From cube-lovers-errors@curry.epilogue.com Tue Jul 2 00:37:39 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id AAA03069; Tue, 2 Jul 1996 00:37:39 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <31D8CDCC.7AB9@dis.on.ca> Date: Tue, 02 Jul 1996 00:20:44 -0700 From: Mark Longridge Organization: Computer Creations X-Mailer: Mozilla 2.01 (Win16; U) Mime-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Cube Moves Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="MARK1.TXT" > My name is Isidro Costantini, I'm a cube lover since '81. Welcome to cube lovers the mailing list. > ( Where's a place to check for those formulas? ) Well, I'm not finished yet, but I do archive all the cube formulas I get a hold of or compose. Some of the work is with the assistance of computers and/or mathematical insight. http://www.dis.on.ca/~cubeman > Another good example is (xchg 3 edges,noFlip) (12) R2 U1 F1 B3 R2 > F3 B1 U1 R2 (9 moves using your way of counting) and another > equivalent: B3 U3 R3 U1 R1 B1 followed by F1 R1 U1 R3 U3 F3 > (6+6 moves, same position). > Another way of counting could be adding the suffix (1,2 or 3) > (counting only clockwise moves) which would preserve parity as well. > I would be pleased if some one can tell me about this subject. The sequence X = (B3 U3 R3 U1 R1 B1 F1 R1 U1 R3 U3 F3) is a very interesting one. Note that X = B3 [U3 R3] B1 + F1 [R1 U1] F3 The above makes use of conjugates and commutators. The following is a top view of a megaminx (magic dodecahedron): /\ / \ / \ \ U / L \ / R \____/ F Then the very similar sequence R+ F+ U+ F- U- R- L- U- F- T+ F+ L+ ...suffices to also 3-cycle the edges (uf, lf, rf) on the megaminx. In this case I don't like the U3 = U- or U' notation. Clearly on the megaminx U3 <> U' Note that each turn of a face is always turned one way and then back. The 5-period rotation of a face is never used. In special cases like these cube moves from the standard 3x3x3 are directly transferable to the megaminx. I have found that isoflips and isotwists work very well on the megaminx. The shortest flip of 2 adjacent edges uses the same 4 sides (so I say "this sequence has face-index 4), is the following: Note use of L-- and L++ etc to denote 2 one-fifth turns of a face! It is of the form P U1 P' U' which is another commutator. L-- R++ F+ U- R+ U+ L++ R++ U+ R-- L-- U- R- U+ F- R-- L++ U- = 18 face turns or 26 one-fifth turns. Perhaps there is some improvement to this sequence. -> Mark <- From cube-lovers-errors@curry.epilogue.com Wed Jul 3 04:41:56 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id EAA07109; Wed, 3 Jul 1996 04:41:55 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 3 Jul 1996 05:26:31 -0300 From: FERNANDO VON REICHENBACH Message-Id: <199607030826.FAA03614@cnea.edu.ar> To: Cube-Lovers@ai.mit.edu 3/7/96 Hi!, I'm Isidro, yesterday Mark wrote: > The shortest flip of 2 adjacent edges uses the same 4 sides > (so I say "this sequence has face-index 4), is the following: > Note use of L-- and L++ etc to denote 2 one-fifth turns of a face! > It is of the form P U1 P' U' which is another commutator. > > L-- R++ F+ U- R+ U+ L++ R++ U+ > R-- L-- U- R- U+ F- R-- L++ U- > > = 18 face turns or 26 one-fifth turns. Perhaps there is some > .improvement to this sequence. I have a megaminx borrowed and solve it using some of the 3x3x3 knowledge that I have. It took me two years to solve all cases of the 4x4x4 (borrowed from the same friend), but not trying so hard... I suppose you already have this, but just in case I'll send my own flip edges formulas in the standard cube to see if they could help with the other (I guess not): R'F'L'U B'U B L F R U'B U'B' (14) (flips F & B edges) the same formula starting in the 4th move (UB'UBLFRU'BU'B' R'F'L') flips B & L edges. A longer, (but faster in my hands) R B R'L U L'B F'D L'D'UB'U'F B' (16) (flips L & R) R' U2 R2 U R' U' R' U2 L F R F' L' (16) (13 counting ^2 as 1) flips F&R L'B'U R'U'R B L followed by R B U'L U L'B'R' (16) flips B&R and maybe works on megaminx... Hope I'm sending something you don't have. I'm looking for improving this formulas (all of them exchanges (no fliping) 2 corners and 2 edges on the top face: R2 B' R' U' R U R U'B R B'U B R (15/14) xchg BL-FL corn & F-R edg F R'U'R F'L'B U'B'L R'U R / L U F U'F'L' (19) xchg BL-BR corn & R-L edg F R'F'R U R U'R2U'R U R B'R'B U (17/16) xchg Bl-FR corn & L-B edg L'UR'U L U'R U'L F'L' [F F'] U'L'U L F U (17) xchg BL-FR corn & L-F edg R U R'U'R'F R [F' F] R U'R'U'R U R'F' (15/14) xchg RF-RB corn & L-R edg (I've some more but it's enough, besides I must tranlate them from spanish) PS: What a coincidence, my first mail were intended to you, Mark. From cube-lovers-errors@curry.epilogue.com Wed Jul 3 19:36:59 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA08757; Wed, 3 Jul 1996 19:36:59 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 3 Jul 1996 15:17:59 -0300 From: FERNANDO VON REICHENBACH Message-Id: <199607031817.PAA08199@cnea.edu.ar> To: Cube-Lovers@ai.mit.edu Subject: Spanish moves Hi! I'm Isidro, (I'm telling so cause I share my mail address) This is the spanish moving convetion: Up = Arriba Dwn = Bajo (abajo) Lft = Izquierda Rgt = Derecha Front = Frente (Same letter :) Back = Tras (atras) (This is the one which is confusing, cause the same letter "B" means differnt things. And I don't have no formulas on my PC, only some sheets of papers from the '81-'82 and in my head, so a trnaslating program (very easy to do) it's useless, these last days I started thinking formulas movements in english, I guess it's the best choice if we want to exchange things, though I have to recheck everything to avoid mistakes... PS: By the way: Which is the preferred convention: U' U3 U- or what? Where can I obtain programs for trying to find formulas? (I started one myself in Pascal, just interprets moves) Isidro Costantini Zappa/Hendrix/King Crimson music lover Olivos, Bs.As. PC hard/soft technician From cube-lovers-errors@curry.epilogue.com Wed Jul 3 19:39:39 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA08766; Wed, 3 Jul 1996 19:39:39 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 3 Jul 1996 15:46:44 -0300 From: FERNANDO VON REICHENBACH Message-Id: <199607031846.PAA08649@cnea.edu.ar> To: Cube-Lovers@ai.mit.edu Subject: Moves: David wrote: > I chose to count R2 as one move as it takes one hand movement, hence > it takes about the same time as R, rather than twice as long. So my > counting is more appropriate to questions of time or efficiency, Although it is may be one hand move, R2 takes longer than R, if we start thinking that way, I suggest to give different values te each move (ie: R=1 R2=1.3 R'=1.2 L'=1 L=1.2 ...) depending on how long it would take to make that move, it could also depend on the previous move... We could have an 'efficiency coeficient' of a given formula, but I guess that would depend on many subjective factors (ie: if you are right or left-handed). I disagree with that, in our own cube meetings we used to have back in '82 (I was 18 then), we accepted the Q method cause it gave a lot of coherence in ALL formulas, and I'm not a mathemacian or group theorist, (a program will probably do R2=R+R+ taking exactly twice the time of one single move), eventually I rather put both counts in parenthesis, but I definetely choose the Q method. (Look at the samples in my 30/6 mail, I have a LOT more) Isidro Costantini Zappa/Hendrix/King Crimson music lover Olivos, Bs.As. PC hard/soft technician From cube-lovers-errors@curry.epilogue.com Fri Jul 5 16:34:40 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA16099; Fri, 5 Jul 1996 16:34:40 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199607051722.AA11168@foxtrot.rahul.net> To: Cube-Lovers@ai.mit.edu Subject: Cube Moves Date: Fri, 05 Jul 96 10:22:26 -0700 From: jmc@rahul.net Over the last several days I noticed a number of posts about people experimenting with moves and different combinations of moves on the cube, using different notation. I just thought I'd mention a java applet a wrote for a class. The applet allows you to enter moves in Singmaster notation and view the results on a cube. The applet uses fairly standard notation, and I put rather complete instructions on the page. It was fun to write, and neat to play with. If you are interesting in finding out new moves, give it a try. Just playing around, making stuff up, I came up with (r^b[u,l^f]r^b)^4, which uses commutators, conjugates and exponentiation, and translaes to (fu,lu,lr) (I think that's the right answer, but I'm not sure how it's written. Basically the move switches around three edges). The applet also supports capital letter moves, which is a clockwise or counter-clockwise