floor.
I don't remember who won... but it was a huge amount of fun.
K.
kristin@wunderland.com
http://www.wunderland.com/wts/kristin
To all the fishies in the deep blue sea, Joy.
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 13 16:02:47 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA24990; Fri, 13 Mar 1998 16:02:46 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Mar 12 16:09:47 1998
Date: Thu, 12 Mar 1998 22:09:22 +0100
MessageId: <199803122109.WAA06383@dataway.ch>
To: CubeLovers@ai.mit.edu
From: Geir Ugelstad
Subject: Rules for speedcubing
Hello,
What are the exact rules for speed cubeing?
I have seen that in the Worldcampionship it was legal to look at
the cube 15 seconds and then put it back on the table. How long time
did it take from puting it back on the table (after looking) and the
real start???
Ys Geir Ugelstad
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 13 17:08:16 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA25241; Fri, 13 Mar 1998 17:08:16 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Mar 12 23:37:59 1998
Date: Thu, 12 Mar 1998 22:34:54 0600 (CST)
From: "J. David Blackstone"
Subject: Oddz On website
InReplyTo: <009C2062.FA899020.3@ice.sbu.ac.uk>
To: David Singmaster
Cc: skouknudsen@email.dk, cubelovers@ai.mit.edu
MessageId:
MimeVersion: 1.0
ContentType: TEXT/PLAIN; charset=USASCII
On Thu, 19 Feb 1998, David Singmaster wrote:
> common knowledge that it was not Rubik's mechanism. One may be able
> to get details from the web site that Oddz On (sp??) has set up. Tom
I may have missed it, but could someone provide the URL of this website?

J. David Blackstone
jxb9451@utarlg.uta.edu
http://www.geocities.com/Athens/Acropolis/1341

From cubeloverserrors@mc.lcs.mit.edu Tue Mar 17 10:14:50 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA04592; Tue, 17 Mar 1998 10:14:50 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Mar 13 14:48:44 1998
From: Phil Servita
Sender: meister@khitomer.epilogue.com
To: cubelovers@ai.mit.edu
Subject: not quite blind cubing
Date: Fri, 13 Mar 98 14:48:43 0500
MessageId: <9803131448.aa12167@khitomer.epilogue.com>
whuang@ugcs.caltech.edu (WeiHwa Huang) writes:
>
> Is there anyone who knows some good techniques for blindfold cubesolving?
>
> I can solve the cube in about 7 "peeks" or so, but that's still quite
> a ways from looking at the cube once and solving it behind one's back.
Back when i was still in college, myself and a friend would occasionally
perform our "geek party trick", which was that we would sit on the floor,
backtoback, and someone would toss one of us a scrambled cube. Whoever
caught it would look at it, make a single quarterturn on it, and pass it
over their shoulder to the other person, who would look at it and make another
quarter turn, pass it back, and so on. We could solve it in this fashion in
just under 2 minutes.
phil
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 17 10:46:50 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA04737; Tue, 17 Mar 1998 10:46:50 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sat Mar 14 03:49:32 1998
MessageId: <3.0.3.32.19980313231431.00835810@netcom13.netcom.com>
Date: Fri, 13 Mar 1998 23:14:31 0800
To: Mark Atwood
From: Ray Tayek
Subject: Re: Blindfold Cubesolving
Cc: cubelovers@ai.mit.edu
InReplyTo:
At 01:47 PM 3/12/98 0500, Mark Atwood wrote:
>...
>I have heard of something like "cubes for the blind". Probably either
>have a different textured material attached to each cubie face, or a
>Braille glyph embossed into each cubie face.
>...
my wife teaches blind kids. do you know where i could get some braile cubes?
thanks
Ray (will hack java for food) http://home.pacbell.net/rtayek/
hate Spam? http://www.compulink.co.uk/~netservices/spam/
[ Moderator's note: There are quite a few notes in the archives about
adding tactile labels to cubes. Adding characters in Braille should
be about the easiest thing to doI'm sure she has a DYMO embosser. ]
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 17 11:02:50 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA04827; Tue, 17 Mar 1998 11:02:49 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 15 15:27:20 1998
From: roger.broadie@iclweb.com (Roger Broadie)
To: "Cube Lovers Submissions"
Subject: Ideal's patent for 4^3
Date: Sun, 15 Mar 1998 20:29:20 0000
MessageId: <19980315202713.AAA21006@home>
On 19 Feb 1998 David Singmaster wrote:
In my Cubic Circular 1 (Autumn 1981), I recorded that Wim
Osterholt, of the Netherlands, had made and patented a 4^3
which he showed me. I don't remember it and I'm not sure when
he brought it to London  perhaps Summer 1981? I also recorded
that Rainier Seitz (product manager of Arxon which was Ideal's
German agent) showed me some German patents and applications
for the 4^3 and 5^3. In Cubic Circular 2 (Spring 1982), I
record talking with another person who had devised a 4^3
mechanism. In Cubic Circular 3/4 (Spring/Summer 1982), I
describe playing with examples. However, I don't recall ever
knowing who devised the mechanism that was produced for Ideal.
It was common knowledge that it was not Rubik's mechanism
I have just come across Ideal's patent for its 4^3. It is US Patent No
4,421,311. The inventor was Peter Sebesteny, and the original application
was made in Germany on 8 Feb 1981, so it may have been one of the patents
David Singmaster was shown. It can be viewed at the IBM patent site from
http://www.patents.ibm.com/details?patent_number=4421311
One of the references cited by the US Patent Examiner was to page 29 of
David Singmaster's "Notes on Rubik's Magic Cube"  undoubtedly the remark
"One can imagine the 4x4x4 cube or the 3x3x3x3 hypercube. The first might
be makeable but its group seems to be much more complicated. The second is
unmakeable, but its group structure may be determinable."
The corresponding European patent application was taken through to the
point where it was ready for grant, but then allowed to lapse. The next
stage would have been quite expensive and have required Ideal to translate
the specification into the languages of the European countries in which it
was to be in force. And the US was not renewed when the first renewal fees
became due in 1986. Presumably by then Ideal had lost interest in the
patent  they may have calculated there was zero chance of anyone launching
an imitation, given the number of 4^3s that had been left unsold.
I don't have ready access to information about the German application, but
I suspect it was applied for by Sebesteny on his own behalf, and he then
interested Ideal in it.
Roger Broadie
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 17 11:43:37 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA05013; Tue, 17 Mar 1998 11:43:37 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 15 06:19:03 1998
To: cubelovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (WeiHwa Huang)
Subject: Re: Blindfold Cubesolving
Date: 15 Mar 1998 11:17:40 GMT
Organization: California Institute of Technology, Pasadena
MessageId: <6egdck$cvj@gap.cco.caltech.edu>
References:
The Moderator wrote:
>[ Moderator's note: You'll notice this is a different topic. Perhaps
> WeiHwa Huang should consider his problem "memory solving" rather
> than "blindfold solving". I've heard that John Conway has a good
> memory method, I think requiring five peeks (cf Roger Frye, 20 Oct
> 1981). There are also several mentions of tactile cubes in the
> archives. ]
I used the term "blindfold solving" patterned after "blindfold chess",
where two players merely recite moves to each other, using no actual
pieces or board.
As far as "solving in the dark" goes, it reminds me that I have a cube
in which under certain lamps, the yellow and white colors are
indistinguishable. Solving such a cube can occasionally give a few
tripups!

WeiHwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/

Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 17 14:28:15 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA05547; Tue, 17 Mar 1998 14:28:14 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue Mar 17 13:30:17 1998
Date: Tue, 17 Mar 1998 13:30:10 0500 (Eastern Standard Time)
From: Jerry Bryan
Subject: Re: Blindfold Cubesolving
InReplyTo: <6egdck$cvj@gap.cco.caltech.edu>
To: WeiHwa Huang
Cc: cubelovers@ai.mit.edu
MessageId:
On Sun, 15 Mar 1998, WeiHwa Huang wrote:
> As far as "solving in the dark" goes, it reminds me that I have a cube
> in which under certain lamps, the yellow and white colors are
> indistinguishable. Solving such a cube can occasionally give a few
> tripups!
I have had the same problem with orange and red, especially on my 2x2x2.
I have a "latter day" 2x2x2 (my kids lost my first one), and the colors
in general do not seem quite true to the colors on my 3x3x3 and 4x4x4
cubes.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 24 12:51:28 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA24618; Tue, 24 Mar 1998 12:51:28 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue Mar 24 11:52:07 1998
MessageId: <3517E49F.DF5B21BF@mail.retina.ar>
Date: Tue, 24 Mar 1998 13:51:44 0300
From: Isidro
ReplyTo: isidroc@usa.net
Organization: Frank Zappa's Fan Club
To: Cube Lovers Submissions
Subject: 5^3 quiz
I need to know the answers for these questions:
Who invented 5^3?
What is the commercial name?
How many cubies it has?

Isidro: isidroc@usa.net
[ Moderator's note: There was a note last July mentioning "Rubik's
Wahn (5x5x5) (maybe also called Professor's cube, Ultimate or Master
Revenge)"any other names?
The number of cubies is obviously 98why didn't you just count them? ]
From cubeloverserrors@mc.lcs.mit.edu Wed Mar 25 10:09:36 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA27206; Wed, 25 Mar 1998 10:09:36 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue Mar 24 16:14:21 1998
MessageId: <3.0.5.16.19980324220550.0bd76334@vip.cybercity.dk>
Date: Tue, 24 Mar 1998 22:05:50
To: cubelovers@ai.mit.edu
From: Philip Knudsen
Subject: RE: 5^3 quiz
To my knowledge, the 5x5x5 was invented by Udo Krell. It was produced by
Uwe Meffert in 1983. I read somewhere that Dr. Chr. Bandelow had the Hong
Kong factory finish extra puzzles from previously manufactured parts around
1990, don't know if this is true. Bandelow is still selling this puzzle,
under the name "Giant Magic Cube". It also seems Meffert reissued the 5x5x5
one or two years ago, under the name "Professor's Cube". This new version
might have other colors than the original. I have seen the puzzle under the
name "Ultimate Cube" several times, the name "Master Revenge" however is
new to me.
Since Meffert is the manufacturer, the "most" official name for the 5x5x5
is probably "Professor's Cube".
Philip K
recording and performing artist
Vendersgade 15, 3th
DK  1363 Copenhagen K
Phone: +45 33932787
Mobile: +45 21706731
Email: skouknudsen@email.dk
Email: philipknudsen@hotmail.com
Sms: 4521706731@sms.tdk.dk (short message, no subject)
From cubeloverserrors@mc.lcs.mit.edu Wed Mar 25 12:56:17 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA27652; Wed, 25 Mar 1998 12:56:16 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue Mar 24 16:46:37 1998
Date: Tue, 24 Mar 1998 22:46:49 +0100 (MET)
MessageId: <199803242146.WAA06298@relay.euronet.nl>
To: CubeLovers@ai.mit.edu
From: Sytse <4xs2fs@euronet.nl>
Subject: Re: 5^3 quiz
Isidro,
Who invented 5^3?
At least I did. In 1982 I designed and built a 5^3 cube, all in plywood.
Although I did not aplly for a patent or other registration, as I was only a
schoolboy by then, the local newspaper recorded this event. As the wooden
prototype was not as speedy as necessary, I later designed a simulator for
the Sinclair ZX Spectrum (a then so called 'personal computer' with an
amazing 48K RAM memory). This simulator also included a 6^3 cube. 7^3 was
not possible as this did not fit in the screen, which was my parents
television set. Oh, those were the days!
Nowadays I am an architect.
Kind regards,
Sytse de Maat
P.S. If you happen to know other designers of 5^3, please mail me.
[ Moderator's note: Can you describe the design that held the plywood
model together while allowing it to turn? ]
From cubeloverserrors@mc.lcs.mit.edu Wed Mar 25 15:19:48 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA28021; Wed, 25 Mar 1998 15:19:48 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 25 02:37:53 1998
Date: Wed, 25 Mar 1998 08:37:42 +0100
MessageId: <199803250737.IAA30286@dataway.ch>
To: CubeLovers@ai.mit.edu
From: Geir Ugelstad
Subject: Jiri's system for solving Rubiks's cube
hello cubelovers
For all of you that haven't been into Jiri's home page at
http://ssie.binghamton.edu/~jirif, you should realy look into it!
Bouth the method and presentation is of very high standard!
I bought myself a system in 1982 but I was so dissapointed that I
trow it In the garbage just after. With the system I bought in 1982
it was not possible to make it faster than 23 minutes. With Jiri's
system it should be possible in about 17 sec.!
Ys Geir Ugelstad
PS: Question to Jiri. How far are you able to do the foreplanning
the 15 sec. before the time start to run? Hopefully longer than
"Place the four edges from the first layer"?
From cubeloverserrors@mc.lcs.mit.edu Thu Mar 26 11:46:40 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA00652; Thu, 26 Mar 1998 11:46:40 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 25 16:14:40 1998
Date: Wed, 25 Mar 1998 16:10:54 0500 (EST)
From: Jiri Fridrich
To: Geir Ugelstad
Cc: CubeLovers@ai.mit.edu
Subject: Re: Jiri's system for solving Rubiks's cube
InReplyTo: <199803250737.IAA30286@dataway.ch>
MessageId:
On Wed, 25 Mar 1998, Geir Ugelstad wrote:
> it was not possible to make it faster than 23 minutes. With Jiri's
> system it should be possible in about 17 sec.!
Yes, you are right  with my system AND a lot of time on your hands :) I
am pretty sure that the systems of other top speed cubists are at
least as as good as mine. The system is only half of the secret.
> PS: Question to Jiri. How far are you able to do the foreplanning
> the 15 sec. before the time start to run? Hopefully longer than
> "Place the four edges from the first layer"?
Nope. 15 seconds is not a long time to plan more than the four edges. Of
course, as you proceed, you will usually be able to spot the corners with
their appropriate cubies from the second layer in some nice position and
continue without delays ...
Jiri
*********************************************
Jiri FRIDRICH, Research Scientist
Center for Intelligent Systems
SUNY Binghamton
Binghamton, NY 139026000
Ph/Fax: (607) 7772577
Email: fridrich@binghamton.edu
http://ssie.binghamton.edu/~jirif/jiri.html
*********************************************
From cubeloverserrors@mc.lcs.mit.edu Thu Mar 26 12:46:25 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA00805; Thu, 26 Mar 1998 12:46:24 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 25 18:23:23 1998
MessageId: <9803252324.AA16745@jrdmax.jrd.dec.com>
Date: Thu, 26 Mar 98 08:24:24 +0900
From: Norman Diamond 26Mar1998 0817
To: cubelovers@ai.mit.edu
Subject: RE: 5^3 quiz
I bought my first 5^3 from a department store in Japan in 1985, while it
was alongside the 3^3 and 4^3 on the mass market. Bought my second one
from Dr. Bandelow some time later. In Japan it was called "Professor Cube"
which could be taken as "Professor's Cube" because it would be a bit too
awkward to pedantically insert the syllable for possessive form (in
Japanese grammar) between two polysyllabic foreign words.
(Tangential details:
purofuesoru kyuubu is 5 + 3 syllables, while
purofuesoru no kyuubu would be 5 + 1 + 3 syllables.)
The magic dodecahedron reached the mass market around 1989 or so.
Those were the days. Some time around 1993, the mass market shifted to
computer games.
 Norman Diamond diamond@jrdv04.enet.decj.co.jp
[Speaking for Norman Diamond not for Digital]
From cubeloverserrors@mc.lcs.mit.edu Thu Mar 26 15:10:34 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA01187; Thu, 26 Mar 1998 15:10:33 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Mar 26 10:47:00 1998
Date: Thu, 26 Mar 1998 15:36:25 +0000
From: David Singmaster
To: skouknudsen@email.dk
Cc: cubelovers@ai.mit.edu
MessageId: <009C3C55.665587E6.39@ice.sbu.ac.uk>
Subject: RE: 5^3 quiz
Bandelow's leaflet, which he encloses with the 5^3, states that the
mechanism was invented by Udo Krell, of Hamburg(?). I haven't seen
the patent but perhaps Bandelow has details.
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171815 7411; fax: 0171815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cubeloverserrors@mc.lcs.mit.edu Thu Mar 26 15:58:05 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA01386; Thu, 26 Mar 1998 15:58:04 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Mar 25 19:10:32 1998
MessageId: <01BD5821.7C9449E0@jburkhardt.ne.mediaone.net>
From: John Burkhardt
To:
Subject: new to list
Date: Wed, 25 Mar 1998 19:09:07 0500
Hi, I just found and joined this list.
So I am looking for any and all oddball cube variations I can find.
Does anyone have anything to sell or trade. I can trade for a "Magic
Dodecahedron" which is the start shaped Hungarian version of the
Megaminx and I might be willing to part with a 5x5x5 cube for anything
really interesting. I'm looking for an original Tomy Megaminx. Also
the octahedron puzzle which is like two Pyraminx's glued together
(there might be an official name). I am also searching for a 4x4x4
but I know they are really hard to find these days (mostly because
they tend to break).
The Dodecahedron puzzle is really amazing. It was actually harder
than the 5x5x5 cube. IT took me about 3 hours to work it out! I
think once you know the 3x3x3 then all the same moves do similar
things and you can easily solve 4x4x4 or 5x5x5 with variations. Of
course there are some cool things you can do with these.
I must say that I was disappointed with one web page that listed a
bunch of moves for the 3x3x3 cube. I was trying some of them out and
thinking, my god, how did anyone figure this out, only to then
discover that a computer had figured them out. OK, that's certainly
an interesting problem, but I have much more fun discovering them on
my own. Interstingly enough, solving the dodecahedron led me to some
neat new moves for the original cube!
So where can we go from here? Have we made all the regular polyhedra
into puzzles? Is there hope of actually building 6x6x6 and beyond
cubes? Is there really any point to doing it? I suppose they would
allow for some nice patterns. Does anyone know of any puzzles that are
not in George Helm's collection? I just bought a Magic Cube puzzle at
Walgreens for $3. It's a 3x3x3 with psychedelic stickers on it...
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 27 09:48:24 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id JAA03279; Fri, 27 Mar 1998 09:48:24 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Mar 27 06:50:30 1998
MessageId: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net>
From: John Burkhardt
To:
Subject: Stickers
Date: Fri, 27 Mar 1998 06:45:47 0500
Does anyone know where to find cube stickers? They must come from
somewhere! I found some vinyl lettering once and the periods were
exactly the right size for a 5x5x5 cube. But they don't come in
orange. There must be a way to buy sheets of the stuff. Any ideas?
From cubeloverserrors@mc.lcs.mit.edu Fri Mar 27 13:44:27 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id NAA03643; Fri, 27 Mar 1998 13:44:27 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Mar 27 11:01:41 1998
Date: Fri, 27 Mar 1998 11:02:04 0500 (EST)
From: Nichael Cramer
To: John Burkhardt
Cc: cubelovers@ai.mit.edu
Subject: Re: Stickers
InReplyTo: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net>
MessageId:
John Burkhardt wrote:
> Does anyone know where to find cube stickers? They must come from
> somewhere! I found some vinyl lettering once and the periods were
> exactly the right size for a 5x5x5 cube. But they don't come in
> orange. There must be a way to buy sheets of the stuff. Any ideas?
Ah, yes, the orange stickers on the 5X .... ;)
Anyway, don't they have sticker sets in any colors other than in the
standard cubepallette? Black or grey come to mind. Not quite the
optimal solution, of course, but it would still give you a useable cube.
Nichael