rotation of the whole cube, reorienting which face is f,l,r.. etc. Read the instructions and enjoy. The URL is: http://www.reed.edu/~jmc/project/ Tell me what you think and what needs clarification. Justin -- Cthulhu For President, why vote for the lesser of two evils? http://www.cthulhu.org/jmc/ From cube-lovers-errors@curry.epilogue.com Fri Jul 5 20:09:33 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id UAA16504; Fri, 5 Jul 1996 20:09:32 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Sender: ltaylor@pop.kaiwan.com Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 5 Jul 1996 17:02:18 -0700 To: Cube-Lovers@ai.mit.edu From: "Larry A. Taylor" Subject: Rubik's Cube mailing list I would be interested in descriptions of algorithms and heuristics used for solutions of the Rubik's Cube problem. I have a copy of the Rubik's Math book in which (Freimaster?) the author describes some computer work done in England or Wales (was it Thistlewaite?) There seemed to be no way to contact the author, trace the paper, and even a letter to the publishers of the book go no clues to the "newsletter" mentioned. It would be a great benefit to find out more about these computer methods. A portion of my dissertation work is based on search over the Rubik's cube domain. What is the status of the legal dispute? I was able to buy a cube in a regular store a short while ago, after apparently being absent for many years. LAT Larry A. Taylor, . UCLA Computer Science Dept., Ph.D. candidate . DBA North Circle Software, 13104 Philadelphia St, Suite 208, Whitter, CA 90601. Bus. phone, (310) 698-2739. Fax (310) 698-8164. <75176.1071@compuserve.com>, From cube-lovers-errors@curry.epilogue.com Sat Jul 13 04:22:06 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id EAA00266; Sat, 13 Jul 1996 04:22:06 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <31E75D1A.2754@durham.net> Date: Sat, 13 Jul 1996 01:23:54 -0700 From: Steve Huff Organization: Huff Corp X-Mailer: Mozilla 2.01 (Win16; U) Mime-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Megaminx Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="MEGAMINX.TXT" Well I was on the right track long ago, but now I have confirmation. The megaminx has a slice group, analagous to the cube slice group. All the possible spot patterns are in the megaminx's slice group, e.g. the 10 spot and the 12 spot patterns. With process M1 we may easily generate any spot pattern, although there is much room for improvement. The slice group of the megaminx is generated by turning the faces opposite to each other in the same direction (i.e. same direction looking at a face head-on!) It is a small enough group to seach from head to tail, although the exact details are still being worked on. In the case of process M1, L is opposite to R, not just separated by a face F, as in processes M2 and M3. My original diagram is rather limited, but it does illustrate the idea of L & R separated by F only (as opposed to a real opposites but I have no satisfactory notation). /\ / \ / \ \ U / L \ / R \____/ F Moves for the Magic Dodecahedron (Megaminx) ------------------------------------------- C_U = Rotate entire dodecahedron clockwise via the U face suffix notation: f = face turns u = unit turns M1 10 spot (L1 R3 C_U)^36 (slice group) (72f) M2 3 cycle of edges (uf, lf, rf) R+ F+ U+ F- U- R- L- U- F- U+ F+ L+ (12f) M3 2 flip L-- R++ F+ U- R+ U+ L++ R++ U+ (18f, 26u) R-- L-- U- R- U+ F- R-- L++ U- From cube-lovers-errors@curry.epilogue.com Fri Jul 26 16:39:05 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA19300; Fri, 26 Jul 1996 16:39:05 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <01BB7B0E.AA2487E0@dialup-17.flnet.com> From: Christopher Pelley To: "'Cube Lovers'" Subject: Ray-traced cubes Date: Fri, 26 Jul 1996 16:22:18 -0400 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable For those interested in three-dimensional graphics of Rubik's Cubes, I = have made a few nice-looking images and put them at: http://www.flnet.com/~ck1 The images are 1024x768 JPEG files. They were inspired by a ray-traced = image I found on America Online a couple years ago. There is also a = scanned image of cube advertisements from the early 80's. Enjoy! Chris Pelley ck1@flnet.com From cube-lovers-errors@curry.epilogue.com Mon Aug 5 22:41:01 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA22841; Mon, 