Nichael Cramer
work: ncramer@bbn.com
home: nichael@sover.net
http://www.sover.net/~nichael/
(The cool bit about letters, of course, is that on the 5X5 face in
question, you could, say, put almost all the letters of the alphabet or
some other personalized message(s) of your choice and give yourself a
little something extra to shoot for as you solve the cube.)
From cubeloverserrors@mc.lcs.mit.edu Mon Mar 30 14:54:27 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA04824; Mon, 30 Mar 1998 14:54:26 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Mar 27 20:28:38 1998
Date: Fri, 27 Mar 1998 20:28:57 0400 (EDT)
From: Jerry Bryan
Subject: All the Isoglyphs [long]
To: CubeLovers
MessageId:
Dan Hoey introduced glyphs and isoglyphs on 5 August 1997. A glyph is a
cube face containing no more than two colors, and an isoglyph is a cube
position where every face contains the same glyph. Isoglyphs tend to be
very striking and pretty patterns.
Each corner and edge facelet of a glyph can be the same or a different
color than the center facelet, so there are 2^8 or 256 possible glyphs.
Dan reported that there are 51 glyphs unique up to symmetry (70 if chiral
pairs are distinguished).
On 8 August 1997, Herbert Kociemba reported that there are 35 continuous
isoglyphs unique up to symmetry (including Start). A continuous isoglyph
is one for which each glyph matches the neighboring glyph along the edge.
Herbert did not include the noncontinuous glyphs because there are so
many, and because noncontinuous glyphs are sometimes not so striking and
pretty as the continuous glyphs.
On 9 August, Dan Hoey classified Herbert's isoglyphs according the their
respective glyphs, and provided the usual name for the isoglyphs where a
usual name existed. Where a usual name did not exist, Dan provided a
reasonable name based on the names of closely related isoglyphs.
On 27 August, Mike Reid gave minimal maneuvers for all the continuous
isoglyphs in both the quarterturn and faceturn metrics.
I have now calculated all the isoglyphs, using Herbert's Cube Explorer 1.5
program. All I really did was to put each of the 51 glyphs into the
program in turn. I can only guess, but this has to be more or less what
Herbert did to obtain his results. The only difference is that I asked
the program to calculate both continuous and noncontinuous isoglyphs, so
the task was a bit bigger. My report is much in the spirit of Herbert's
original report. I have made no effort to calculate minimal maneuvers,
nor have I made any attempt to associate names with the maneuvers.
However, my report does include all the glyphs along with their associated
isoglyphs. In fact, for each glyph I have included the entire equivalence
class of glyphs under the rotations and reflections of the square (either
1, 2, 4, or 8 glyphs in each equivalence class). There is, of course, no
necessary relationship between the number of glyphs in the equivalence
class and the number of isoglyphs. You only need to put one glyph from
the equivalence class into Cube Explorer 1.5 to create the isoglyph, and
any one glyph from the equivalence class will do as well as any other.
I can report that of the 51 glyphs unique up to symmetry, 8 of them
produce only continuous isoglyphs, 17 of them produce only noncontinuous
isoglyphs, 14 of them produce both continuous and noncontinuous
isoglyphs, and 12 of them produce no isoglyphs.
In addition to confirming Herbert's figure of 35 continuous isoglyphs, I
can report that there are 249 noncontinuous isoglyphs. In the category
of "most isoglyphs", one glyph has 2 continuous and 49 noncontinuous
isoglyphs, and another has 4 continuous and 46 noncontinuous isoglyphs.
The only other thing that probably requires explanation about the chart
that follows is that there is a two character code below each glyph. This
is a hexadecimal representation of a binary number based on the following
pattern,
765
4X3
210
where the number includes 2^k if facelet k is the same color as the center
facelet. This is not intended as a new classification to replace Dan's.
It is just a bookkeeping technique I used (a 16x16 matrix) to keep track
of the 256 glyphs.
000
0X0
000
00
D' U L' R B' F D' U (8) * continuous
000 000 00X X00
0X0 0X0 0X0 0X0
00X X00 000 000
01 04 20 80
R2 D L2 U' B2 D' U2 R' F' U R B' L' D' F L2 B2 R U' (19) continuous
B2 D F2 U' L2 B' D2 B U B' D2 F L R' D U F' (17) continuous
000 000 000 0X0
0X0 0XX XX0 0X0
0X0 000 000 000
02 08 10 40
D' U B D' L' R F D' B' D' U L (12) * continuous
L2 D' B' F L' D U' F L' R U B' F' (13) not
F2 D' L2 B' D' U' R B L F L F U' F' (14) not
F2 D L2 R2 U' B' U L D L D2 R' F' D' B' D (16) not
U R2 D B2 D F D' B' L' B' D2 F' L' F U F' R' (17) not
R2 U2 B' F D B2 L' R D2 F' R2 F2 D' U (14) not
B2 D2 R2 B' F D' F2 L' R U2 F' L2 D U (14) not
D' U' L2 F' D2 L R' B2 D' B F' R2 D2 B2 (14) not
B2 D U' L2 F' D2 L R' B2 D' B F' R2 (13) not
R2 D2 R' B' L' B D' R2 B' R B' D' R (13) not
000 000 000 000 00X 0XX X00 XX0
0X0 0X0 0XX XX0 0XX 0X0 XX0 0X0
0XX XX0 00X X00 000 000 000 000
03 06 09 14 28 60 90 C0
F2 D F2 D B2 L2 U L2 D' L D L' B' L U' F' U R' U' (19) not
F2 U L2 U L2 U' F U' F' D2 B L R U' B' D' R F' (18) not
D' R2 D2 B2 U' F2 U' L2 B D R D' U F U' B' U2 B' R' U' (20) not
000 00X X00 X0X
0X0 0X0 0X0 0X0
X0X 00X X00 000
05 21 84 A0
F2 U2 L' R D2 F2 L' R (8) not
F2 U2 B2 L2 U' B2 U' B2 L2 D2 L2 U R2 U' (14) not
U' L2 D' L2 D B2 F2 L2 R2 D F2 U' F2 U' (14) not
U2 L2 F2 D U' B2 L2 D' U' (9) not
000 00X X00 XXX
0X0 0XX XX0 0X0
XXX 00X X00 000
07 29 94 E0
(none)
000 000 0X0 0X0
0XX XX0 0XX XX0
0X0 0X0 000 000
0A 12 48 50
F2 D' R2 D' L' U' L' R B D' U B L F2 L U2 (16) continuous
U B2 L D B' F L' D U' L' R F' D2 R' (14) continuous
U' B2 R2 U2 F' D2 L' F2 U' F2 D2 F U2 R' U2 (15) continuous
D2 U B2 D U' R' D2 B' R2 D2 L' R' D' B2 L B (16) not
B2 U2 F2 D2 F2 U R' F' L2 U2 L R U' L2 F2 L' F (17) not
U F2 L2 U2 B' U2 L F2 U B2 D2 B D2 R' U2 (15) not
F2 D2 B2 D' B2 L2 U2 B D2 R F2 D F2 D2 B' U2 L F2 (18) not
U2 B2 F2 D' F2 R2 D2 B U2 L' U2 B' D2 F2 D F2 R' U2 (18) not
D2 U' B2 U2 F D2 L D2 F' D2 L2 F2 U B2 R' U2 (16) not
F2 U R2 D U B' D' B' D' F L' F D' U2 L U2 (16) not
D2 U' B2 U2 F D2 L D2 F U2 R2 B2 U' B2 R' U2 (16) not
U2 F2 D F2 L2 U2 F' U2 L F2 D B2 D2 F D2 R' F2 (17) not
F2 D L2 D2 B2 R2 B2 L2 F2 U2 R B U L U B U' L' U (19) not
000 000 0XX XX0
0XX XX0 0XX XX0
0XX XX0 000 000
0B 16 68 D0
U2 F2 R2 U' L2 D B R' B R' B R' D' L2 U' (15) continuous
000 000 00X 00X 0X0 0X0 X00 X00
0XX XX0 0X0 XX0 0X0 0X0 0X0 0XX
X00 00X 0X0 000 00X X00 0X0 000
0C 11 22 30 41 44 82 88
D U2 L2 U R2 U' L2 U R' B2 L2 F' L2 B' R' F' L D U' (19) continuous
U' F2 L2 D2 U F2 U2 F' L' D2 B2 R' D' B R' U L2 B2 F' (19) not
000 000 00X 0XX X00 X0X X0X XX0
0XX XX0 0X0 0X0 0X0 0XX XX0 0X0
X0X X0X 0XX 00X XX0 000 000 X00
0D 15 23 61 86 A8 B0 C4
D2 L2 F2 R2 U2 B2 D2 F2 R2 U2 R2 U2 (12) not
U2 L2 B2 L2 U2 F2 U2 F2 L2 U2 R2 U2 (12) not
U2 R2 B2 L2 U2 F2 U2 F2 R2 U2 R2 U2 (12) not
D2 R2 F2 R2 U2 B2 D2 F2 L2 U2 R2 U2 (12) not
000 000 00X 0X0 0X0 0XX X00 XX0
0XX XX0 0XX 0XX XX0 XX0 XX0 0XX
XX0 0XX 0X0 00X X00 000 0X0 000
0E 13 2A 49 54 70 92 C8
(none)
000 000 00X 0XX X00 XX0 XXX XXX
0XX XX0 0XX 0XX XX0 XX0 0XX XX0
XXX XXX 0XX 00X XX0 X00 000 000
0F 17 2B 69 96 D4 E8 F0
D2 R2 F2 U2 F2 U2 F2 U2 R2 B2 (10) not
F2 L2 D2 B F R2 B F' R2 (9) not
F2 U2 L2 F2 D U R2 F2 D U' B2 (11) not
U2 L2 B2 D2 F2 U2 F2 U2 R2 B2 (10) not
U2 L2 B2 U2 B2 D2 F2 U2 R2 B2 (10) not
U2 F2 L2 B2 U2 B2 D2 F2 U2 R2 (10) not
L2 D2 F2 L2 U' B2 L2 R2 F2 D' R2 (11) not
D2 R2 F2 D2 B2 D2 F2 U2 R2 B2 (10) not
U2 L2 R2 D F2 U' R2 F2 U2 F2 D' U2 F2 U' (14) not
D' R2 D' B2 U2 B2 F2 R2 B2 F2 U' F2 U' (13) not
U' B2 U' F2 D2 B2 F2 R2 B2 F2 U' F2 U' (13) not
B2 U B2 U' L2 D2 F2 U' R2 U F2 (11) not
D' R2 D' F2 D2 L2 R2 U' B2 F2 D R2 U' F2 U' (15) not
L2 D F2 U' R2 F2 U2 F2 D B2 U' B2 U2 (13) not
F2 D F2 U' R2 U2 F2 U' R2 D B2 (11) not
D2 B2 U' L2 U B2 U B2 D' R2 D' R2 U' (13) not
D' B2 D L2 D2 B2 U B2 U' (9) not
D' L2 B2 D L2 D2 B2 U B2 R2 U' (11) not
D' U' L2 D2 U2 B2 D' U' (8) not
L2 U2 B2 L2 D B2 L2 R2 F2 U' (10) not
L2 U2 R2 D' U' B2 R2 B2 D' U' (10) not
L2 D2 L2 B2 U2 F2 D2 F2 R2 F2 R2 U2 (12) not
L2 D2 B2 D2 F2 R2 B2 R2 F2 U2 R2 U2 (12) not
B F D2 L2 B F (6) * not
000 0X0
XXX 0X0
000 0X0
18 42
L2 U2 L2 R2 U2 L' R' (7) * not
000 000 00X 0X0 0X0 0XX X00 XX0
XXX XXX XXX 0X0 0X0 0X0 XXX 0X0
00X X00 000 0XX XX0 0X0 000 0X0
19 1C 38 43 46 62 98 C2
D2 L2 D R2 U B2 U2 B R' B' D B2 R' F R2 F' U R' (18) not
000 0X0 0X0 0X0
XXX 0XX XX0 XXX
0X0 0X0 0X0 000
1A 4A 52 58
D F2 R2 F2 R2 U F2 R F2 R D2 U' F L' F' L D (17) continuous
B2 L2 U' B2 F2 D2 B2 R B' F2 U' B' D2 L' B' U L2 D' U' (19) not
D' B2 U' B2 F2 D F2 D2 F L2 U L F' D' F2 L' U' (17) not
L2 F2 U B2 U2 F2 R2 B2 R2 F R2 D F2 R' D' B' D' B' R' U (20) not
F2 D2 R2 B2 D2 F2 D' F2 L' B2 U' L B' L D L B' R2 U' F2 (20) not
L2 D R2 U' L2 F2 L2 D' B2 F' L R B D2 R' B F L F2 U' (20) not
L2 D2 U2 L' U2 L' R2 D2 U2 R' U2 R' (12) not
U2 R2 B2 F2 D2 U2 L' B2 F2 R' D2 L' R (13) not
000 000 0X0 0X0 0XX 0XX XX0 XX0
XXX XXX 0XX XX0 0XX XXX XX0 XXX
0XX XX0 0XX XX0 0X0 000 0X0 000
1B 1E 4B 56 6A 78 D2 D8
U R2 U' F' U2 F2 U2 F R F2 R' U R2 U' (14) continuous
B2 D U2 R2 D F2 B' L2 D2 F L F2 L' R2 F' U' F' U (18) not
000 0XX X0X XX0
XXX 0X0 XXX 0X0
X0X 0XX 000 XX0
1D 63 B8 C6
F2 L' R B2 U2 L R' D2 (8) not
D' U' B2 L2 D' U R2 F2 U2 (9) not
000 0XX XX0 XXX
XXX 0XX XX0 XXX
XXX 0XX XX0 000
1F 6B D6 F8
(none)
00X X00
0X0 0X0
X00 00X
24 81
(none)
00X X00 X0X X0X
0X0 0X0 0X0 0X0
X0X X0X 00X X00
25 85 A1 A4
D' B2 L2 F2 R2 F2 U R2 U2 F' R B L D B U' F R' U2 R (20) continuous
B2 L2 R2 U R2 B2 U L2 U' B F D2 L' B2 R2 D' U B' L' R' (20) continuous
F2 R2 U2 B2 D' R2 D L2 D2 R2 F2 U' R2 U' (14) not
00X 00X 00X 0XX X00 X00 X00 XX0
0X0 0XX XX0 0X0 0X0 0XX XX0 0X0
XX0 X00 X00 X00 0XX 00X 00X 00X
26 2C 34 64 83 89 91 C1
(none)
00X 00X X00 X00 X0X X0X XXX XXX
0X0 0XX 0X0 XX0 0XX XX0 0X0 0X0
XXX X0X XXX X0X 00X X00 00X X00
27 2D 87 95 A9 B4 E1 E4
L2 U' R2 D U2 L' B2 F' D' R' B2 D L2 R2 U2 F' L U (18) continuous
B R2 B' F2 L2 B' L' D2 R D' L2 U F' R2 B' L B U (18) not
00X 0XX X00 XX0
0XX XX0 XX0 0XX
XX0 X00 0XX 00X
2E 74 93 C9
(none)
00X X00 XXX XXX
0XX XX0 0XX XX0
XXX XXX 00X X00
2F 97 E9 F4
U' L2 R2 F2 U L2 U' F2 R2 L' U B' R D' B2 D2 B' R' (18) continuous
R2 B2 D B2 D U R2 D' B' D' R F2 R' D B U' (16) continuous
D' L2 U' F2 U F2 U2 F2 D' L2 U B2 U' (13) not
00X 0X0 X00 X0X
XX0 0X0 0XX 0X0
00X X0X X00 0X0
31 45 8C A2
(none)
00X 0X0 0X0 X00
XX0 0XX XX0 0XX
0X0 X00 00X 0X0
32 4C 51 8A
D U L2 B2 D U' F' U F' R F2 R' F D' B2 L2 D' U' (18) continuous
R2 B2 D2 L2 U L2 D B L2 U2 B2 L' R2 F2 D' U R U2 R' (19) continuous
00X 0X0 0X0 0XX X00 X0X X0X XX0
XX0 0XX XX0 XX0 0XX 0XX XX0 0XX
0XX X0X X0X 00X XX0 0X0 0X0 X00
33 4D 55 71 8E AA B2 CC
R2 B2 D U L' R' D2 L' R D U (11) not
B2 L2 D2 L2 R2 B' F' R2 B F' R2 (11) not
B2 R2 B2 R2 F2 U2 B2 R2 U2 R2 (10) not
R2 F2 D U L' R' U2 L' R D' U' (11) not
R2 F2 D' U' L' R' U2 L R' D U (11) not
L2 D' U F2 L R B2 L R D U (11) not
F2 L2 B2 R2 F2 U2 B2 R2 D2 R2 (10) not
R2 B2 D' U' L' R' D2 L R' D' U' (11) not
B2 L2 B2 U R2 U' B2 R2 U2 R2 U B2 U' (13) not
R2 F2 U' B2 D' R2 U2 F2 U' F2 D' B2 L2 (13) not
B2 F2 R2 D F2 D' L2 U2 B2 U' B2 U R2 (13) not
D' R2 D2 B2 R2 B2 U B2 D U B2 U' F2 U' (14) not
B2 R2 U2 L2 D' F2 R2 U' L2 D2 B2 U' R2 F2 U' (15) not
B2 R2 F2 R2 F2 D F2 D' L2 U2 B2 U' B2 U R2 (15) not
F2 U R2 U F2 D2 L2 U B2 U L2 R2 F2 (13) not
B2 F2 L2 R2 D' B2 D B2 D2 R2 U F2 U' (13) not
B2 R2 D' R2 U F2 D2 L2 U B2 D' L2 F2 (13) not
U F2 R2 U2 F2 D' R2 U2 B2 D F2 R2 U' (13) not
B2 F2 L2 R2 D U R2 F2 D' U' (10) not
F2 L2 F2 R2 F2 U2 F2 D2 U2 R2 (10) not
L2 D2 R2 U2 B2 F2 R2 F2 R2 U2 (10) not
L2 D2 R2 F2 U2 B2 D2 F2 R2 F2 R2 U2 (12) not
L2 D2 B2 U2 F2 L2 B2 R2 F2 U2 R2 U2 (12) not
B2 L2 F2 L2 F2 D2 F2 R2 D2 U2 (10) not
00X 0XX X00 X0X X0X X0X X0X XX0
XX0 0X0 0XX 0X0 0X0 0XX XX0 0X0
X0X X0X X0X 0XX XX0 X00 00X X0X
35 65 8D A3 A6 AC B1 C5
B2 L' D2 L' B2 L U2 F2 U2 R' F' L2 D' L F2 U F' L B R (20) not
D2 L2 U2 F' L2 F R2 F U B' D2 F2 R U' F2 L' U2 B2 D F (20) not
L2 U' L2 B2 U' R2 F R B' D L U B' U' R2 D R2 F L F (20) not
00X 0XX X00 XX0
XX0 0XX 0XX XX0
XX0 X00 0XX 00X
36 6C 8B D1
(none)
00X 0XX X00 X0X X0X XX0 XXX XXX
XX0 0XX 0XX 0XX XX0 XX0 0XX XX0
XXX X0X XXX 0XX XX0 X0X X00 00X
37 6D 8F AB B6 D5 EC F1
D2 R2 F2 L2 F2 D R2 D' R2 U2 F2 U' R2 U' (14) not
D2 B2 D' L2 D F2 U2 R2 U R2 U F2 (12) not
U' L2 U' L2 D2 F2 U' F2 U' F2 R2 B2 L2 (13) not
R2 D F2 U R2 D2 L2 B2 D' B2 U L2 F2 U2 (14) not
F2 U R2 D' F2 R2 U2 L2 D B2 U' (11) not
F2 R2 D' B2 U F2 D2 F2 R2 D' F2 U F2 (13) not
R2 B2 L2 U B2 U R2 D2 F2 U L2 U' B2 U2 (14) not
U2 L2 F2 L2 F2 U F2 U' F2 U2 L2 U' L2 U' (14) not
00X 0X0 X00 XXX
XXX 0X0 XXX 0X0
00X XXX X00 0X0
39 47 9C E2
B2 D2 L R' D2 B2 L R' (8) not
U2 R2 F2 D' U B2 L2 D' U' (9) not
00X 0X0 0X0 0X0 0X0 0XX X00 XX0
XXX 0XX XX0 XXX XXX XX0 XXX 0XX
0X0 XX0 0XX 00X X00 0X0 0X0 0X0
3A 4E 53 59 5C 72 9A CA
D' U2 B2 U2 L2 U B U' L2 B2 R' B2 R F2 D2 F2 D F' (18) not
F2 D' B2 U2 F2 U R2 D B L2 B' R B2 U2 F D2 L' U' F (19) not
D F2 D U2 F2 L2 D' B2 F D L' B2 L2 F D F D2 U2 F2 R (20) not
00X 0X0 0X0 0XX X00 XX0 XXX XXX
XXX 0XX XX0 XXX XXX XXX 0XX XX0
0XX XXX XXX 00X XX0 X00 0X0 0X0
3B 4F 57 79 9E DC EA F2
D2 R2 B2 L2 U2 F2 U2 B2 R2 U2 R2 U2 (12) not
U2 R2 F2 R2 U2 B2 D2 B2 L2 U2 R2 U2 (12) not
U2 L2 F2 R2 U2 B2 D2 B2 R2 U2 R2 U2 (12) not
D2 L2 B2 L2 U2 F2 U2 B2 L2 U2 R2 U2 (12) not
00X 0XX X00 XX0
XXX 0X0 XXX 0X0
X00 XX0 00X 0XX
3C 66 99 C3
(none)
00X 0XX X00 X0X X0X XX0 XXX XXX
XXX 0X0 XXX XXX XXX 0X0 0X0 0X0
X0X XXX X0X 00X X00 XXX 0XX XX0
3D 67 9D B9 BC C7 E3 E6
L2 F2 L2 U R2 D' F2 U' R2 D R2 U R2 U' (14) not
D2 R2 B2 D B2 U R2 B2 D2 F2 D' B2 U B2 (14) not
F2 L2 U2 F2 D' R2 D L2 D2 R2 F2 U' R2 U' (14) not
D2 B2 R2 U B2 U F2 R2 D2 F2 R2 U L2 U' (14) not
B2 L2 R2 D' F2 L2 U' B' F' L D2 F2 R D' U' F' L' R' (18) not
00X 0XX 0XX 0XX X00 XX0 XX0 XX0
XXX 0XX XX0 XXX XXX 0XX XX0 XXX
XX0 XX0 XX0 X00 0XX 0XX 0XX 00X
3E 6E 76 7C 9B CB D3 D9
(none)
00X 0XX X00 XX0 XXX XXX XXX XXX
XXX 0XX XXX XX0 0XX XX0 XXX XXX
XXX XXX XXX XXX 0XX XX0 00X X00
3F 6F 9F D7 EB F6 F9 FC
L2 D' L B2 U' B' L B U B2 L' B D (13) not
U' F2 D' L' U' F U2 L U2 F D R2 F' R' U2 (15) not
U R2 D2 B2 U' F L2 B R D R B R' D' F2 U2 (16) not
0X0
XXX
0X0
5A
U B2 U2 L2 U F2 R2 B2 U' L2 D2 F2 U' B L2 R2 D2 U2 F' (19) continuous
L2 R' B2 F2 D2 B2 F2 L2 R2 U2 R' (11) continuous
0X0 0X0 0XX XX0
XXX XXX XXX XXX
0XX XX0 0X0 0X0
5B 5E 7A DA
D U2 R2 D' U' R D B2 R2 B2 R2 D B2 D2 R U' (16) continuous
0X0 0XX X0X XX0
XXX XX0 XXX 0XX
X0X 0XX 0X0 XX0
5D 73 BA CE
(none)
0X0 0XX XX0 XXX
XXX XXX XXX XXX
XXX 0XX XX0 0X0
5F 7B DE FA
B2 D2 B2 R2 F2 L2 U2 L2 F2 R2 (10) not
B2 D2 B2 L2 U' F2 U' F2 R2 U2 L2 U R2 U' (14) not
B2 L2 D2 L2 U' F2 U B2 U2 F2 R2 U' R2 U' (14) not
D2 L2 D U' L2 F2 D U' F2 U2 (10) not
0XX X0X X0X XX0
XX0 0XX XX0 0XX
X0X XX0 0XX X0X
75 AE B3 CD
D2 U R2 D' F2 U F2 R2 B R2 F2 U2 L' F2 D2 B2 D B' U' (19) continuous
U2 L2 F2 R2 F2 U B2 U' B2 D2 L2 U' L2 U' (14) not
0XX 0XX X0X X0X XX0 XX0 XXX XXX
XX0 XXX XXX XXX 0XX XXX 0XX XX0
XXX X0X 0XX XX0 XXX X0X XX0 0XX
77 7D BB BE CF DD EE F3
L2 U' L2 D' L2 D F' L2 R' U B D2 B' D' U' R U (17) continuous
L2 B2 F2 D2 L2 B2 U R2 U B2 F D' B2 U2 L F2 L D' B' (19) not
0XX XX0
XXX XXX
XX0 0XX
7E DB
(none)
0XX XX0 XXX XXX
XXX XXX XXX XXX
XXX XXX 0XX XX0
7F DF FB FE
U' L2 U F' R2 F U' L2 U F' R2 F (12) continuous
R2 D B2 D' B2 U' B2 U B2 U R2 U' (12) not
X0X
0X0
X0X
A5
D2 F2 U' B2 F2 L2 R2 D R' B F D' U L R D2 U2 F' U2 (19) continuous
R' D2 U2 L2 B2 F2 L' F D' U R2 B2 F2 R' L' B F' U' (18) continuous
B2 F2 L2 R2 D2 U2 (6) * continuous
X0X X0X X0X XXX
0X0 0XX XX0 0X0
XXX X0X X0X X0X
A7 AD B5 E5
F2 U2 B2 F2 L2 U' B2 L D2 F' R2 B L2 R U' R' D' F' R (19) continuous
L2 R2 D2 L2 D' U F L' R U2 B2 U B F' R' D2 R' U2 (18) not
B2 U2 F2 U2 L2 D U' B' U2 L' R B2 D' R2 B' F' L' R (18) not
B2 R2 D U' F2 U2 R2 B D U B' F' L' R' F' D' U R2 (18) not
B2 L2 D' U2 L2 B2 R' B U2 R B L' D' F2 R (15) not
R2 D L2 B2 F2 R2 U' R2 D2 U2 (10) not
R2 F2 D B2 L2 B F L B2 L2 D U' F' L' R D' (16) not
R2 U2 L2 R2 U' R2 D B F' L' B2 R2 D' U B L' R' (17) not
U L2 U2 F2 U F2 R2 F' L F U B' D' R' D R2 D2 F' D (19) not
U B2 F2 R2 D2 U2 R D2 U2 R2 B D U' L R2 B' D' (17) not
D2 U B2 U2 B2 R2 U2 B' R2 D U F U' L R' B L F U' (19) not
U2 L2 B2 R2 U2 B2 R2 D U B U B' L' F2 U' L' U R' D U' (20) not
U F2 L2 R2 F2 L2 R' B D B2 F2 U' B' L' U' (15) not
D' B2 D' L2 F2 D' L' R F D2 L2 F D' U' R B F' (17) not
R2 U' B2 L2 F2 U' L2 D F2 L2 F D F U L U B' U' L' (19) not
D' U' B2 F2 L' R F R2 D' U F2 L' B' F U2 (15) not
R2 D L2 F2 U F2 D R2 B R' D R D2 R2 B' F2 D F R2 (19) not
F2 D' B2 D F2 L2 D2 U L2 U R D R2 D2 L' F' U2 B2 U F (20) not
U2 B U2 R2 D2 B L2 U' B' U L F2 U R F' D' R2 B' R' (19) not
U2 L2 R2 D' F2 U' B2 R F' D R' B' D F R' U L' D2 F U (20) not
D2 B2 U2 R2 B2 R2 U2 F2 R2 U2 (10) not
X0X X0X XXX XXX
0XX XX0 0XX XX0
XXX XXX X0X X0X
AF B7 ED F5
F2 L2 D' R2 B2 L2 R2 F U2 L2 D' L' D' R2 F' D' L' F2 (18) continuous
D2 L2 D' F2 D U F' R2 D' L' R F' L' R' B' U' R2 U2 (18) continuous
D U F2 R' B D2 U2 F' D2 U2 R F2 D' U' (14) continuous
U B2 L B F' L2 R' B' F D U2 L' B2 U' (14) continuous
B2 L2 U' L2 U2 B2 R2 U' B U R' D' L' D2 L D B D U' (19) not
B2 F2 L2 D2 R2 U L F D' L2 R2 D2 U' F' R' D L2 U2 (18) not
R2 U' L2 R2 D B2 D R F' R' B L' R U L' U' F R' (18) not
F2 D2 B2 U L2 B2 L2 D' R' B R' D' L2 B' D' B L2 R2 U' (19) not
B2 D' R2 D R2 D' B R' F R' D L2 F' U L B' L U' (18) not
R2 F2 L2 D' B2 U' B2 L2 F2 L B' L' U' B' D' B D B' D' (19) not
B2 L2 F2 D' L2 B2 D' L B L D F2 D B' D U2 F' U2 (18) not
D L2 F2 D U2 B2 R2 F' D R U2 L2 F' L2 U' R' U B (18) not
U2 R2 F2 D U' B2 R2 B2 R2 D' U' (11) not
D2 L2 B2 D' U B2 R2 B2 R2 D' U' (11) not
D2 B2 D2 L2 D2 L2 U2 F2 R2 F2 R2 U2 (12) not
D2 B2 U2 R2 U2 L2 U2 F2 R2 F2 R2 U2 (12) not
B2 F2 D2 U L2 D2 R D' L2 R D' B2 D F D F' U' (17) not
B2 F2 D' F2 D U R D' L2 R U' L2 U F D F' U' (17) not
R2 U2 B2 L2 F2 R2 D2 F2 U2 F2 R2 U2 (12) not
L2 F2 U2 B2 U2 R2 B2 R2 F2 U2 R2 U2 (12) not
L2 U2 F2 R2 F2 R2 U2 B2 U2 F2 R2 U2 (12) not
R2 F2 D2 F2 D2 R2 B2 L2 F2 U2 R2 U2 (12) not
D2 B2 L2 U2 R2 U2 B2 L2 U2 F2 R2 U2 (12) not
D2 B2 L2 D2 L2 D2 B2 L2 U2 F2 R2 U2 (12) not
U2 R2 F2 D R2 F2 R2 F L D2 L' D' F' L' U2 B2 R' (17) not
B2 L2 U R2 D U' L2 B L R2 D' L' B D L' R2 B L' (18) not
F2 L2 B2 R2 D2 F2 D2 F2 R2 U2 R2 U2 (12) not
F2 R2 B2 R2 U2 B2 U2 F2 L2 U2 R2 U2 (12) not
L2 U2 R2 F2 U2 B2 U2 R2 F2 R2 F2 U2 (12) not
L2 D2 L2 B2 D2 B2 U2 R2 F2 R2 F2 U2 (12) not
F2 R2 B2 R2 D2 F2 D2 F2 L2 U2 R2 U2 (12) not
D' R2 B2 R2 D' R2 B' D' F' L' U' B' U L F D R2 (17) not
L2 U2 R2 D2 R2 U2 B D' U2 R F D2 U2 B' L' D' B' (17) not
L2 B2 D' B2 L2 B2 F R B R2 U F2 R U B U F' (17) not
R2 D F2 D U' R2 L B D L2 R2 D' L' B F2 D' R2 (17) not
B2 D U' L2 D2 F' D U' R F D U' R' D' U' (15) not
L2 D2 L2 D' U' F2 L' D' U B L D' U B' D' U' (16) not
U' L2 F2 L2 D F2 L2 D' U L B' L' D' L2 D B D L (18) not
R2 B2 R2 D' F2 L2 U L' D' L2 R F' R' D R F R' U' (18) not
U' F2 D' F2 L2 D2 U B' L' B D L' U' L' F2 L' U (17) not
D' L2 D' L2 B2 D' B' L' B D L' U' L' F2 L' U (16) not
R2 U F2 L2 B2 U L2 D' R2 F2 R2 U' (12) not
D L2 D' F2 D' R2 U2 R2 U2 B D2 F' R' U R' D2 U B (18) not
R2 B2 D2 F2 D2 R2 F2 L2 F2 U2 R2 U2 (12) not
U F2 R2 F2 D' L2 D U2 B2 L2 F D' R' F R' D R F R' (19) not
L2 B2 L2 B2 D2 R2 U2 R2 B2 U2 F2 U2 (12) not
B2 R2 U2 R2 D2 R2 B F' R2 B' F' (11) not
B2 R2 D2 L2 U2 R2 B F' R2 B' F' (11) not
U' B2 D' L2 U' L2 R2 D L D U' F D F D' U L D (18) not
B2 D' B2 F2 D' L2 U2 B2 R' B R U2 L' F' L' U' F2 L2 U (19) not
X0X XXX
XXX 0X0
X0X XXX
BD E7
L2 B F' L2 R2 B F' R2 (8) not
F2 L2 U2 L2 R2 B' D2 U2 F U2 R2 F2 (12) not
R2 B2 F2 R2 U' B2 F2 D2 L2 R2 U' (11) not
L2 R2 D B2 F2 R2 B2 F2 R2 U' (10) not
U2 B2 R2 D2 U2 R2 F2 U2 (8) not
X0X XXX XXX XXX
XXX 0XX XX0 XXX
XXX XXX XXX X0X
BF EF F7 FD
U R' D' U F2 D U' R' U' (9) * continuous
L2 U' F2 B D' R' D2 R B' F2 L U (12) * continuous
D' B' R2 B' D U' L B2 D2 U2 R D2 U' (13) not
D2 U R' D2 U2 B2 L' D' U B R2 B D (13) not
F2 D' L2 R2 B' L' R D B2 D L' R F' D' F2 (15) not
U F2 U2 F L' U' B' U2 B L U F' U (13) not
U' F2 U F2 R B' U' R' U R B U2 R' F2 (14) not
U' R U L' R B' R' B U' L R' F (12) not
U L2 B2 D' F2 R2 B2 U' F2 U' (10) not
L2 R2 U' L2 R2 D' L' R F2 L R' (11) not
L2 R2 U L2 B2 R2 D' R2 B' L2 U2 R2 F L2 (14) not
F2 D U2 R2 B R2 U2 R2 B R2 D' F2 (12) not
D2 R2 B2 R2 D' R2 B2 R2 D (9) not
F2 R2 U L2 U' R2 F2 L R F' U2 F L' R' (14) not
R2 D B R' D' R' B' R' D B R' (11) * not
R2 D2 B D2 R2 B2 L B2 U2 F2 R F' U2 (13) not
D2 B2 U2 L' U2 B' D2 R2 U2 F U2 R (12) not
L2 D2 F2 D2 R D2 F' R2 D2 L2 B U2 R (13) not
F2 U' F2 D2 B2 D2 F2 U' F2 (9) not
R2 D R2 B R2 D R2 B' R2 D' R2 B' (12) not
D2 B2 R2 D2 R2 B R2 D2 R2 B' D2 (11) not
U2 F2 U2 L2 U2 F U2 L2 U2 F' U2 (11) not
R2 D2 B2 D2 U2 F D2 L2 D2 F' U2 (11) not
D2 R2 B2 L2 U' R2 F2 R2 U L2 R2 (11) not
U F' U2 L D2 B2 U2 R D2 F' U' (11) * not
D' U' B D B U R2 D' L R2 B' L' (12) not
R2 D F D' L' B F L' B' U L F2 R2 (13) not
F2 D' L2 R2 U L R' U2 L R' (10) not
D' R2 B2 R2 D R2 B2 R2 D2 (9) not
D2 B2 R2 D2 R2 D2 R2 U2 F2 U2 (10) not
U R' F L R' D' R D' L' R F U' (12) not
F2 L2 F' L2 R2 B L2 B L2 R2 F (11) not
D2 R2 U2 F2 D2 U2 B' U2 L2 U2 B (11) not
R2 D' L2 B2 L2 D R2 U2 R' F2 D2 F2 R U2 (14) not
D2 R2 U' R2 F2 L2 D R2 B L2 U2 R2 F' D2 (14) not
D U B' R' D2 B' D' R B D' U' B D' (13) not
R2 F2 L2 D' B2 L B2 D2 F2 R' B2 U (12) not
U' R B U' R' U' B' U' R B U2 (11) * not
L2 U L2 F2 R2 D' L2 B' L2 D2 R2 F (12) not
U2 R2 B' L' R' B2 L' R' F' L2 (10) * not
D2 B D2 U2 F D U' R2 D' U' (10) * not
R2 U' L' U' B' L' B U L U B R2 (12) * not
D2 L D2 F' R2 D2 L2 B D2 R' U2 F2 (12) * not
U' L D' U F L R' U' L' R F' D (12) * not
R2 F2 L' F' L' R U L U L R' F R2 (13) not
U' L2 D' L' D' U B D B D U' L U (13) not
D U R' D' U F' D U' R D' U F U2 (13) not
U' L2 B L' R D' L D' L R' B L U (13) not
D U F2 U2 B2 D' U' F D2 U2 B' R2 (12) not
D U2 B2 U B U' B' U' B2 F L' D L F' D' U2 (16) not
B2 F2 R2 D' F2 R2 F2 D' L' U2 F' L2 U2 L2 F' D2 R F2 (18) not
XXX
XXX
XXX
FF
(0) continuous (this is Start)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Mon Mar 30 15:45:45 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA05004; Mon, 30 Mar 1998 15:45:45 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Mar 27 21:26:20 1998
MessageId:
InReplyTo: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net>
Date: Fri, 27 Mar 1998 21:26:52 0500
To: cubelovers@ai.mit.edu
From: Charlie Dickman
Subject: Re: Stickers
>Does anyone know where to find cube stickers? They must come from
>somewhere! I found some vinyl lettering once and the periods were
>exactly the right size for a 5x5x5 cube. But they don't come in
>orange. There must be a way to buy sheets of the stuff. Any ideas?
I have found some adhesive backed vinyl sheets at a local Art Emporium but
they are mostly irridescent shades and you have to cut the pieces to size
yourself. I seem to recall that there was an orange color but I'm not sure.
Charlie Dickman
charlied@erols.com
From cubeloverserrors@mc.lcs.mit.edu Mon Mar 30 16:23:46 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA05149; Mon, 30 Mar 1998 16:23:46 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 29 04:29:11 1998
MessageId: <3.0.5.16.19980329094205.097f34b6@vip.cybercity.dk>
Date: Sun, 29 Mar 1998 09:42:05
To: cubelovers@ai.mit.edu
From: Philip Knudsen
Subject: Eclipse and Pyramorphix
There are two new puzzles out,
by the two most prominent veterans respectively:
1) Rubik's Eclipse, which is some sort of twoplayer game and,
according to the people who have it, a real gem.
2) Pyramorphix, by Meffert.
David Byrden's Twisty Puzzles page shows a picture of
a 2x2x2 Pyraminx together with the text
"A solid version of this amazing puzzle is now available
from Uwe Meffert, called the Pyramorphix".
Now the 2x2x2 pyraminx looks like an old east german puzzle,
which was a 2x2x2 cube in tetrahedral shape. The shape changed
when the puzzle was scrambled, so the name Pyramorphix would
apply. However the east german puzzle was not by Meffert.
Now if anyone knows more about these new puzzles,
or where to get them, please reply.
Philip K
recording and performing artist
Vendersgade 15, 3th
DK  1363 Copenhagen K
Phone: +45 33932787
Mobile: +45 21706731
Email: skouknudsen@email.dk
Email: philipknudsen@hotmail.com
Sms: 4521706731@sms.tdk.dk (short message, no subject)
From cubeloverserrors@mc.lcs.mit.edu Tue Mar 31 10:02:23 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA07294; Tue, 31 Mar 1998 10:02:22 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon Mar 30 21:08:16 1998
MessageId: <19980331020806.13788.qmail@hotmail.com>
XOriginatingIp: [206.114.5.101]
From: "HADER MESA"
To: zot@ampersand.com, rtayek@netcom.com
Cc: cubelovers@ai.mit.edu
Subject: i need information!!!
Date: Mon, 30 Mar 1998 18:08:05 PST
Hello, I am a fond of the cube of Rubik, but in my country it is very
difficult to get it.
She/he would want to know if you can give me information about where I
can get the cube and their different variants.
For the information that you can to give, I thank him a lot.
Cordially: Hader Mesa
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 1 10:55:58 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA10617; Wed, 1 Apr 1998 10:55:57 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 29 18:20:22 1998
Date: Sun, 29 Mar 1998 18:20:42 0400 (EDT)
From: Jerry Bryan
Subject: All the Partial Isoglyphs
To: CubeLovers
MessageId:
I have been able to calculate all the partial isoglyphs a little more
quickly than expected. I can report that there are 10 continuous partial
isoglyphs and 130 noncontinuous partial isoglyphs, unique up to symmetry.
Here is a breakdown of how the solid faces can be arranged.
97  two solid faces, opposite to each other
11  two solid faces, adjacent to each other
25  one solid face
1  three solid faces, mutually adjacent to each other
2  three solid faces, not mutually adjacent to each other
3  four solid faces, other two opposite to each other
1  four solid faces, other two adjacent to each other