5 Aug 1996 22:41:00 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com From: bagleyd Message-Id: <199608060132.VAA14816@hertz.njit.edu> Subject: panex puzzle To: cube-lovers@ai.mit.edu Date: Mon, 5 Aug 96 21:32:18 EDT X-Mailer: ELM [version 2.3 PL11] Hi I just made a new puzzle for the X Window System and MS Windows 3.1 or greater. The new puzzle is Panex which is very similar to (but a lot harder than) the Tower of Hanoi. In fact there is a Hanoi mode in the puzzle. My wife who is usually indifferent towards puzzles, liked this one. I also made updates to my other puzzles; the rubik, dino, and skewb puzzles. You can pick this stuff up at http://hertz.njit.edu/~bagleyd/ Source code and README files are also supplied. Cheers, /X\ David A. Bagley // \\ bagleyd@hertz.njit.edu http://hertz.njit.edu/~bagleyd/ (( X xlockmore, new stuff for xlock @ ftp.x.org//contrib/applications \\ // altris, tetris games for x @ ftp.x.org//contrib/games/altris \X/ puzzles, magic cubes for x @ ftp.x.org//contrib/games/puzzles From cube-lovers-errors@curry.epilogue.com Tue Aug 6 14:37:19 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA24897; Tue, 6 Aug 1996 14:37:18 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Tue, 06 Aug 1996 08:58:23 -0500 (EST) From: Jerry Bryan Subject: Commuting Sets To: Cube-Lovers Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 7BIT If X and Y are sets of permutations, we define XY to be the set {xy | x in X and y in Y}. In my various search programs, I have encountered a number of cases where we have XY=YX, even though we do not in general have xy=yx. For example, let Q[n] be the set of all positions which are n quarter turns from Start. My standard breadth first search is essentially Q[n+1] = Q[n]Q[1] - Q[n-1]. But we could just as well say Q[n+1] = Q[1]Q[n] - Q[n-1] because Q[n]Q[1] and Q[1]Q[n] are the same set. I have been wondering, what are the necessary and sufficient conditions for XY = YX? Note that X and Y are not necessarily groups. I really don't know the answer, and I wondered if anybody out there does. I have some suspicions it has something to do with conjugacy. In all the cases I have worked with, it it the case that if x in X and y in Y, then all the K-conjugates of x are also in X and all the K-conjugates of y are also in Y -- where K is usually M, the set of 48 rotations and reflections of the cube. For other searches such as , K is the symmetry group associated with the group being searched. It is trivial to make an X and Y that don't "commute" in this matter. That is, pick x and y that don't commute and have sets X and Y containing only the single elements x and y, respectively. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7127 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@curry.epilogue.com Fri Aug 30 15:44:49 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id PAA13576; Fri, 30 Aug 1996 15:44:49 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Organization: Centro di Calcolo - Dipartimento di Informatica di Pisa - Italy From: Mario Velucchi Message-Id: <199608301638.SAA11650@helen.cli.di.unipi.it> Subject: Chameleon CUBE (I think a NEW <<< from Hungary) To: cube Date: Fri, 30 Aug 1996 18:38:42 +0200 (MET DST) Cc: Mario VELUCCHI X-Mailer: ELM [version 2.4 PL24] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Dear "Cube fans" friends, in these days I have received from the Hungary the "Chameleon Cube". I am not very expert of cubes but I think this is a Magyar new. For more information/references write to: --------------- Blazsik ZOLTAN 6701 SZEGED Pf.:1298 HUNGARY --------------- I think this a news but ... if this cube is best known ... i am sorry for the trouble! Best, Mario VELUCCHI -- Best Regards, MV \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// Mario Velucchi University of PISA Via Emilia, 106 Department of Computer Science I-56121 Pisa e-mail:velucchi@cli.di.unipi.it ITALY talk:velucchi@helen.cli.di.unipi.it http://www.cli.di.unipi.it/~velucchi/intro.html \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// From cube-lovers-errors@curry.epilogue.com Sat Aug 31 16:01:10 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA18411; Sat, 31 Aug 1996 16:01:09 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Authentication-Warning: coronado.nadn.navy.mil: wdj owned process