140
The partial isoglyphs are all included in the chart which follows. If
nothing is listed with respect to the manner in which the solid faces are
arranged, then there are two solid faces opposite to each other.
Otherwise, the arrangement of the solid faces is listed explicitly.
This chart follows the same format as the previous one I posted for all
the isoglyphs, except that this time I included only a single
representative glyph for each partial isoglyph, rather than the complete
equivalence class of glyphs.
000
0X0
000
00
D B2 F2 D U' L2 R2 U' (8) continuous
000
0X0
00X
01
(none)
000
0X0
0X0
02
D B2 L2 R2 F2 U' L2 B2 F2 R2 (10) not
D' B2 F2 D U L2 R2 U' (8) not
F2 D2 B2 D U L2 D' U' (8) not
B2 D U' L2 D U' (6) * not
D B2 D L2 . B F' D2 R' B2 R2 D' U F L R' (15) not
000
0X0
0XX
03
L2 U2 B2 D' B2 R2 D2 B2 R2 U' B2 L2 U2 R2 U' (15) not
000
0X0
X0X
05
D U F2 D' L2 . B' F U2 L' D U' F2 R2 F' L R' (16) not
D B2 L2 R2 F2 U' L2 B2 F2 R2 U2 (11) not
L2 F2 U B2 F2 U2 B2 F2 U' F2 R2 (11) not
000
0X0
XXX
07
D' B2 F2 D' U L2 R2 U' (8) continuous
000
0XX
0X0
0A
L2 D2 R2 F2 U2 R2 F2 U2 F2 U2 (10) not
D B2 D' U' L2 R2 U2 R2 U' (9) not
000
0XX
0XX
0B
B2 L2 U2 L2 U' L2 B2 D2 F2 U F2 R2 U2 R2 U' (15) not
000
0XX
X00
0C
F2 L2 U2 L2 U L2 F2 D2 B2 U' B2 R2 U2 R2 U' (15) not
000
0XX
X0X
0D
L2 D2 R2 F2 U2 R2 F2 U2 F2 (9) not
B2 R2 F2 D U B2 L2 F2 L2 D U (11) not
F2 R2 B2 D' U' B2 R2 F2 R2 D' U' (11) not
000
0XX
XX0
0E
L2 U2 F2 D F2 R2 D2 F2 R2 U F2 L2 U2 R2 U' (15) not
R2 U2 F2 D' F2 L2 U2 B2 R2 U' B2 R2 U2 R2 U' (15) not
F2 U2 R2 U F2 D2 F2 R2 D' F2 R2 D2 F2 R2 U' (15) not
000
0XX
XXX
0F
B2 D U' L2 D U (6) * not
000
XXX
000
18
D2 U2 (2) * continuous
D' U (2) * continuous
L2 F2 L2 R2 F2 R2 (6) * not
F2 U2 B2 F2 U2 F2 (6) * not
000
XXX
00X
19
(none)
000
XXX
0X0
1A
D F2 R2 B2 F2 R2 D' U R2 U' (10) not
D' B2 U2 B2 L2 R2 D2 F2 L2 R2 D' (11) not
L2 B2 F2 R2 D' U2 L2 B2 F2 R2 U' (11) not
U' L2 R2 U2 L2 R2 U' (7) * not
000
XXX
0XX
1B
B2 D2 L2 U' F2 D2 F2 L2 D F2 L2 U2 B2 R2 U' (15) not
000
XXX
X0X
1D
L2 B2 F2 R2 D U2 L2 B2 F2 R2 U' (11) not
F2 U L2 R2 D2 L2 R2 U' F2 (9) not
000
XXX
XXX
1F
D (1) * continuous
D2 (1) * continuous
00X
0X0
X00
24
D' L2 F2 U2 B2 R2 U2 L2 D U' R2 U' (12) not
00X
0X0
X0X
25
(none)
00X
0X0
XX0
26
B2 L2 D U F2 L2 D U F2 R2 (10) not
L2 D' U' F2 D U . L R' U2 L R (11) not
D' U' F2 D' U . L R' U2 L R (10) not
U' B2 L2 D2 R2 F2 U2 F2 R2 U' (10) not
00X
0X0
XXX
27
L2 D2 R2 B2 U R2 B2 D2 L2 F2 U' F2 D2 R2 U' (15) not
00X
0XX
XX0
2E
(none)
00X
0XX
XXX
2F
L2 D2 B2 D B2 L2 U2 B2 L2 D' B2 R2 U2 R2 U' (15) not
00X
XX0
00X
31
L2 F2 L2 R2 F2 R2 U2 (7) * not
00X
XX0
0X0
32
F2 D2 L2 U B2 D2 B2 L2 D' B2 L2 U2 F2 R2 U' (15) not
00X
XX0
0XX
33
D F2 R2 B2 F2 R2 D' U R2 U (10) not
00X
XX0
X0X
35
F2 L2 D2 R2 D' R2 F2 D2 B2 D' B2 R2 U2 R2 U' (15) not
00X
XX0
XX0
36
(none)
00X
XX0
XXX
37
L2 U2 F2 D F2 R2 U2 F2 R2 D' F2 L2 D2 R2 U' (15) not
B2 U2 F2 L2 D L2 D2 R2 B2 D B2 D2 F2 R2 U' (15) not
F2 R2 D2 L2 D L2 F2 U2 F2 D F2 R2 U2 R2 U' (15) not
00X
XXX
00X
39
U2 F2 U L2 . B' F U2 R' F2 R2 D U' B L R' (15) not
B2 L2 R2 F2 D' L2 B2 F2 R2 U' (10) not
B2 U' B2 L2 R2 F2 D' F2 (8) not
00X
XXX
0X0
3A
B2 U2 R2 U' B2 D2 B2 R2 D B2 R2 D2 B2 R2 U' (15) not
00X
XXX
0XX
3B
L2 D2 L2 F2 U2 R2 B2 U2 F2 (9) not
U' R2 B2 R2 D F2 D' R2 B2 R2 U (11) not
B2 R2 F2 D' U' B2 L2 F2 R2 D' U' (11) not
00X
XXX
X00
3C
D' L2 B2 U2 F2 R2 U2 L2 D' U R2 U' (12) not
F2 R2 D U L2 B2 R2 B2 D' U' F2 R2 (12) not
D' R2 B2 U2 B2 R2 U2 R2 D' U R2 U' (12) not
00X
XXX
X0X
3D
(none)
00X
XXX
XX0
3E
B2 L2 D2 B2 R2 F2 L2 U2 F2 R2 (10) not
D' U' L2 D' U . B F' D2 B F (10) not
U2 L2 . B' L2 D2 U2 R2 F' R2 (9) not
D2 L2 . B' D2 L2 R2 U2 F' R2 (9) not
00X
XXX
XXX
3F
R2 U2 F2 D' F2 L2 D2 B2 R2 D B2 L2 D2 R2 U' (15) not
0X0
XXX
0X0
5A
D F2 R2 F2 D' U R2 F2 R2 U' (10) not
B2 F2 D2 L2 R2 D B2 F2 U2 L2 R2 U' (12) not
0X0
XXX
0XX
5B
(none)
0X0
XXX
X0X
5D
B2 F2 L2 R2 D B2 F2 L2 R2 (9) not
B2 F2 L2 R2 D2 B2 F2 L2 R2 (9) not
0X0
XXX
XXX
5F
D' L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not
L2 B2 D' B2 L2 R2 F2 U' F2 R2 (10) not
0XX
XX0
X0X
75
F2 D2 B2 L2 D L2 U2 L2 F2 D F2 U2 F2 R2 U' (15) not
0XX
XX0
XXX
77
R2 D2 B2 D' B2 R2 D2 F2 L2 D F2 R2 U2 R2 U' (15) not
0XX
XXX
XX0
7E
D' R2 F2 U2 F2 R2 U2 R2 D U' R2 U' (12) not
0XX
XXX
XXX
7F
(none)
X0X
0X0
X0X
A5
D2 R2 U2 L2 R2 U2 R2 U2 (8) not (four solid, other two opposite)
D2 L2 F2 L2 R2 F2 R2 U2 (8) not
X0X
0X0
XXX
A7
D' B2 U' L2 . B' F U2 R' D U' B2 L2 B L R' (15) not
B2 L2 R2 F2 D' U2 L2 B2 F2 R2 U' (11) not
U2 F2 D' U' R2 U2 . R B2 F L' R D L' B2 F2 R B (17) not *1
D2 L2 B2 R2 U' F2 L2 D U' . R B2 U' F' D2 U' R F2 D L' R2 (20) not *2
B2 R2 F2 U' L2 U . R B D2 B' R' D' R' F2 L R2 B2 U' (18) not *2
D F2 L2 F2 D' U' R2 D' R2 . B' D2 B' D' L' U L2 R' U' R' (19) not *2
D' L2 R2 D' U' B2 F2 U' (8) not
F2 L2 D2 B2 U2 B2 F2 R2 F2 U2 (10) not
F2 D' F2 D B2 U B2 F2 U2 L2 U F2 . R B U' B2 U B' R' (19) not
D' F2 R2 B2 F2 R2 D' U R2 U (10) not
*1 two solid faces, adjacent
*2 one solid face
X0X
0XX
XXX
AF
D . F' D2 U2 B R B' D2 U2 F L' D' (12) * continuous *1
R2 U2 . L B L U R' U R' D' F' D' (12) * not *2
L2 U2 R2 D2 R2 U2 (6) * not *4
U F2 D U2 L2 U' F2 . L' U' F D2 U L' F2 U2 B' R (17) not *3
D B2 D' U2 . F D F U' R' U F' U' R' U' B2 (15) not *2
F2 U' L2 U L2 U' B2 . L' U' B U2 B' U L' F2 D' R2 (17) not
R2 D' R2 U' R2 U . R U L F2 D2 L' U' B' D B D (17) not *2
L2 U2 R2 D2 R2 . B' L' B' U' F U' F D R D F2 (16) not *2
B2 D2 B2 U2 F2 L2 D' . F' D' L' U L' U R B R (16) not *2
L2 U2 R2 F2 D2 R2 F2 D2 F2 U2 (10) not
F2 L2 U2 B2 D R2 . B' L' D' L' D' B' U' B' F U' B U2 (18) not
R2 D2 B2 D2 F2 U . F' D' L' D' B D L D F' R2 (16) not *3
U F2 L2 U2 L2 F2 R2 D' U' . B' D' L' B' R' B' R B L (18) not *2
B2 L2 . B' D' L' U B' R2 U' L F' D' F (13) not *2
B2 R2 F2 L2 U2 F2 R2 U2 F2 U2 (10) not
*1 three solid faces mutually adjacent
*2 one solid face
*3 two solid faces adjacent
*4 four solid, other faces opposite
X0X
XXX
X0X
BD
D B2 F2 D' U L2 R2 U' (8) continuous
D2 B2 D2 U2 F2 U2 (6) * not *4
R2 . F' U2 L2 D2 B2 L2 U2 R2 F' R2 (11) * not *2
F2 D2 L2 . F D2 U2 B' R2 U2 F2 (10) * not *3
F2 R2 U2 . B' D2 U2 F D2 R2 F2 (10) * not *1
D2 U2 B2 U2 R2 . F' D2 U2 B L2 U2 F2 (12) not *5
L2 D' U' B2 F2 D' U' R2 (8) not
L2 D2 R2 . B' U2 F U2 L2 U2 F R2 F' (12) not *5
B2 F2 L2 R2 U' B2 F2 D2 L2 R2 U' (11) not
L2 R2 U2 B2 F2 U B2 F2 U2 L2 R2 U' (12) not
L2 R2 D2 L2 B2 U B2 F2 D' F2 R2 U2 (12) not
D B2 L2 B2 D U' R2 F2 R2 U' (10) not
*1  three solid faces, not mutually adjacent
*2  four solid faces, other two faces adjacent
*3  two solid faces, adjacent
*4  four solid faces, other two faces opposite
*5  one solid face
X0X
XXX
XXX
BF
D' U R2 F2 D U' . R' D U' B' L2 B D' U R' (15) continuous
L2 D U . B D' B' U' L2 D L D L' D2 (13) continuous *1
B2 D U' L2 D' U (6) * not
D B2 U2 . L' U2 B2 D2 R' D (9) * not *2
R2 D U' . B D' B' D' U R D R (11) * not *2
F L R' D2 L' R F (7) * not *2
L2 U2 . B U2 L2 D2 F D2 (8) * not *1
L2 . F L R' D2 L' R F L2 (9) * not *2
U2 L2 D2 . B' L2 U2 R2 F' (8) * not *1
L2 D U' . F' L F D' U L' B' L' (11) * not *2
F2 U2 L2 D2 . B' L2 U2 R2 F (9) * not *2
D2 . B' L' R D2 L R' B' D2 (9) * not *3
R2 U2 . B D2 L2 U2 F D2 (8) * not *1
U2 B2 U2 L2 U2 . B D2 R2 U2 F' D2 (11) not *2
D . R B2 F2 L' U' L B2 F2 R' (10) * not *2
D . R' B F' U R' U' B' F R (10) * not *2
D . F' R' B' L' D' L B R F (10) * not *1
B2 D L2 U . R U R' F U2 L D' L B' (13) not *2
R2 D . F D' F' R2 D' B' D B (10) * not *1
D F2 D R2 . F R2 D2 R2 F R2 D F2 D' (13) not *2
D' F2 U2 B2 U2 F2 D' (7) * not
F2 R2 U2 . B' U2 R2 U2 B' U2 F2 (10) * not *2
F2 D2 . F D2 R2 D2 F D2 R2 F2 (10) * not *2
B2 R2 U' L2 U R2 B2 R2 U F2 U' R2 (12) not
D L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not
F' L2 R2 B2 L2 R2 F' (7) * not
L2 . B L' B' D2 R' B' R B D2 L' (11) * not *1
D U' . B F' U' B' F R2 D' U F' (11) * not
*1  2 solid, adjacent
*2  1 solid
*3  3 solid, not mutually adjacent
XXX
XXX
XXX
FF
(none)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Fri Apr 3 17:18:01 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA16814; Fri, 3 Apr 1998 17:18:01 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 1 06:05:52 1998
To: cubelovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (WeiHwa Huang)
Subject: Re: new to list
Date: 1 Apr 1998 09:19:33 GMT
Organization: California Institute of Technology, Pasadena
MessageId: <6ft0r5$6kj@gap.cco.caltech.edu>
References:
John Burkhardt writes:
>The Dodecahedron puzzle is really amazing. It was actually harder
>than the 5x5x5 cube. IT took me about 3 hours to work it out! I
>think once you know the 3x3x3 then all the same moves do similar
>things and you can easily solve 4x4x4 or 5x5x5 with variations. Of
>course there are some cool things you can do with these.
Really?? I found the Dodecahedron significantly easier than the 4x4x4.
The Dodecahedron gives more "space" for moves...

WeiHwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/

Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.
From cubeloverserrors@mc.lcs.mit.edu Fri Apr 3 18:45:49 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id SAA16932; Fri, 3 Apr 1998 18:45:48 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 29 18:56:43 1998
Date: Sun, 29 Mar 1998 18:57:08 0400 (EDT)
From: Jerry Bryan
Subject: Re: All the Partial Isoglyphs
InReplyTo:
To: CubeLovers
MessageId:
On Sun, 29 Mar 1998, Jerry Bryan wrote:
> Here is a breakdown of how the solid faces can be arranged.
>
> 97  two solid faces, opposite to each other
> 11  two solid faces, adjacent to each other
> 25  one solid face
> 1  three solid faces, mutually adjacent to each other
> 2  three solid faces, not mutually adjacent to each other
> 3  four solid faces, other two opposite to each other
> 1  four solid faces, other two adjacent to each other
> 
> 140
>
As this table shows, the vast majority of partial isoglyphs involve two
solid faces opposite to each other. The basic reason for this is the
corners. If the corners are not fixed, then the only partial isoglyphs
which are possible have two solid faces opposite to each other.
Conversely, the 43 partial isoglyphs which do not have two solid faces
opposite to each other do fix the corners.
In fact, 67 of the partial isoglyphs derive from just 5 of the glyphs,
namely those which fix the corners. If the corners of the partial
isoglyph are fixed, you can think of the edges as consisting of a set of
strongly constrained edge flips and swaps. (Be careful  if the corners
are fixed, then *any* resultant position can be thought of as just a bunch
of edge flips and swaps. But for partial isoglyphs, the possible edge
flips and swaps are strongly constrained.)
The glyph which yields the most partial isoglyphs is the one my charts
call BF, whick looks like the following.
X0X
XXX
XXX
With this glyph, each face of a partial isoglyph can have at most one edge
cubie which is swapped or flipped, but on a cubewide basis there are
quite a few different ways to arrange for this to happen.
Another interesting glyph which fixes the corners is called BD on my
charts, and which appears as follows.
X0X
XXX
X0X
As an isoglyph, this glyph yields five different patterns on the 6H
theme. As a partial isoglyph, this glyph yields a number of pretty 2H,
3H, 4H, and 5H patterns. You may also think of the H patterns as
complicated edge swappers/flippers, with exactly zero or two edges
swapped/flipped on each face, and with the coloring requirements for
partial isoglyphs being maintained.
The following two glyphs (A7 and AF in my charts) are in the same spirit
as the H, except that the configuration of the edges on each face which
are swapped/flipped is slightly different than for the H.
X0X X0X
0X0 0XX
XXX XXX
Finally, for completeness in the list of glyphs which fix the corners, the
glyph called A5 on my charts appears as follows.
X0X
0X0
X0X
However, this glyph only yields two partial isoglyphs.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Fri Apr 3 19:32:23 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id TAA16989; Fri, 3 Apr 1998 19:32:23 0500 (EST)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Mar 29 19:35:16 1998
Date: Sun, 29 Mar 1998 19:35:43 0400 (EDT)
From: Jerry Bryan
Subject: Re: partial isoglyphs
InReplyTo: <199708210441.AAA22489@life.ai.mit.edu>
To: CubeLovers
MessageId:
On Thu, 21 Aug 1997, michael reid wrote:
> dan recently introduced the concept of "partial isoglyphs", in which
> some faces are solid, and the others are glyphs of the same pattern.
> i looked into this a little and didn't find much. only the case
> of two opposite solid faces seems to have many possible glyph types,
> although some of these possible types may have many solutions.
>
> here's what i found
Note that all the glyph types which Mike lists (01, 02, 0D, 04, and 03 in
Dan's taxonomy) fix the corners. Thus, his note below points out that in
order to have anything other than two solid faces opposite to each other,
you must fix the corners.
The correspondence between Dan's taxonomy and my charts is 01=BF, 02=AF,
03=A7, 04=A5, and 0D=BD. As I said earlier, the identfication numbers on
my charts are not a taxonomy. Rather, they provide a unique
identification for each of the 2^8 glyphs.
>
> 6 solid faces: start
> 5 solid faces: no possibilities
> 4 solid faces:
> other two faces opposite: types 02, 0D and 04 are possible
All three possibilities do occur in my chart.
> other two faces adjacent: type 0D is possible
This possibility does occur in my chart.
> 3 solid faces:
> mutually adjacent: type 02 is possible
This possibility does occur in my chart.
> not mutually adjacent: types 01 and 0D are possible
Both possibilities do occur in my chart.
> 2 solid faces:
> adjacent: types 01, 02, 0D and 03 are possible
All four possibilities do occur in my chart.
> opposite: many possible types
Indeed!
> 1 solid face: types 01, 02 and 0D are possible
>
All three possibilities do occur in my chart. In addition, I found three
partial isoglyphs of type 03 with one solid face.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Sun Apr 5 16:13:15 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA20772; Sun, 5 Apr 1998 16:13:15 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon Mar 30 20:40:51 1998
Date: Mon, 30 Mar 1998 20:41:13 0400 (EDT)
From: Jerry Bryan
Subject: Pretty vs. NotSoPretty Isoglyphs
To: CubeLovers
MessageId:
After looking at a lot of isoglyphs and partial isoglyphs in the last
little while, I wonder if it's not the case that some of the
noncontinuous isoglyphs are prettier than some of the continuous ones,
and that some of the partial isoglyphs are prettier than some of the
isoglyphs?
Continuous isoglyphs do *in general* seem prettier than noncontinuous
ones, and isoglyphs do *in general* seem prettier than partial isoglyphs.
But consider the following two (counter?) examples.
The glyph
000
XXX
000
yields (among other things) L2 F2 L2 R2 F2 R2, which is a noncontinuous
partial isoglyph. It looks about as follows (quite pretty and striking,
in my opinion):
XXX
XXX
XXX
0X0 0X0 0X0 0X0
0X0 0X0 0X0 0X0
0X0 0X0 0X0 0X0
XXX
XXX
XXX
On the other hand,
U B2 R2 F2 L2 U L2 F2 U2 R' B' R F' L' U2 B2 R2 B' D' U'
is a real mess in my opinion, even though it is a continuous isoglyph. It
looks something like the following.
X00
0X0
XXX
XOX XXX X00 00X
XX0 0X0 0X0 0XX
X00 00X XXX X0X
00X
0XX
X0X
Notice that the partial isoglyph which was my first example "looks" fairly
continuous, even though it really isn't. The reason it looks that way is
that it is continuous along all the edges where the nonsolid glyphs come
together. Call such a noncontinuous partial isoglyph quasicontinuous.
I think your eye tends to ignore the solid faces anyway, so that a
quasicontinuous partial isoglyph tends to be very striking and very
pretty. For example, there are a number of 4H and 4T patterns among the
partial isoglyphs which are quasicontinuous and which are very pretty.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Sun Apr 5 23:28:33 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id XAA21469; Sun, 5 Apr 1998 23:28:32 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun Apr 5 18:06:04 1998
Date: Sun, 5 Apr 1998 18:05:59 0400 (EDT)
From: der Mouse
MessageId: <199804052205.SAA03822@Twig.Rodents.Montreal.QC.CA>
To: cubelovers@ai.mit.edu
Subject: Re: Pretty vs. NotSoPretty Isoglyphs
> On the other hand,
> U B2 R2 F2 L2 U L2 F2 U2 R' B' R F' L' U2 B2 R2 B' D' U'
> is a real mess in my opinion, even though it is a continuous
> isoglyph.
I think this (the pattern, not the operator to produce it) is actually
rather striking and pretty  provided you look at the cube along the
URBLDF cornertocorner axis.
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 8 12:17:06 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA28530; Wed, 8 Apr 1998 12:17:05 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 8 11:04:08 1998
To: CubeLovers@ai.mit.edu
Date: Wed, 8 Apr 1998 07:55:06 0700
Subject: A workable 6x6x6 cube design (probably)
MessageId: <19980408.075506.7150.0.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)
I have found that the 6x6x6 cube can only be made practical if the
outer rows of cubies are slightly larger (about 3mm or 1/8 inch).
If the rows are all the same size then some crosssections of pieces
(e.g. the corner pieces) are less than 3 sqmm, and other pieces are
extremely thin (0.6mm in some places).
If the plastic is black (or white) and the stickers are all the same
size then the inequality in the size of the cubies will be effectively
masked. The stickers would have to be spaced evenly. The cube will
look as if it has a small 'border' but the perception will be that the
cubies are the same size.
This design is actually almost as strong as the 4x4x4 cube. It contains
an internal frame plus 256 movable pieces of ten different types. No
cross section of a piece is smaller than 7 sqmm (the 4x4x4 has center
pieces with 9 sqmm cross section). Two of the types of piece
(FACE EDGE PIECE, SPACER PIECE 2) come in two mirror image forms, so the
number of molds that would be needed to produce this is 14 (counting two
for the internal frame). The internal mechanism would need to be greased
to allow it to turn smoothly, but it should be no worse than the 5x5x5.
The following is an exact geometric description of each piece. To be
able to understand this you need to know how to use Cartesian and Polar
coordinates. All pieces are intersections of planes, spheres, and
hyperboloids (which can probably be approximated as cones).
The SPACER PIECE 2 could probably be replaced by some sort of rectangular
but rounded bloblike thing, it does not need to be an exact shape and
the cube might turn more smoothly if it is rounded. It also might then
be possible to make it symmetrical so they could be produced with a
single mold, which would slightly reduce production cost.
Comments, suggestions and quibbles are welcome.
LEGEND  x,y,z are Cartesian coordinates, r is distance from origin
Dx, Dy, Dz is distance from x, y, z axis respectively
NO TOLERANCES  pieces must be shrunk away from all sides a little bit
DIMENSIONS assume that the size of an inner CUBIE is 100 and the size of
an outer CUBIE is 125, this allows the pieces to be much stronger than if
the cubies were all the same size. The puzzle occupies the space such
that
325175, y>175, z>175, 280320
AND all points such that 0175, z>175, 280360
AND all points such that 100175, z>175, 320175, z>175, 280360
AND all points such that 0175, 320175, 280360
all points such that 100175, 320175, 280360
AND all points such that 0175, 2800, 24060, y>60, z>0, 20030, y>30, z>0, 1000, 2000, 100sqrt(x^2+60^2), Dy>sqrt(y^2+60^2),
Dz>sqrt(z^2+60^2), x>0, y>0, z>0,
200sqrt(x^2+30^2), Dy>sqrt(y^2+30^2),
Dz>sqrt(z^2+30^2), x>0, y>0, z>0,
100120, y>120, z>120, 240120, y>120, 0175, 320
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA01718; Thu, 9 Apr 1998 16:30:17 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 8 18:05:08 1998
To: CubeLovers@ai.mit.edu
Date: Wed, 8 Apr 1998 13:45:07 0700
Subject: A workable 6x6x6 cube design (probably)  correction
MessageId: <19980408.144131.8926.2.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)
Yikes, there were errors in my geometric description.
Here is a (hopefully) correct version:
CORNER PIECE consists of:
all points such that 200320
AND all points such that 0360
AND all points such that 100175, z>175, 280360
AND all points such that 0175, 280360
all points such that 100175, 280360
AND all points such that 00, 24060, y>60, z>0, 20030, y>30, z>0, 1000, 2000, 100sqrt(x^2+60^2), Dy>sqrt(y^2+60^2),
Dz>sqrt(z^2+60^2), x>0, y>0, z>0,
200sqrt(x^2+30^2), Dy>sqrt(y^2+30^2),
Dz>sqrt(z^2+30^2), x>0, y>0, z>0,
100120, y>120, z>120, 240120, y>120, 0
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA11586; Mon, 13 Apr 1998 12:07:53 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sat Apr 11 21:10:52 1998
MessageId: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net>
From: John Burkhardt
To: "CubeLovers@ai.mit.edu"
Subject: RE: A workable 6x6x6 cube design (probably)  correction
Date: Sat, 11 Apr 1998 21:05:25 0400
So who gets to try and make one? I understood that the dies for the
5x5x5 cube are too expensive to build now due to "lack of interest". On
the other hand, we should try to build one because we can. If we can
that is :)
JRB
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 15 15:02:21 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA17097; Wed, 15 Apr 1998 15:02:20 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 15 13:07:56 1998
Date: Wed, 15 Apr 1998 18:07:57 +0100
From: David Singmaster
To: cubelovers@ai.mit.edu
MessageId: <009C4C21.E208C3B3.8@ice.sbu.ac.uk>
Subject: Hamiltonian circuits on the cube
The discussion of isoglyphs, etc., has reminded me of a problem which I
worked on in the early 1980s but never resolved. I took an all white cube and
traced a Hamitonian circuit through all the 54 facelets. If you jumble this
up, it is essentially impossible to restore. Indeed there are probably many
solutions to the problem. This led me to ask some questions about such
Hamiltonian circuits through the 54 facelets.
A. How many are there?
B. Are there any such circuits where the pattern is the same on each
face? I thought I could prove that such did not exist, but I think I assumed
that the circuit entered and left each face once, but this need not be the
case.
I was able to find a circuit with two types of face pattern and the two
types were mirror images. If you index the facelets on a face by 11, 12, ...,
33, then the path on the face is: 11, 12, 22, 21, 31, 32, 33, 23, 13.
If the circuit enters and leaves each face just once, then the sequence
of faces visited forms a Hamiltonian circuit on the faces of the cube, which is
better viewed as the vertices of an octahedron. It is easy to see that there
are just two such circuits on the octahedron (up to isomorphism). One of these
circuits has two kinds of vertex behavior and hence is not suitable.
Does this question interest anyone? The reason for the second question
was that if just one type of face pattern could be used, then it would be easy
to print up stickers for sale  one would just do the same pattern six times!
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171815 7411; fax: 0171815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 15 16:23:24 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA17354; Wed, 15 Apr 1998 16:23:24 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 15 15:35:06 1998
Date: Wed, 15 Apr 1998 15:38:00 0400 (Eastern Daylight Time)
From: Dale Newfield
ReplyTo: DNewfield@cs.virginia.edu
To: cubelovers@ai.mit.edu
Subject: Re: Hamiltonian circuits on the cube
InReplyTo: <009C4C21.E208C3B3.8@ice.sbu.ac.uk>
MessageId:
On Wed, 15 Apr 1998, David Singmaster wrote:
> The discussion of isoglyphs, etc., has reminded me of a problem which I
> worked on in the early 1980s but never resolved. I took an all white cube and
> traced a Hamitonian circuit through all the 54 facelets. If you jumble this
> up, it is essentially impossible to restore. Indeed there are probably many
> solutions to the problem. This led me to ask some questions about such
> Hamiltonian circuits through the 54 facelets.
This is quite reminiscent of "Oddmaze,"
(http://www.edoc.com/zarf/customcubes.html) which is a creation by Andrew
Plotkin realized using Kristin Looney's "Custom Cube Technology"
(http://www.wunderland.com/WTS/Kristin/Technology.html).
On its surface is a labyrinth with no branches or dead ends. Each
facelet has exactly two paths through it. In the "start" position, at
least, the path obeys the Celtic knotwork property (over/under
alternations). It is really quite interesting, and well described on the
above mentioned page.
(This doesn't help answer your questions, but might put you in contact
with another that has given them some thought.)
Dale Newfield
Dale@Newfield.org
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 15 17:12:21 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA17446; Wed, 15 Apr 1998 17:12:21 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 15 16:25:48 1998
Date: Wed, 15 Apr 1998 16:29:26 0400 (EDT)
From: Nicholas Bodley
To: John Burkhardt
Cc: "CubeLovers@ai.mit.edu"
Subject: RE: A workable 6x6x6 cube design (probably)
InReplyTo: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net>
MessageId:
Am I missing something? The geometrical description seemed plausible
and fine, but unless I'm far off base, it seems that some quiteclever
mechanical design is essential. Fairly sure that Douglas Hofstadter
noted in passing (I think in Go"del (G"odel ? :), Escher, Bach...) that
a physical prototype of the 6^3 has been built.
I have pulled apart and studied all "sizes" from the 2^3 to the 5^3,
and the innards of each are rather different; the 5 is based on the 3,
but the 4 (Rubik's Revenge) has a ball inside, as probably most List
readers know. The innards of the 2 are quite distinctive, again; (also,
borderline impossible to assemble/disassemble!). It's remarkable how a
simple increment of one, so to speak, has such a profound effect on the
basic internal design.
My awareness of most abstruse corners of math. is quite comparable with
that of, let's say, a turtle. However, I do know modest bits about
formal kinematics, fourbar linkages, and some underlying principles of
the linkage variety of mechanical analog computers, for instance, so my
ignorance is somewhat better that that of a rock. I also know the
innards of mechanical calculators rather well.
However, with such nonqualifications, I suspect that there is no
theory of such mechanisms as we find inside our cubes and related
puzzles. Mathematicians seem to be able to handle braids (Emil Artin?)
rather well, and knots seem to be doing well, but I really doubt that
there's any significant theory that can be used to develop a design such
as the innards of a 5^3.
Ordinary geometry, I feel fairly confident, is of relatively little
help. One can at least define the geometry of the requisite constraints
and "freedoms" of motion, but to create the requisite shapes, seems to
me, requires a special and clever kind of mind.
Honestly, I'd welcome having big holes figuratively shot through my
contentions! I'm sure I'd learn something.
For limited (and probably very costly) prototype runs, the technology
that goes by various names such as 3D printing, rapid prototyping, and
(ugh!) stereolithography should do well to create the shapes. (Seems to
me it's a fairly formidable challenge to a CAD program to create some
of the weird shapes, but I plead ignorance!
(The "stereo" part of that long word is fine, but it's really
stretching a point to think of it as writing on stone.)
My best to all,
* Nicholas Bodley ** Electronic Technician {*} Autodidact & Polymath
* Waltham, Mass. ** 
* nbodley@tiac.net ** When will the nonword "alot" first be listed
* Amateur musician ** in a dictionary? Maybe 2030?

From cubeloverserrors@mc.lcs.mit.edu Wed Apr 15 18:36:13 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id SAA17618; Wed, 15 Apr 1998 18:36:12 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 15 16:32:14 1998
Date: Wed, 15 Apr 1998 16:35:57 0400 (EDT)
From: Nicholas Bodley
To: John Burkhardt
Cc: "CubeLovers@ai.mit.edu"
Subject: RE: A workable 6x6x6 cube design (probably)  another comment
InReplyTo: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net>
MessageId:
On Sat, 11 Apr 1998, John Burkhardt wrote:
}So who gets to try and make one? I understood that the dies for the
}5x5x5 cube are too expensive to build now due to "lack of interest". On
Does anyone know if the dies still exist? I wouldn't be a bit surprised
if the whole set weighs several tons, even if they are singlecavity
types. Tooling for injection molding is fiercely expensive! (Tooling for
a decent ("serious") plastic soprano recorder runs probably a third to a
half $US million, for instance. (Mostly bigger parts, a few very
critical tolerances, and far fewer parts, also.))
Best,
* Nicholas Bodley ** Electronic Technician {*} Autodidact & Polymath
* Waltham, Mass. ** 
* nbodley@tiac.net ** I might need to switch to shore.net, but will
* Amateur musician ** do my best to minimize the nuisance if so.