doing -bs Date: Sat, 31 Aug 1996 08:42:30 -0400 (EDT) From: Assoc Prof W David Joyner X-Sender: wdj@coronado To: cube-lovers@ai.mit.edu cc: Assoc Prof W David Joyner Subject: cube programs, etc In-Reply-To: <199608301638.SAA11650@helen.cli.di.unipi.it> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Hello cube lovers: Several things: 1. I have written computer programs in maple for (a) the skewb,(b) the rainbow masterball, (c) the 3x3 Rubik's cube, and (d) the 4x4 Rubik's cube. You must have maple (http://www,maplesoft.com) to run it and the program simulates any move of one of the above puzzles using maple's 3-d graphics. The idea is that, using one of these programs, you can "virtually" make a move, the program draws the cube in 3-space, and maple allows you to rotate the cube around with your mouse (assuming you have the windows version of maple). These programs do not solve the puzzle, only simulates the moves. It appears to be possible, with some work, to link these programs with gap to provide a solution as well, but I don't have the time to do that. 2. Andrew Southern from London (whom I've lost touch with) and I worked out a fairly simple collection of moves to help solve the rainbow masterball. These are available. Apparently 2-cycle exist on the masterball, unlike the Rubik's cube. We do not know of a relatively short expression for one. If anyone out there knows of one please let me know. 3. This stuff can be found on my www page http://www.nadn.navy.mil/MathDept/wdj/myhome.html under "computer programs" and "Rubik's cube like puzzles". If there are any problems loading them I'll try to help. - David PS: FYI, Ishi Press International has moved recently. They are having a puzzle sale as well (their phone is (800)859-2086 or (415)323-6996). I think they have some cheap skewbs. Also, on a recent business trip I stopped by Puzzletts store in downtown Seattle - the best puzzle or game store I've ever seen. Their www address has been posted recently in this list so I won't repeat it. From cube-lovers-errors@curry.epilogue.com Mon Sep 2 19:33:10 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA03526; Mon, 2 Sep 1996 19:33:09 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Mon, 2 Sep 1996 22:50:07 +0300 (IDT) From: Rubin Shai X-Sender: s2394459@csc To: Assoc Prof W David Joyner cc: cube-lovers@ai.mit.edu, Assoc Prof W David Joyner Subject: Re: cube programs, etc In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Hi all I have a computer program that learn to solve the 2X2X2 cube. I mean that after several hours of 'learning' the program knows to solve any legal start position of this cube. Before learning the program solve the cube after about 15 minuets, after learning it takes about 5 seconds. The letter from Prof Joyner made me think about the following things: 1. Does anyone have a program (in C) that can take a move (a string or a line from a file) and show it on the display. 2. Does anyone know about similar programs to my. Program that 'learn' to solve the cube by themselves. Shai From cube-lovers-errors@curry.epilogue.com Fri Sep 6 13:06:11 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id NAA02196; Fri, 6 Sep 1996 13:06:10 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Fri, 6 Sep 1996 07:31:48 -0400 (EDT) From: Assoc Prof W David Joyner X-Sender: wdj@coronado Reply-To: Assoc Prof W David Joyner To: Rubin Shai cc: cube-lovers@ai.mit.edu Subject: Re: cube programs, etc In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Mon, 2 Sep 1996, Rubin Shai wrote: > Hi all > I have a computer program that learn to solve the 2X2X2 cube. I mean that > after several hours of 'learning' the program knows to solve any legal > start position of this cube. Before learning the program solve the cube > after about 15 minuets, after learning it takes about 5 seconds. > The letter from Prof Joyner made me think about the following things: > 1. Does anyone have a program (in C) that can take a move (a string or a > line from a file) and show it on the display. I have no C programs for the cube but MAPLE has a MAPLE-to-C conversion, but one would have to write their own display. I don't have a 2x2 program in MAPLE but I'm saving that project for a student since it is relatively easy, given that I have one for the 3x3 and 4x4 cubes. > 2. Does anyone know about similar