From cubeloverserrors@mc.lcs.mit.edu Mon Apr 20 15:57:40 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA00263; Mon, 20 Apr 1998 15:57:40 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon Apr 20 11:51:47 1998
To: cubelovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (WeiHwa Huang)
Subject: Re: Hamiltonian circuits on the cube
Date: 20 Apr 1998 15:55:44 GMT
Organization: California Institute of Technology, Pasadena
MessageId: <6hfr60$lfq@gap.cco.caltech.edu>
References:
David Singmaster writes:
> The discussion of isoglyphs, etc., has reminded me of a problem which I
>worked on in the early 1980s but never resolved. I took an all white cube and
>traced a Hamitonian circuit through all the 54 facelets. If you jumble this
>up, it is essentially impossible to restore. Indeed there are probably many
>solutions to the problem. This led me to ask some questions about such
>Hamiltonian circuits through the 54 facelets.
> A. How many are there?
> B. Are there any such circuits where the pattern is the same on each
>face? I thought I could prove that such did not exist, but I think I assumed
>that the circuit entered and left each face once, but this need not be the
>case.
The answer to B is "Yes"!!
I was pretty surprised to come up with this within ten minutes of reading
the question:
++++
424344
++++
474645
++++
54 3 4
+++++++++++++
 1 2 5 6 7 8262740414853
+++++++++++++
1413121110 9252839384952
+++++++++++++
151617182122242936375051
+++++++++++++
333219
++++
343120
++++
353023
++++
X=====X=====X=====X
H H H H
+ H
H H H  H
X=====X=====X====X
H H H  H
+ H
H H H H
X=====X=====X=====X
H H H H
+ H ++ H
H  H  H  H
X====X====X====X=====X=====X=====X=====X=====X=====X====X====X====X
H  H  H  H H H H H H H  H  H  H
H ++ H ++ H ++ H ++ H  H  H
H H H H H H  H  H  H  H H  H  H
X=====X=====X=====X=====X=====X====X====X====X====X=====X====X====X
H H H H H H  H  H  H  H H  H  H
H ++ H  H  H ++ H  H  H
H  H H H H H H  H  H H  H  H  H
X====X=====X=====X=====X=====X=====X====X====X=====X====X====X====X
H  H H H H H H  H  H H  H  H  H
H ++ H ++ H  H  H ++ H ++ H
H H H H  H  H  H  H  H  H H H H
X=====X=====X=====X====X====X====X====X====X====X=====X=====X=====X
H H H H
H ++ H +
H  H  H  H
X====X====X====X
H  H  H  H
H  H  H +
H  H  H H
X====X====X=====X
H  H  H H
H  H  H +
H  H  H  H
X====X====X====X

WeiHwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/

Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 22 11:53:02 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA00421; Wed, 22 Apr 1998 11:53:02 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 22 11:43:05 1998
Date: Wed, 22 Apr 98 11:42:49 EDT
MessageId: <9804221542.AA10123@sun28.aic.nrl.navy.mil>
From: Dan Hoey
To: whuang@ugcs.caltech.edu
Cc: cubelovers@ai.mit.edu
InReplyTo: <6hfr60$lfq@gap.cco.caltech.edu>
Subject: Re: Hamiltonian circuits on the cube
whuang@ugcs.caltech.edu (WeiHwa Huang) writes:
> I was pretty surprised to come up with this within ten minutes of reading
> the question:
Wow, I'm impressed. I thought I'd have to write a program to find
them, and here's a nice symmetric solution. The symmetry is more
visible in a different unfolding:
+@+@+@++++
 @@@@@ @@@@@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@@@@ @@@@@ 
++++@+@+@++++
 @@@@@ @@@@@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@@@@ @@@@@ 
++++@+@+@++++
 @@@@@ @@@@@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@@@@ @@@@@ 
++++@+@+@+
It shouldn't be that hard to solve a cube with these markingsthere
are only two different kinds of corner cubies, three kinds of edge
cubies, and the face centers need only be oriented mod 180 degrees.
Working from one of the symmetric corners, it's not hard to see that
this is the only continuous solution.
I've noticed a minor modification to your pattern that also admits an
isoglyphic Hamiltonian path:
+@+@+@+@+++
@@ @@@@@  @@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@  @@@@@ @@
+++@+@+@+@+@+++
@@ @@@@@  @@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@  @@@@@ @@
+++@+@+@+@+@+++
@@ @@@@@  @@@@@@@@@ 
+ + + + + + @ +
 @@@@@@@@@@@@@@@@@@@@ 
+ @ + + + + + +
 @@@@@@@@@  @@@@@ @@
+++@+@+@+@+
Anyone who's working on an exhaustive search to see if there are any
others, send me email before I hack again!
Dan
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 22 12:36:05 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA00597; Wed, 22 Apr 1998 12:36:05 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 22 12:19:02 1998
MessageId: <353E1961.6231@sgi.com>
Date: Wed, 22 Apr 1998 09:22:57 0700
From: Derek Bosch
To: Dan Hoey
Cc: cubelovers@ai.mit.edu
Subject: Re: Hamiltonian circuits on the cube  kind of
References: <9804221542.AA10123@sun28.aic.nrl.navy.mil>
On a similar note, has anyone stickers with:

/
 
/

or





(or any of those rotations?) Kind of a cross between a
rubik's Tangle and a rubik's cube? Especially if each of
the lines has a different color?
D

Derek Bosch "A little nonsense now and then
(650) 9332115 is relished by the wisest men"... W.Wonka
bosch@sgi.com
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 22 14:41:52 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA01093; Wed, 22 Apr 1998 14:41:51 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 22 14:20:19 1998
Date: Wed, 22 Apr 1998 14:24:21 0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: Hamiltonian circuits on the cube
InReplyTo: <9804221542.AA10123@sun28.aic.nrl.navy.mil>
To: Dan Hoey
Cc: whuang@ugcs.caltech.edu, cubelovers@ai.mit.edu
MessageId:
On Wed, 22 Apr 1998, Dan Hoey wrote:
> whuang@ugcs.caltech.edu (WeiHwa Huang) writes:
>
> > I was pretty surprised to come up with this within ten minutes of reading
> > the question:
>
> Wow, I'm impressed. I thought I'd have to write a program to find
> them, and here's a nice symmetric solution. The symmetry is more
> visible in a different unfolding:
>
Not to minimize the difficulty of the problem or the beauty of the
solution (quite the contrary), but the solution seems almost trivial
when viewed in the light of Dan's particular unfolding of the surface of
the cube. The same comment is true of Dan's isoglyphic solution.
It makes me wonder of you actually saw Dan's unfolding in your mind's
eye, as it were, as you worked out your solution. Or another way to put
it, did you work out your solution in 2D or in 3D? It also makes me
wonder if there is any other unfolding that would lead as naturally to a
Hamiltonian circuit. I tend to think not, but I could well be wrong.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Thu Apr 23 11:51:20 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA04842; Thu, 23 Apr 1998 11:51:19 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Apr 23 11:42:59 1998
From: whuang@ugcs.caltech.edu (WeiHwa Huang)
MessageId: <199804231547.IAA09346@gluttony.ugcs.caltech.edu>
Subject: Re: Hamiltonian circuits on the cube
To: jbryan@pstcc.cc.tn.us (Jerry Bryan)
Date: Wed, 22 Apr 1998 16:58:41 0700 (PDT)
Cc: cubelovers@ai.mit.edu
InReplyTo: <9804231425.AA10935@sun28.aic.nrl.navy.mil> from "Dan Hoey" at Apr 23, 98 10:25:30 am
ReplyTo: whuang@ugcs.caltech.edu
Jerry Bryan typed something like this in a previous message:
> It makes me wonder of you actually saw Dan's unfolding in your mind's
> eye, as it were, as you worked out your solution. Or another way to put
> it, did you work out your solution in 2D or in 3D? It also makes me
> wonder if there is any other unfolding that would lead as naturally to a
> Hamiltonian circuit. I tend to think not, but I could well be wrong.
>
Actually, I didn't visualize any unfolding at all, so I guess I
did it in 3D. Here's approximately the line of reasoning that
led to my solution.
As Dr. Singmaster notes, there is only one way to draw a Hamiltonian on
a 1x1x1 cube where all the faces are identical, and that is with a right
angle on each face. Naturally one's first impulse is to find a path that
enters each 3x3 face in one place and exits in another  and these two
ends must be on edges 90degree apart. One quickly sees that the two exits
must be on edge cubies, since if any were on corner cubies there would
be a parity problem between "inner corners" and "outer corners." But if
they were edge cubies, then no Hamiltonian path exists (as the inner corner
must join to the ends already).
However, another extension is the "three parallel paths" pattern: put this
on each face:
A B C
  
  +D
 +E
+F
This leads to three paths on the cube, where the center one is the
traditional 1x1x1 Hamiltonian. If this can be rearranged to a solution,
we must try to reconnect the ends so that there is some "interaction"
between the three paths. C must connect to D, but we can connect A to B
instead  and this leads to a solution, which surprised me when I
visualized it on a 3d cube. (I most definitely find visualizing in
3D easier than visualizing the links in an unfolded cube.)

WeiHwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/

Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.
From cubeloverserrors@mc.lcs.mit.edu Thu Apr 23 20:24:58 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id UAA05974; Thu, 23 Apr 1998 20:24:58 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Apr 23 20:22:45 1998
Date: Thu, 23 Apr 98 20:21:11 EDT
MessageId: <9804240021.AA11374@sun28.aic.nrl.navy.mil>
From: Dan Hoey
To: cubelovers@ai.mit.edu
Cc: whuang@ugcs.caltech.edu
Subject: Re: Hamiltonian circuits on the cube
I wrote:
"...send me email before I hack again!"
Too late. The only chiral Hamiltonian isopaths are the two we've
already seen, and:
+++@++@++
 @@@@@ @@@@@@@@ @@
+ @ + @ + + + + @ +
 @ @@@@@@@@@@@@ @ 
+ @ + + + + @ + @ +
@@ @@@@@@@@ @@@@@ 
+++@++@++@+++
 @@@@@ @@@@@@@@ @@
+ @ + @ + + + + @ +
 @ @@@@@@@@@@@@ @ 
+ @ + + + + @ + @ +
@@ @@@@@@@@ @@@@@ 
+++@++@++@+++
 @@@@@ @@@@@@@@ @@
+ @ + @ + + + + @ +
 @ @@@@@@@@@@@@ @ 
+ @ + + + + @ + @ +
@@ @@@@@@@@ @@@@@ 
++@++@+++
I actually generated all the continuous chiral isopaths, and the
following is the other extremethe only one with nine disjoint paths.
Yet one of the paths goes through one third of the facelets.
+@+@++@+@+@+
@@ @@@@@@@@ @ @ 
+ + + + + @ + @ +
@@ @@@@@@@@@@@@ @@
+ @ + @ + + + + +
 @ @ @@@@@@@@ @@
+@+@++@+@+@++@+@+
@@ @@@@@@@@ @ @ 
+ + + + + @ + @ +
@@ @@@@@@@@@@@@ @@
+ @ + @ + + + + +
 @ @ @@@@@@@@ @@
+@+@++@+@+@++@+@+
@@ @@@@@@@@ @ @ 
+ + + + + @ + @ +
@@ @@@@@@@@@@@@ @@
+ @ + @ + + + + +
 @ @ @@@@@@@@ @@
+@+@+@++@+@+
Dan
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Fri Apr 24 09:41:36 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id JAA07001; Fri, 24 Apr 1998 09:41:36 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Apr 24 09:38:22 1998
Date: Fri, 24 Apr 98 09:38:06 EDT
MessageId: <9804241338.AA11821@sun28.aic.nrl.navy.mil>
From: Dan Hoey
To: cubelovers@ai.mit.edu
Cc: whuang@ugcs.caltech.edu
Subject: Re: Hamiltonian circuits on the cube
I wrote:
> I actually generated all the continuous chiral isopaths, and the
> following is the other extremethe only one with nine disjoint paths.
Which was bogus. I actually generated only the continuous chiral
isopaths in which no circuit lies entirely on one face. That's fine
for the Hamiltonian circuit problem, but for the maximum number of
disjoint circuits we probably want the 14circuit pattern
+@+@+@++++
@@ @@@@@  @@@@@ @@
+ + + + @ + @ + @ +
@@ @@@@@  @@@@@ @@
+ @ + @ + @ + + + +
@@ @@@@@  @@@@@ @@
+@+@+@++++@+@+@+
@@ @@@@@  @@@@@ @@
+ + + + @ + @ + @ +
@@ @@@@@  @@@@@ @@
+ @ + @ + @ + + + +
@@ @@@@@  @@@@@ @@
+@+@+@++++@+@+@+
@@ @@@@@  @@@@@ @@
+ + + + @ + @ + @ +
@@ @@@@@  @@@@@ @@
+ @ + @ + @ + + + +
@@ @@@@@  @@@@@ @@
++++@+@+@+
which should be familiar to Tartan fans.
Dan
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Sat Apr 25 20:15:48 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id UAA10540; Sat, 25 Apr 1998 20:15:47 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri Apr 24 14:24:49 1998
Date: Fri, 24 Apr 1998 14:21:43 0400 (Eastern Daylight Time)
From: Dale Newfield
ReplyTo: DNewfield@cs.virginia.edu
To: cubelovers@ai.mit.edu
Subject: 4x4x4 pieces, and in quantity
MessageId:
[ Moderators note: Dale Newfield passes on this notice.
Contact Mike Green for details. ]
Date: Fri, 24 Apr 1998 01:17:15 0700
From: Mike Green
To: Dale Newfield
Cc: Dale Newfield , Dale Newfield
Subject: "Rubik's Revenge"  4x4x4
Dale,
Thank you for your inquiry. We do have a limited number of "Rubik's
Revenge" parts for those of you who have a broken cube:
ITC030a 4x4x4 Center Cubie  Ideal Toy Co. $ 2.50 each
ITC030b 4x4x4 Ball Center  Ideal Toy Co. $10.00 each
ITC030c 4x4x4 Corner Cubie  Ideal Toy Co. $ 2.00 each
ITC030d 4x4x4 Edge Cubie  Ideal Toy Co. $ 2.00 each
ITC030e 4x4x4 Sticker  Ideal Toy Co. $ .50 each
You want 1 corner and 2 centers? You will reuse your stickers? How will
you pay? Postage will probably be $2.00.
Recently the price of a "Rubik's Revenge" has hit as high as $200.00 each
on the "Web". Can you believe that! The last five we sold, fortunately for
our customers, went for $65.00 each. How would you like to see it back in
the market for less than $30.00? Possibly even less than $25.00. Would you
buy more than one? For us to bring it back we have to place a minimum
order of between 10,000 to 30,000 pieces and pay for new tooling  all up
front. Tell your friends and have them tell their friends, and their
friend's friends to get on our wish list. Have your local puzzle retailer
contact us as well. By using the power of the "Internet", email, and word
of mouth I'm sure we can get the numbers up there and make this happen in
less than a year. I'm ready and willing are you?
In the meantime, we also carry as standard stock the Rubik's 2x2x2 for
$5.99, Rubik's 3x3x3 for $10.99, 3x3x3 Magic Cube for $6.99, 5x5x5 for
$38.99, Square 1 for $14.99, and Skewb for $32.
We also pull in on a fairly regular basis Megaminx, Impossiball, Pyraminx,
Mickey's Challenge, Masterballs, and various other sequential movement
puzzles when we can. Prices and quantities vary, but we're always on the
hunt.
We'd very much like to bring the 4x4x4 back to market. You can help
greatly by spreading the word.
Thank you.
Sincerely,
Mike D. Green
President
From cubeloverserrors@mc.lcs.mit.edu Sat Apr 25 21:20:14 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id VAA10646; Sat, 25 Apr 1998 21:20:14 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sat Apr 25 20:50:39 1998
Date: Sat, 25 Apr 98 20:50:27 EDT
MessageId: <25Apr1998.202137.Hoey@AIC.NRL.Navy.Mil>
From: Dan Hoey
To: bosch@sgi.com
Cc: cubelovers@ai.mit.edu
InReplyTo: <353E1961.6231@sgi.com> (message from Derek Bosch on Wed, 22 Apr
1998 09:22:57 0700)
Subject: Re: Hamiltonian circuits on the cube  kind of
Derek Bosch asks for a cross between a Rubik's tangle
and a Rubik's cube. Here's a Hamiltonian chiral isotangle.
.__._____._____.__.__._____._____.__.
 \ : / : \  \ :  : \ 
+. `+' .+. `+. `+++. `+
..\..:../..:..\....\..:....:..\..
  : / : /  / : / :  
+++' .+' .+' .+' .+++
....:../..:../..../..:../..:....
 \ :  : \  \ : / : \ 
+. `+++. `+. `+' .+. `+
.__._____._____.____\__:____:__\____\__:__/__:__\__
 \ : / : \  \ :  : \ 
+. `+' .+. `+. `+++. `+
..\..:../..:..\....\..:....:..\..
  : / : /  / : / :  
+++' .+' .+' .+' .+++
....:../..:../..../..:../..:....
 \ :  : \  \ : / : \ 
+. `+++. `+. `+' .+. `+
.__._____._____.____\__:____:__\____\__:__/__:__\__
 \ : / : \  \ :  : \ 
+. `+' .+. `+. `+++. `+
..\..:../..:..\....\..:....:..\..
  : / : /  / : / :  
+++' .+' .+' .+' .+++
....:../..:../..../..:../..:....
 \ :  : \  \ : / : \ 
+. `+++. `+. `+' .+. `+
__\__:____:__\____\__:__/__:__\__
There's only one path, so it's all one color.
Dan
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Wed Apr 29 10:54:31 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA00312; Wed, 29 Apr 1998 10:48:07 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Wed Apr 29 00:53:36 1998
MessageId: <3546B17A.3419@idirect.com>
Date: Wed, 29 Apr 1998 00:50:02 0400
From: Mark Longridge
To: cubelovers@ai.mit.edu
Cc: cubeman@idirect.com
Subject: Various Cube Thoughts
Ok, I'm back into cubing again... a few interesting, if somewhat
disjoint observations:
Summary of the 3 different types of optimal superflip sequences:
1) Superflip with minimal q turns & symmetric moves
Process has central reflection symmetry
R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 D3 U1
(24q, 22f)
2) Superflip with minimal q turns & asymmetric moves
U1 R2 F3 R1 D3 L1 B3 R1 U3 R1 U3 D1 F3 U1 F3 U3 D3 B1 L3 F3 B3 D3 L3
(24q, 23f)
3) Superflip with minimal f turns & asymmetric moves
U1 R2 F1 B1 R1 B2 R1 U2 L1 B2 R1 U3 D3 R2 F1 D2 B2 U2 R3 L1
(28q, 20f)

No matter which cube you start searching from, e.g. pons asinorum,
12 flip, or any random cube, the dispersion of cubes is the same:
1, 12, 114, 1068, 10011... etc
So much for trying to search backwards from the 12flip to number
the positions from (perhaps) antipode to start!

I have got Mike Reid's optimal solver to work under the dos shell
in windows 95. I finally managed to compile it using WATCOM 11.0
thusly:
wcl386 /k10000000 search.c
I had to give it a 10 megabyte stack for it to work!
It found the sequence ( F R B L )^5 to require 20 q turns, so there
is nothing better. Next I tried ( F R B L )^6 to see if that would
be 24 q but a 20 q solution was found. Mike Reid confirmed the
result on another computer running Linux.

Lastly, some nonmathematical ideas on how to do optimal searches
of rubik's cube patterns. Using my own human solving algorithm
I solve the 4 down edge cubes last. One of the patterns I
get was solved optimally by Mike's program thusly:
D' R' D' F B' D' L' D L D F' B D R
If we assign a value of 1 to each face and add them we get:
D = 6 U = 0
F = 2 B = 2
L = 2 R = 2
Note that most of the action occurs with the D face, which I find
suggestive. After all, nothing is moved except the 4 bottom edge cubes.
Also all the other faces have an even number of turns!
My idea is perhaps with some preprocessing of a goal state it is
possible to prune the number of moves down to such a degree that
the number of moves actually tried is quite small. Also note that
this particular goal state has only 2 pairs of cubes swapped, and
all the other cubes are in place.
Now I may be using too much hindsight, but to me it is counter
intuitive that it is possible to have 3 separate turns of the
D face! So, sequences with 3 uses of the D face should not be
considered. My theory is that ultimately with enough preprocessing
only the optimal sequences will be even considered!
> Mark <
From cubeloverserrors@mc.lcs.mit.edu Thu Apr 30 10:09:06 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA02688; Thu, 30 Apr 1998 10:09:06 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Apr 30 09:59:19 1998
Date: Thu, 30 Apr 1998 09:57:20 0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: Various Cube Thoughts
InReplyTo: <3546B17A.3419@idirect.com>
To: Mark Longridge
Cc: cubelovers@ai.mit.edu
MessageId:
On Wed, 29 Apr 1998, Mark Longridge wrote:
> 
>
> No matter which cube you start searching from, e.g. pons asinorum,
> 12 flip, or any random cube, the dispersion of cubes is the same:
>
> 1, 12, 114, 1068, 10011... etc
>
> So much for trying to search backwards from the 12flip to number
> the positions from (perhaps) antipode to start!
>
> 
This has been discussed before on CubeLovers. There are several ways
to look at why it is true. I think at the most basic level that it
depends on the inverse property of groups. Let A be any nonempty
subset (not necessarily a subgroup) of G, and let x be any element of G.
Then xA contains the same number of elements as A. Hence, if A is (for
example) the set of all positions which are n moves from Start, then xA
is the set of all positions which are n moves from x, and xA is the same
size as A (remember that the distance from Start to a is the same as the
distance from x to xa for any a in A).
Notice that if A is a subgroup of G rather than just being a subset,
then xA is a coset. The fact that cosets are either equal or disjoint,
combined with the fact that A is the same size as xA, constitute the
basis for the proof that the size of a subgroup must divide evenly the
size of the group.
The inverse property is involved in showing that A and xA are the same
size as follows. Suppose we have A={a,b,c} which contains three
elements. Then we have xA={xa,xb,xc} which also appears to contain
three elements. The only way that xA would not have three elements
would be if some of the apparently distinct elements were really the
same, for example if xa and xc were really two different names for the
same element. But if xa=xc, then we have x'(xa)=x'(xc) so that
(x'x)a=(x'x)c so that ia=ic so that a=c. We know by definition that a
and c are distinct. Hence, xa and xc must be distinct.
Just to give one more illustration of the importance of the inverse
property in showing that A and xA are the same size, here is a false
counterexample. Consider the multiplicative group of the real numbers
or of the rational numbers. Suppose A={ 2/3, 3/4, 7} and x=4. Then,
xA={ 8/3, 3, 28}. So far, so good because both A and xA have three
elements. But suppose x=0. Then xA={0, 0, 0}={0} which has only one
element. Here we have A with three elements and xA with only one
element. So what is wrong. The problem is that any multiplicative
group of what we might call "normal" numbers (e.g., real or rational or
complex) must omit zero because 0 does not have a multiplicative
inverse. That is, there is no solution to the equation 0*x=1. So when I
let x=0, I was cheating by multiplying by a number which is not in the
multiplicative group and which does not have a multiplicative inverse.
The reason I know that this has been discussed before was that I was
involved in the discussion. At one point I incorrectly asserted that
what you are calling "the dispersion of the cubes" did depend on which
position was at the root of the search. CubeLovers was quick to
correct me, of course. However egregious was my error, it was still an
honest error. The reason for the honest error is that I accomplish
nearly all my searches by counting patterns (Mconjugacy classes) rather
than by counting positions. And when you count by patterns, "the
dispersion of the cubes" does depend upon which pattern is at the root
of the search. So my mistake was to make a statement about positions
which should have been applied only to patterns.
Your note reminded me of a question I have thought about off and on ever
since that previous discussion. Suppose you are searching by patterns.
Under what circumstances can you start the search with two different
patterns and still have the "dispersion of the cubes" be the same? I
suspect that there is a very simple answer, but I am having trouble
ascertaining what it is. I suspect that the only possibility is if the
two positions differ by superflip, that is if one of them is x then the
other one must be xf=fx, where f is the superflip. But I am simply not
sure if there are any more possibilities. Note that having the two
different patterns be Mconjugate is not an answer to the question
because if two patterns are Mconjugate then they are really just one
pattern.
As a last comment, readers of CubeLovers should be familiar with the
sequence 1, 12, 114... for positions in quarter turn searches. A search
for patterns in quarter turns begins 1, 1, 5... The first 1 is Start.
The second 1 (1q from Start) is Q, the set of twelve quarter turns. The
5 (2q from Start) represents the following five patterns: 1) any face
twisted twice in the same direction, 2) any two opposite faces twisted
once each in the same direction (an antislice), 3) any two opposite
faces twisted once each in the opposite direction (a slice), 4) any two
adjacent faces twisted once each in the same direction (e.g., UF or
U'F'), and 5) any two adjacent faces twisted once each in the opposite
direction (e.g., UF' or U'F). Beyond 2q from Start, it becomes too
complicated to calculate the patterns in my head.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 5397198
10915 Hardin Valley Road (423) 6946435 (fax)
P.O. Box 22990
Knoxville, TN 379330990
From cubeloverserrors@mc.lcs.mit.edu Thu Apr 30 14:16:03 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA03343; Thu, 30 Apr 1998 14:16:03 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu Apr 30 13:01:01 1998
MessageId:
Date: Thu, 30 Apr 1998 12:59:19 0500
To: cubelovers@ai.mit.edu
From: kristin@wunderland.com (Kristin Looney)
Subject: Garden Variety Rubik's Cube
Cube Lovers 
a new cube pic on the image wall... for your viewing pleasure...
http://wunderland.com/EBooks/ImageWall/Pages/GardenVarietyCube.html
Peace 
K.
kristin@wunderland.com
http://www.wunderland.com/wts/kristin
To all the fishies in the deep blue sea, Joy.
From cubeloverserrors@mc.lcs.mit.edu Fri May 1 10:45:38 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA06031; Fri, 1 May 1998 10:45:37 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 1 01:54:06 1998
From: Andrew John Walker
MessageId: <199805010552.PAA00579@wumpus.its.uow.edu.au>
Subject: Square like groups
To: cubelovers@ai.mit.edu
Date: Fri, 1 May 1998 15:52:34 +1000 (EST)
Does anyone have any information on patterns where each
face only contains opposite colours, but are not in the square
group? L' R U2 L R' may be an example. If square moves
are applied to such patterns to form new groups, how many such groups
exist?
Andrew Walker
From cubeloverserrors@mc.lcs.mit.edu Fri May 1 19:58:06 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id TAA07381; Fri, 1 May 1998 19:58:05 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 1 19:57:13 1998
Date: Fri, 1 May 98 19:56:56 EDT
MessageId: <9805012356.AA16835@sun28.aic.nrl.navy.mil>
From: Dan Hoey
To: ajw01@uow.edu.au
Cc: cubelovers@ai.mit.edu
InReplyTo: <199805010552.PAA00579@wumpus.its.uow.edu.au> (message from
Andrew John Walker on Fri, 1 May 1998 15:52:34 +1000 (EST))
Subject: Re: Square like groups
Andrew Walker asks:
> Does anyone have any information on patterns where each
> face only contains opposite colours, but are not in the square
> group?
We may call this the "pseudosquare" group P. It consists of
orientationpreserving permutations that operate separately on the
three equatorial quadruples of edge cubies and the two tetrahedra of
corner cubies, and for which the total permutation parity is even. So
Size(P) = 4!^5 / 2 = 3981312.
> L' R U2 L R' may be an example.
No, that's in the square group, says GAP. Also, Mark Longridge
noticed (8 Aug 1993) that the square group is mapped to itself under
conjugation by an antislice (though I don't recall a proofis there
an easy one?). Your position is (L R)' R2 T2 R2 (L R), so this result
would apply. Does anyone have a square process for it?
> If square moves are applied to such patterns to form new groups, how
> many such groups exist?
Consider the subgroup of P consisting of positions in which the
parity of the corner permutation is even. (The edge permutation will
then also be even, and the parity of the permutations of the two edge
tetrahedrons will be equal). Call it AP, for "alternating P".
Size(AP) = Size(P)/2 = 1990656.
The square group S is a subgroup of index 3 in AP, so
Size(S)=Size(AP)/3=663552. I don't have a very criterion for choosing
elements of AP to be in S, except that it has to do with a correlation
between the permutations of the two tetrahedrons of corners, provided
those permutations are of the same parity (as they must be for the
position to be in AP).
According to GAP, these are the only three possibilities. To be
explicit, let us label the cube's corners
1 D B 3
C 2 4 A
Then we can partition S4 into six cosets:
C1 = { (), (3,4)(1,2), (1,4)(2,3), (2,4)(1,3) }
C3 = { (1,2,3), (1,4,2), (1,3,4), (2,4,3) }
C2 = { (1,3,2), (1,4,3), (2,3,4), (1,2,4) }
C4 = { (1,2), (1,4,2,3), (1,3,2,4), (3,4) }
C5 = { (2,3), (1,4), (1,3,4,2), (1,2,4,3) }
C6 = { (1,3), (2,4), (1,4,3,2), (1,2,3,4) }
and similarly D1,D2,...,D4 for S4 acting on {A,B,C,D}. Now let c be
an arbitrary permutation in P that fixes {A,B,C,D} elementwise, and
let Coset(c) be the coset to which c's operation on {1,2,3,4} belongs.
Let d be an arbitrary permutation in P that fixes {1,2,3,4}
elementwise, and let Coset(d) be the coset to which d's operation on
{A,B,C,D} belongs. Then the group generated by depends only
on Coset(c) and Coset(d):
Coset(d)
D1 D2 D3 D4 D5 D6
Coset(c)
C1 S AP AP P P P
C2 AP S AP P P P
C3 AP AP S P P P
C4 P P P S AP AP
C5 P P P AP S AP
C6 P P P AP AP S
There may be some wisdom to be gained in seeing that C1 is normal in
S4, so S4/C1 is isomorphic to S3. We can represent the Ci and Di by
their action on {1,2,3,A,B,C}. The above table shows whether the
group ,
has order 6, 18, or 24.
I'd love to hear a more explanatory description of this phenomenon,
especially if it explains the absence of a subgroup of index 3 in P.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Sat May 2 17:23:32 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA09204; Sat, 2 May 1998 17:23:31 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 1 22:41:24 1998
MessageId: <354A8671.730D@idirect.com>
Date: Fri, 01 May 1998 22:35:29 0400
From: Mark Longridge
ReplyTo: cubeman@idirect.com
To: Dan Hoey
Cc: cubelovers@ai.mit.edu
Subject: Re: Square like groups
References: <9805012356.AA16835@sun28.aic.nrl.navy.mil>
Dan Hoey wrote:
>
> Andrew Walker asks:
>
> > Does anyone have any information on patterns where each
> > face only contains opposite colours, but are not in the square
> > group?
>
> We may call this the "pseudosquare" group P. It consists of
> orientationpreserving permutations that operate separately on the
> three equatorial quadruples of edge cubies and the two tetrahedra of
> corner cubies, and for which the total permutation parity is even. So
> Size(P) = 4!^5 / 2 = 3981312.
>
> > L' R U2 L R' may be an example.
R2 F2 R2 U2 R2 F2 R2 U2 F2
>
> No, that's in the square group, says GAP. Also, Mark Longridge
> noticed (8 Aug 1993) that the square group is mapped to itself under
> conjugation by an antislice (though I don't recall a proofis there
> an easy one?). Your position is (L R)' R2 T2 R2 (L R), so this result
> would apply. Does anyone have a square process for it?
I almost forgot about all that info back in 1993!
But I hardly think a proof is necessary. After the moves (L' R) all the
following moves are in the square's group. Then we are just doing
the inverse of (L` R) at the end. Not very rigourous, but...
I'll search for a counterexample.
> Mark <
From cubeloverserrors@mc.lcs.mit.edu Sat May 2 18:35:38 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id SAA09349; Sat, 2 May 1998 18:35:38 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sat May 2 18:31:11 1998
Date: Sat, 2 May 98 18:30:59 EDT
MessageId: <9805022230.AA17631@sun28.aic.nrl.navy.mil>
From: Dan Hoey
To: cubelovers@ai.mit.edu
Subject: Re: Square like groups
With respect to the square group, I wrote:
> I'd love to hear a more explanatory description of this phenomenon,
> especially if it explains the absence of a subgroup of index 3 in P.
I should really have waited until I got back home to Singmaster's
book, which has a marvelous explanation of why the squares group has
index 6 in the pseudosquare group.
First, the edges are permuted in in all possible ways consistent with
1. remaining in their "equators" of four edges,
2. not being flipped, and
3. having a permutation parity equal to that of the corners.
so we need only consider the 2x2x2 cube, and then we fix the BLD
corner in place. Corners don't get twisted, so we consider only the
permutation.
We express the generators as permutations of the seven movable
corners, expressed as follows:
2A
/ / \
/ T / \ F^2 = B^2 = (1,4)(B,C),
/ / \ R^2 = L^2 = (1,3)(A,C),
B1 R 3 T^2 = D^2 = (1,2)(A,B).
\ \ /
\ F \ /
\ \ /
4C
The neat part is to notice that the permutation on {A,B,C} is
determined by the permutation on {1,2,3,4}. We do this by
representing these generators as symmetries on a tetrahedron, labelled
as follows.
1C2
\`. .'/
\ `A. .B' /
\ `. .' /
\ `4' /
\ : /
B : A
\ : /
\ C /
\ : /
\ : /
\:/
3
Notice that the symmetry that permutes the tetrahedron's vertex labels
as (1,4) also permutes the edge labels as (B,C), corresponding to F^2
in the cube's action. Similarly (1,3) implies (A,C) and (1,2) implies
(A,B).
With respect to Mark Longridge's having noticed that the square group
is mapped to itself under conjugation by an antislice (L R), the proof
turns out to be pretty easy. First, we notice that we may consider
conjugation by a slice (L R') since that differs by a square (R^2)
from the antislice. Now we work in the group that includes wholecube
orientations, and perform the slice in the mechanically easy way, as a
4cycle of face centers and an equatorial 4cycle of edges. Note that
all the edges of the equator are flipped (with respect to the
orientation that is preserved by the psueudosquare and square groups)
by the slice.
So if S is a squaregroup process that rotates the edges in an equator
E, the process
Slice' S Slice S'
has the following actions:
1. Identity on the corners and the two equators other than E,
because they are not moved by the slice,
2. Identity on the face centers, because they are not moved by S,
3. Flips each edge of E twice (once in Slice' and once in Slice), so
restores the orientation, and
4. Is an even permutation of the edges in E (odd in Slice, odd in
Slice', and equal in S and S').
The even permutation (4) of the edges in E is a slice group process,
as Mark noted, as for instance the 3cycle (R^2 F^2 R^2 T^2)^2 F^2.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
From cubeloverserrors@mc.lcs.mit.edu Mon May 4 10:31:18 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA12732; Mon, 4 May 1998 10:31:18 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sun May 3 17:19:27 1998
MessageId: <199805032117.RAA07495@life.ai.mit.edu>
Date: Sun, 3 May 1998 17:18:53 0400
From: michael reid
To: cubelovers@ai.mit.edu
Subject: Re: Square like groups
andrew walker asks
> Does anyone have any information on patterns where each
> face only contains opposite colours, but are not in the square
> group? L' R U2 L R' may be an example.
the set of such patterns is what i called the "target subgroup" for my
optimal solver. it is the intersection of the three subgroups
, and
(or the intersection of any two of them).
the position he mentions is in the square group (mark longridge gives
a minimal maneuver for it). dan hoey remarks that the square group has
index 6 in this "pseudosquare" group. christoph bandelow's book
"inside rubik's cube and beyond" gives a nice criterion for a pseudo
square pattern to be in the square group.
bandelow's criterion (slightly paraphrased) is
the four U corners must be coplanar, the four F corners
must be coplanar, and the four R corners must be coplanar.
(equivalently, all twelve sets of four coplanar corners
remain coplanar.)
in fact, this forces the parity of the corner permutation to be
even (and thus the same for the edge permutation).
this reminds me of an interesting idea i had for a puzzle: a 3x4x5 box,
whose faces and slices are restricted to 180 degree turns. this sort
of thing could also be done with any dimensions.
mike
From cubeloverserrors@mc.lcs.mit.edu Mon May 4 11:24:21 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id LAA12818; Mon, 4 May 1998 11:24:20 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon May 4 10:36:17 1998
From: "Noel Dillabough"
To:
Subject: Revenge and the 5x5x5
Date: Mon, 4 May 1998 10:35:52 0400
MessageId: <000001bd7769$f0ced480$02c0c0c0@nat>
Since we all know that Rubik's Revenge (4x4x4) puzzles are nearly impossible
to find (all of mine have long ago broken) and the 5x5x5 cubes fall apart so
easily that they are basically unusable.
Well, as a solution to this, I took a Virtual Cube simulation and added
sizing buttons (the cube program supports 2x2x2 to 5x5x5 sized cubes), a
keyboard interface, and allowed it to receive sequences in standard UDFBLR
notation. I also added locking of the center pieces to make using a paired
up Revenge easier.
The cube is located at http://www.mud.ca/cube/cube.html. Any thoughts,
comments, suggestions about the program should be sent to:
mailto://noel@mud.ca.
Enjoy,
Noel Dillabough
From cubeloverserrors@mc.lcs.mit.edu Mon May 4 14:18:15 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id OAA13472; Mon, 4 May 1998 14:18:14 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon May 4 12:28:19 1998
MessageId: <19980504162440.4037.qmail@hotmail.com>
From: "Philip Knudsen"
To: CubeLovers@ai.mit.edu
Subject: Re: Revenge and the 5x5x5
Date: Mon, 04 May 1998 09:24:39 PDT
Noel writes:
> Since we all know that Rubik's Revenge (4x4x4) puzzles are
> nearly impossible to find (all of mine have long ago broken)
> and the 5x5x5 cubes fall apart so easily that they are basically
> unusable.
You are right about the 4x4x4 availability. I have, however,
never had any problems with my 5x5x5 cube. Actually the 5x5x5
mechanism is quite ingenious. I never heard of any broken one.
The only problem i can think of is the orange sticker tendency
fall off.
Philip K
From cubeloverserrors@mc.lcs.mit.edu Mon May 4 15:28:57 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA13664; Mon, 4 May 1998 15:28:57 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon May 4 12:40:08 1998
MessageId: <199805041642.MAA16954@nineCo.com>
To: CubeLovers@ai.mit.edu
Subject: 4x4x4 (Rubik's Revenge) puzzles for sale
ReplyTo: yanowitz@gamesville.com
Date: Mon, 04 May 1998 12:42:51 0400
From: Jason Yanowitz
Hi,
I have 6 Rubik's Revenge puzzles (in the original packaging) that I'm
considering selling. If people are interested in purchasing one, send
me an offer (yanowitz@gamesville.com).
I apologize for the commercial nature of this post, but I've seen a
few other commercial posts.
thanks,
 Jason
[ Moderator's note: Announcements of ontopic stuff for sale is generally
okay, up until it starts clogging the list. I usually snip any
detailed descriptions of the auction process, catalogues of other
products, corporate history, etc.you can get that from Jason
(though he thoughtfully omitted excess in his message). ]
From cubeloverserrors@mc.lcs.mit.edu Mon May 4 15:58:52 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA13709; Mon, 4 May 1998 15:58:52 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Mon May 4 15:11:31 1998
Date: Mon, 4 May 1998 15:09:33 0400 (EDT)
From: Nichael Cramer
To: CubeLovers@ai.mit.edu
Subject: Re: Revenge and the 5x5x5
InReplyTo: <19980504162440.4037.qmail@hotmail.com>
MessageId:
Philip Knudsen wrote:
> You are right about the 4x4x4 availability. I have, however,
> never had any problems with my 5x5x5 cube.
BTW, for interested (and nearby) folks, Games People Play in Harvard Sq
had several 5Xs on the shelf when I dropped through the store last Thurs.
Nichael
From cubeloverserrors@mc.lcs.mit.edu Wed May 6 09:18:29 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id JAA18760; Wed, 6 May 1998 09:18:18 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue May 5 13:57:15 1998
MessageId: <19980505174802.8836.qmail@hotmail.com>
From: "Philip Knudsen"
To: CubeLovers@ai.mit.edu
Subject: Re: Revenge and the 5x5x5
Date: Tue, 05 May 1998 10:48:00 PDT
I suggest people with 5x5x5 that tend to fall apart try and fasten the
small screw underneath the center caps. This might help, at least it did
on mine. Mine didn't fall apart though, it just got loose, and sometimes
the pieces between the corners and the centres would sort of make a
wrong twist. After i tightened the screws that problem disappeared.
Philip K
From cubeloverserrors@mc.lcs.mit.edu Tue May 12 15:55:03 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA06770; Tue, 12 May 1998 15:55:02 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue May 12 14:24:43 1998
MessageId: <35576405.5EC25A05@frontiernet.net>
Date: Mon, 11 May 1998 16:48:05 0400
From: John Bailey
To: CubeLovers
Subject: Solving a 4 Dimensional Rubik's type Cube
Announcing a web page at
http://www.frontiernet.net/~jmb184/solution.html which gives the
explicit steps to solve a challenge configuration for a 2x2x2x2 (that's
four dimensions) Rubik type cube.
The challenge configuration is available at
http://www.frontiernet.net/~jmb184/Nteract4.html.
These pages do NOT require a Java enabled browser however, they do
require Netscape 4.0 or Microsoft Explorer 4.0.
This note is to solicit your judgments regarding the difficulty of the 4
Dimensional Rubik with no edge cubes (2x2x2x2.) I believe it is
relatively easy, provided only that the simulation provides for the
cyclic permutation move, (NE>SW, SW>NW, NW>NE)
Background:
Posted on rec.puzzles dl April 21, 1998:
A four dimensional articulated cube is on the web at
http://www.frontiernet.net/~jmb184/4cube.html
The result of marrying a Rubik's cube with a tesseract, this cube is
2x2x2x2. It has 16 corners and 24 faces. It does not have edge cubes
and the corners have no orientation requirement. Only 4 colors are
used. The solution space is thus roughly equivalent to that of a 3x3x3
Rubik if not smaller. It is rendered in Javascript and will run on
Netscape 3.0 and 4.0
This posting caused about 25 hits to the page, but got no followup
dialog on the rec.puzzles news dl.
Note that in this first version, the corners are only identified by
color, not by correct position.
I wrote the page without having a clue as to how to solve it. In the
process of just testing code I discovered that it is remarkably
unchallenging, once you get a sense of which corners the various
buttons rotate. (Flipping a glove from lefthanded to righthanded can
be done in 4space, but is impossible in 3space.) I may not be an
unprejudiced solver, but I would rate the challenge only slightly
harder than a 15 square slider puzzle. To increase the level of
difficulty, a second version of the puzzle was developed. In this
version, the solution requires that the corners are returned to their
correct location. They still do not requires 4space orientation.
This version was announced in the following posting.
Posted on the rec.puzzles dl May 2, 1998:
A Four dimensional Rubik's Cube with solution.
At http://www.frontiernet.net/~jmb184/Nteract4.html
Redesigned to allow importing of 3D Rubik methods, this version uses
(a slightly extended version of) standard Rubik cube naming of moves
and positions, has a shortcut button for one of the common permutation
moves and a scramble button to provide a challenge position.
I rate the challenge as equivalent to solving two faces of a 3D Rubik
cube. I am looking forward to your comments, opinions, and
suggestions. I am especially interested in positions which cannot be
solved or cannot be solved without extensive permutation moves other
than the one included.
This page has received about 50 hits. But again, there was no
responding dialog on rec.puzzles news dl.
The difficulty of the second version is higher, but I rated the
challenge as equivalent to solving two layers of a 3x3x3 cube. The only
obstacle, an ordinary solver might face, is finding the longish
sequence required to permutate 3 of 4 corners. That's why I provided
the shortcut button (which applies the actions: L'URU'R'LRUR'U' with one
click.)
Discussion:
My concern is that people assume the puzzle is really hard and not worth
the effort. It may be seen as somewhat like the sequences from one
time pads which would be cryptographers who post and ask if anyone can
decrypt them. To make it clear that a solution is not that difficult, I
have now made a page which gives an explicit solution, with
illustrations of each step and even some animation at
http://www.frontiernet.net/~jmb184/solution.html
There are obviously shorter sequences to obtain a solution, however this
one has the value of providing clear checkpoints along the way, such
that a solver can determine if they have missed a twist.
I want and would welcome your judgment about how easy or hard the puzzle
is.
John Bailey
jmb184@frontiernet.net
http://www.frontiernet.net/~jmb184
May 11, 1998
From cubeloverserrors@mc.lcs.mit.edu Tue May 12 17:33:46 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA06959; Tue, 12 May 1998 17:33:46 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Tue May 12 14:24:43 1998
MessageId: <35576405.5EC25A05@frontiernet.net>
Date: Mon, 11 May 1998 16:48:05 0400
From: John Bailey
To: CubeLovers
Subject: Solving a 4 Dimensional Rubik's type Cube
Announcing a web page at
http://www.frontiernet.net/~jmb184/solution.html which gives the
explicit steps to solve a challenge configuration for a 2x2x2x2 (that's
four dimensions) Rubik type cube.
The challenge configuration is available at
http://www.frontiernet.net/~jmb184/Nteract4.html.
These pages do NOT require a Java enabled browser however, they do
require Netscape 4.0 or Microsoft Explorer 4.0.
This note is to solicit your judgments regarding the difficulty of the 4
Dimensional Rubik with no edge cubes (2x2x2x2.) I believe it is
relatively easy, provided only that the simulation provides for the
cyclic permutation move, (NE>SW, SW>NW, NW>NE)
Background:
Posted on rec.puzzles dl April 21, 1998:
A four dimensional articulated cube is on the web at
http://www.frontiernet.net/~jmb184/4cube.html
The result of marrying a Rubik's cube with a tesseract, this cube is
2x2x2x2. It has 16 corners and 24 faces. It does not have edge cubes
and the corners have no orientation requirement. Only 4 colors are
used. The solution space is thus roughly equivalent to that of a 3x3x3
Rubik if not smaller. It is rendered in Javascript and will run on
Netscape 3.0 and 4.0
This posting caused about 25 hits to the page, but got no followup
dialog on the rec.puzzles news dl.
Note that in this first version, the corners are only identified by
color, not by correct position.
I wrote the page without having a clue as to how to solve it. In the
process of just testing code I discovered that it is remarkably
unchallenging, once you get a sense of which corners the various
buttons rotate. (Flipping a glove from lefthanded to righthanded can
be done in 4space, but is impossible in 3space.) I may not be an
unprejudiced solver, but I would rate the challenge only slightly
harder than a 15 square slider puzzle. To increase the level of
difficulty, a second version of the puzzle was developed. In this
version, the solution requires that the corners are returned to their
correct location. They still do not requires 4space orientation.
This version was announced in the following posting.
Posted on the rec.puzzles dl May 2, 1998:
A Four dimensional Rubik's Cube with solution.
At http://www.frontiernet.net/~jmb184/Nteract4.html
Redesigned to allow importing of 3D Rubik methods, this version uses
(a slightly extended version of) standard Rubik cube naming of moves
and positions, has a shortcut button for one of the common permutation
moves and a scramble button to provide a challenge position.
I rate the challenge as equivalent to solving two faces of a 3D Rubik
cube. I am looking forward to your comments, opinions, and
suggestions. I am especially interested in positions which cannot be
solved or cannot be solved without extensive permutation moves other
than the one included.
This page has received about 50 hits. But again, there was no
responding dialog on rec.puzzles news dl.
The difficulty of the second version is higher, but I rated the
challenge as equivalent to solving two layers of a 3x3x3 cube. The only
obstacle, an ordinary solver might face, is finding the longish
sequence required to permutate 3 of 4 corners. That's why I provided
the shortcut button (which applies the actions: L'URU'R'LRUR'U' with one
click.)
Discussion:
My concern is that people assume the puzzle is really hard and not worth
the effort. It may be seen as somewhat like the sequences from one
time pads which would be cryptographers who post and ask if anyone can
decrypt them. To make it clear that a solution is not that difficult, I
have now made a page which gives an explicit solution, with
illustrations of each step and even some animation at
http://www.frontiernet.net/~jmb184/solution.html
There are obviously shorter sequences to obtain a solution, however this
one has the value of providing clear checkpoints along the way, such
that a solver can determine if they have missed a twist.
I want and would welcome your judgment about how easy or hard the puzzle
is.
John Bailey
jmb184@frontiernet.net
http://www.frontiernet.net/~jmb184
May 11, 1998
From cubeloverserrors@mc.lcs.mit.edu Thu May 14 10:52:20 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id KAA10874; Thu, 14 May 1998 10:52:19 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu May 14 09:18:42 1998
Date: Thu, 14 May 1998 14:03:21 +0100
From: David Singmaster
To: CubeLovers@AI.MIT.Edu
Cc: zingmast@ice.sbu.ac.uk
MessageId: <009C62C9.843B05E9.31@ice.sbu.ac.uk>
Subject: New radio programme
TO: FRIENDS AND COLLEAGUES
I am participating in a new weekly program called 'Puzzle Panel' on
BBC Radio 4, beginning on Thursday, 4 June at 1:30. We recorded a pilot in
January and the commissioning producers were delighted with it. There will be
a group of three to five panelists and we will discuss both mathematical and
verbal puzzles. Some will be sent in by listeners and some will be set to the
listeners by the panellists. At the pilot, the panel was myself, Chris
Maslanka (of the Guardian) as chair, William Hartston (of the Independent,
etc.) and Ann Bradford (compiler of a Crossword dictionary), but the membership
may vary. I'll let you know of any changes of time/date, etc.
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171815 7411; fax: 0171815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cubeloverserrors@mc.lcs.mit.edu Thu May 21 13:24:10 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id NAA13269; Thu, 21 May 1998 13:24:09 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu May 21 12:24:47 1998
MessageId: <35646320.2295@ping.be>
Date: Thu, 21 May 1998 18:23:45 +0100
From: Geoffroy Van Lerberghe
To: CubeLovers
Subject: Cristoph's Jewel internal mechanism
The Christoph's Magic Jewel is a disguised Rubik's cube (cf. Metamagical
Themas by Douglas R. Hofstadter p.339 : Stan Isaacs's coloring scheme)
but what about the internal mechanism? Is it simply a Rubik's cube with
only edge and centre cubes or is the mechanism different from the
classic cube.
I haven't managed to disassemble the Magic Jewel yet.
Geoffroy.VanLerberghe@ping.be
From cubeloverserrors@mc.lcs.mit.edu Thu May 21 17:26:51 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id RAA13891; Thu, 21 May 1998 17:26:50 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Thu May 21 14:06:58 1998
MessageId: <19980521180258.25636.qmail@hotmail.com>
From: "Philip Knudsen"
To: CubeLovers@ai.mit.edu
Subject: Re: Cristoph's Jewel internal mechanism
Date: Thu, 21 May 1998 11:02:57 PDT
The Jewel is basically an octahedron, but the vertex pieces are absent.
This does not make the puzzle easier. Apart from the jewel i also have a
taiwanese and a polish made octahedreon (with vertex pieces). A third
version exists, made by Uwe Meffert, but quite rare. The turning quality
of the jewel is very close to that of the polish made octahedron, so i
believe that is where the jewel originates (Correct me if i'm wrong,
Christoph!)
The disassembled polish octahedron has a mechanism very close to that of
the Pyraminx puzzle, also by Uwe Meffert. It is not a cube mechanism.
>The Christoph's Magic Jewel is a disguised Rubik's cube (cf.
Metamagical
>Themas by Douglas R. Hofstadter p.339 : Stan Isaacs's coloring scheme)
>but what about the internal mechanism? Is it simply a Rubik's cube with
>only edge and centre cubes or is the mechanism different from the
>classic cube.
>I haven't managed to disassemble the Magic Jewel yet.
>
>Geoffroy.VanLerberghe@ping.be
____________________________________
Philip K
recording and performing artist
Vendersgade 15, 3th
DK  1363 Copenhagen K
Phone: +45 33932787
Mobile: +45 21706731
Email: philipknudsen@hotmail.com
Email: skouknudsen@get2net.dk
Email: skouknudsen@email.dk
Email: 4521706731@sms.tdm.dk (leave subject blank!)
From cubeloverserrors@mc.lcs.mit.edu Fri May 22 12:26:20 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id MAA16202; Fri, 22 May 1998 12:26:19 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 22 06:24:16 1998
MessageId: <19980522101332.6763.qmail@hotmail.com>
From: "Philip Knudsen"
To: cubelovers@ai.mit.edu
Subject: spare piece for domino variant
Date: Fri, 22 May 1998 03:13:31 PDT
I just received puzzle from a fellow collector:
It is like a Magic Domino, but only about 47 mm along the long
edges. The pieces are red and white. The 9 red pieces have a
drawing of Superman and the 9 white pieces a drawing of Superwoman!
Unfortunately the puzzle was broken on arrival.
Does anyone on the list have a similar broken puzzle,
and maybe could spare a piece (edge)?
____________________________________
Philip K
recording and performing artist
Vendersgade 15, 3th
DK  1363 Copenhagen K
Denmark
Phone: +45 33932787
Mobile: +45 21706731
Email: skouknudsen@get2net.dk
Email: philipknudsen@hotmail.com
Email: skouknudsen@email.dk (soon to expire)
Email: 4521706731@sms.tdm.dk (leave subject blank!)
From cubeloverserrors@mc.lcs.mit.edu Fri May 22 19:06:21 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id TAA17343; Fri, 22 May 1998 19:06:21 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 22 19:02:24 1998
Date: Fri, 22 May 1998 18:59:24 0400 (EDT)
From: Nicholas Bodley
To: Cube Mailing List
Subject: Magic Jack
MessageId:
Sorry if my memory's faulty, but I don't recall any recent mention of
the Magic Jack. This is a 3cubed, 3D array of 26 small cubes
constrained by an outer cage to slide past their neighbors. At first
glance, it looks like a Rubik's Cube, but immediately one realizes it's
quite different. It's about the same size. Disassembly looks impossible
unless the outer "cage" is cut.
As you'd expect, it's a 3^3 array, but with one position empty. It's a
3D analog of the 15 Puzzle. The individual cubes are not connected in
any sense to their neighbors. While the moves in a 15 Puzzle are in one
plane and easily defined by amateur mathematicians, in the Magic Jack,
there are many more possible ways of moving a given cube to another
position.
Also, not surprisingly, cube moves are strictly translational.
The fun begins when one attempts to create patterns. Each cube has
specific surface markings. The simplest configuration creates an
exterior in which all cubes have a random, finegrained, glittery
diffractiongratinglike surface. More complicated, and difficult, are
the colored patterns, which when solved, create (iirc) a continuous path
around the whole puzzle. There are three, I'm fairly sure; one creates a
message. Solving is made more difficult by the fact that most cube
faces are obscured by their neighbors.
As to its intrinsic mathematical difficulty, I'm not close to being
well informed/educated enough to judge. The practical problem of hidden
faces does add to the practical difficulty, and the number of "degrees"
of freedom for a given cube (from 3 to 6, depending on position)
certainly increases the available choices.
I saw this puzzle at Games People Play in Cambridge; it's a German
import. Quality of construction was good, although there was no
detenting, and it could be easier to move the cubes. It might actually
be easier to constrain potential interferers, and let gravity do the
work. The difficulty was essentially caused by other cubes' getting out
of position, not poor quality. Price in the store is $25.
Not sure whether they're interested in mail orders, but it might be
worth a try. While I have no connections with G.P.P., perhaps it
wouldn't be out of order to give some info.:
The Games People Play
1100 Massachusetts Ave. (Abbreviation = Mass. is OK!)
Cambridge, Mass. 02138
(617) 4920711
Afaik, they had possibly as many as a dozen in stock.
G.P.P. also periodically imports 5^3s from Germany, perhaps not from
Dr. Bandelow. They have a nice collection of movablepiece puzzles.
* Nicholas Bodley ** Electronic Technician {*} Autodidact & Polymath
* Waltham, Mass. ** 
* nbodley@tiac.net ** Are you designing an icon for a GUI?
* Amateur musician ** China has been doing it for millennia.