programs to my. Program that 'learn' to > solve the cube by themselves. This is much more serious than anything I have. My programs are simply "virtual" cubes with no brains. Sounds like your program gives the cube a brain! Maybe you could post more details. I don't understand how it works. - David Joyner > Shai > > > From cube-lovers-errors@curry.epilogue.com Sat Sep 7 19:51:21 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA06720; Sat, 7 Sep 1996 19:51:21 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Sender: ltaylor@pop.kaiwan.com Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 6 Sep 1996 15:08:33 -0700 To: cube-lovers@ai.mit.edu From: "Larry A. Taylor" Subject: Re: cube programs, etc Cc: Rubin Shai >Hi all >I have a computer program that learn to solve the 2X2X2 cube. I mean that >after several hours of 'learning' the program knows to solve any legal >start position of this cube. Before learning the program solve the cube >after about 15 minuets, after learning it takes about 5 seconds. >The letter from Prof Joyner made me think about the following things: >1. Does anyone have a program (in C) that can take a move (a string or a >line from a file) and show it on the display. >2. Does anyone know about similar programs to my. Program that 'learn' to >solve the cube by themselves. >Shai Dr. Richard Korf (korf@cs.ucla.edu) included demonstrations of macro learning on the Rubik's Cube in his dissertation, and in his book on "Learning Macro Operators." He may still have his C language code for this available somewhere. I have used the 2x2x2 and 3x3x3 cube in my work on "Pruning Duplicate Operators in Depth-First Search." Most available format is Proceedings AAAI-93 (Wash. DC), or from my web page area. I do not learn to solve the cube, but learn about the cube state space to speed search. Neither of our programs produce graphical output. I may make a Cube page with a Java applet, unless one of you do it first. LAT Larry A. Taylor, . UCLA Computer Science Dept., Ph.D. candidate . DBA North Circle Software, 13104 Philadelphia St, Suite 208, Whitter, CA 90601. Bus. phone, (310) 698-2739. Fax (310) 698-8164. <75176.1071@compuserve.com>, From cube-lovers-errors@curry.epilogue.com Wed Sep 11 17:04:33 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id RAA05135; Wed, 11 Sep 1996 17:04:33 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Organization: Centro di Calcolo - Dipartimento di Informatica di Pisa - Italy From: Mario Velucchi Message-Id: <199609111154.NAA11311@helen.cli.di.unipi.it> Subject: Chameleon Cube (E-Mail address) To: cube Date: Wed, 11 Sep 1996 13:54:03 +0200 (MET DST) X-Mailer: ELM [version 2.4 PL24] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit after my precedent e-mail i have received more answers/questions for to know the E-mail address of my hungarian friend, this is the old message with the E-mail address: Dear "Cube fans" friends, in these days I have received from the Hungary the "Chameleon Cube". I am not very expert of cubes but I think this is a Magyar new. For more information/references write to: --------------- Blazsik ZOLTAN 6701 SZEGED Pf.:1298 HUNGARY --------------- blazsik@inf.u-szeged.hu --------------- I think this a news but ... if this cube is best known ... i am sorry for the trouble! Best, Mario VELUCCHI -- Best Regards, MV \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// Mario Velucchi University of PISA Via Emilia, 106 Department of Computer Science I-56121 Pisa e-mail:velucchi@cli.di.unipi.it ITALY talk:velucchi@helen.cli.di.unipi.it http://www.cli.di.unipi.it/~velucchi/intro.html \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// From cube-lovers-errors@curry.epilogue.com Thu Sep 26 22:37:26 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA10261; Thu, 26 Sep 1996 22:37:25 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Organization: Centro di Calcolo - Dipartimento di Informatica di Pisa - Italy From: Mario Velucchi Message-Id: <199609261617.SAA20286@helen.cli.di.unipi.it> Subject: WWW devoted to Recreational Mathematics (CUBE, too ...) To: cube Date: Thu, 26 Sep 1996 18:17:51 +0200 (MET DST) X-Mailer: ELM [version 2.4 PL24] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Dear Friend, are you interested to Recreational Mathematics and related fields? If so, see this my new WWW address: http://www.geocities.com/SiliconValley/9174/material.html I think you will find a lot of interesting items. Best Regards, Mario VELUCCHI -- Best Regards, MV \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// Mario Velucchi University of PISA Via Emilia, 106 Department of Computer Science I-56121 Pisa e-mail:velucchi@cli.di.unipi.it ITALY talk:velucchi@helen.cli.di.unipi.it http://www.cli.di.unipi.it/~velucchi/intro.html \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// From cube-lovers-errors@curry.epilogue.com Mon Sep 30 23:21:04 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id XAA12380; Mon, 30 Sep 1996 23:21:03 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Mon, 30 Sep 1996 22:29:36 -0500 (EST) From: Jerry Bryan Subject: Solving One Cubie To: Cube-Lovers Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII Content-transfer-encoding: 7BIT I've been thinking about a simple little problem I thought I would share. Most of the solution is in the archives, but under other guises. Suppose you scramble a cube and give it to a cubemeister with instructions to solve any one cubie. This is a truly trivial problem, but let's see what it can teach us. The most obvious question is -- what is God's algorithm? That is, from any position, what is the minimal solution? The cubemeister would observe that for any position, each of the eight corner cubies and each of the twelve edge cubies has its own individual minimal solution which is easy to discover. The cubemeister would then choose the cubie with the smallest minimal solution and solve it. Given this simple technique for God's algorithm, what is the maximal position? That is, what is the position where the minimal solution is as large as possible? We start with the edges. The solution is in the archives in two separate articles. On 6 August 1980, David Vanderschel introduced the concept of Oriented Distance from Home (ODH). On 7 January 1981, Dan Hoey used the ODH concept to show that the Pons Asinorum position requires exactly twelve quarter turns for solution. But for our purposes, the salient point is that an edge cubie can be at most four quarter-turns from home. There is exactly one such position for each edge cubie. And the only position for which each edge cubie is four quarter-turns from home is the Pons. So for our trivial little problem, the maximal position for the edges is the Pons. I have found little information in the archives concerning the same problem for the corners. (By the way, I have this vision in my mind that the information for the corners is in there somewhere, but I cannot find it, neither in the archives nor in Singmaster. Am I remembering a mirage, or is it in there somewhere and I can't find it?). Vanderschel does not define an Oriented Distance from Home for corners, but the generalization is obvious. The following are the ODH values for the f facelet of the flt cubie. 1+2 +T+ 2+3 l+2 0+1 1+2 2+3 +L+ +F+ +R+ +B+ 2+3 1+2 2+3 3+2 1+2 +D+ 2+3 The maximum distance from Start for any particular corner cubie is therefore three quarter-turns. The question then is whether all eight corner cubies can be three quarter-turns from Start simultaneously. There are probably a number of ways which will work, but the following works very nicely. Place each corner cubie in its diametrically opposed corner cubicle. For example, place the flt cubie in the bdr cubicle. The twist doesn't matter for the individual cubies, except that the overall configuration for the eight corner cubies must conserve twist. The reason that twist doesn't matter is that when a corner cubie is in its diametrically opposed corner cubicle, all three twists are conjugate (see below). The maximal position for the corners can peacefully co-exist with the Pons for the edges. That is, if each corner cubie is in its diametrically opposed corner cubicle, the parity of the corners is even (as is the Pons). In a certain sense, God's algorithm for a single corner cubie is identical to God's algorithm for the 1x1x1 cube, which is to say, it is identical to God's algorithm for the rotation group of the cube (which we normally denote by C). (See my note of 14 Nov 1995.) Here is how it works. Consider any particular corner cubie such as flt, and consider any sequence of quarter-turns such as TL where each quarter-turn moves the cubie in question. Then