From cubeloverserrors@mc.lcs.mit.edu Mon May 25 15:42:05 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id PAA23077; Mon, 25 May 1998 15:42:04 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 22 19:38:36 1998
Date: Fri, 22 May 1998 19:35:00 0400
MessageId: <22May1998.192434.Alan@LCS.MIT.EDU>
From: Alan Bawden
Sender: Alan@lcs.mit.edu
To: nbodley@tiac.net
Cc: Cubelovers@ai.mit.edu
InReplyTo:
(message from Nicholas Bodley on Fri, 22 May 1998 18:59:24 0400
(EDT))
Subject: Re: Magic Jack
Date: Fri, 22 May 1998 18:59:24 0400 (EDT)
From: Nicholas Bodley
...
Not sure whether they're interested in mail orders, but it might be
worth a try. While I have no connections with G.P.P., perhaps it
wouldn't be out of order to give some info.:
The Games People Play
1100 Massachusetts Ave. (Abbreviation = Mass. is OK!)
Cambridge, Mass. 02138
(617) 4920711
Afaik, they had possibly as many as a dozen in stock.
Check your local puzzle outlet first  Magic Jack may be pretty widely
available. When I was in the hospital last summer, my father brought one
of these with him when he came to vist me from Philadelphia. I don't
recall the name of the store there where he purchased it.
I still haven't solved it. The first step would clearly be to just catalog
the 26 different cubies, but I haven't even done that...
 Alan
From cubeloverserrors@mc.lcs.mit.edu Mon May 25 16:14:10 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA23149; Mon, 25 May 1998 16:14:09 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Fri May 22 20:01:43 1998
Date: Fri, 22 May 1998 19:58:40 0400 (EDT)
From: Nicholas Bodley
To: Alan Bawden
Cc: Cubelovers@ai.mit.edu
Subject: Magic Jack website (!)
InReplyTo: <22May1998.192434.Alan@LCS.MIT.EDU>
MessageId:
Sorry, all; the 'Net still has its surprises. Guess what: The Magic
Jack has its own Web site:
www.magicjack.com
They list the retailers who carry it; there are very roughly a dozen or
so. The site looks worth a visit.
Gosh, Alan, I guess we all should welcome you back, if my recollection's
clear! May you continue to be well!
My best to all,
* Nicholas Bodley ** Electronic Technician {*} Autodidact & Polymath
* Waltham, Mass. ** 
* nbodley@tiac.net ** Are you designing an icon for a GUI?
* Amateur musician ** China has been doing it for millennia.

From cubeloverserrors@mc.lcs.mit.edu Mon May 25 16:48:52 1998
ReturnPath:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP
id QAA23216; Mon, 25 May 1998 16:48:52 0400 (EDT)
Precedence: bulk
ErrorsTo: cubeloverserrors@mc.lcs.mit.edu
Mailfrom: From cubeloversrequest@life.ai.mit.edu Sat May 23 03:33:16 1998
From: canttype@earthlink.net
MessageId:
InReplyTo:
Date: Sat, 23 May 1998 00:34:41 0700
To: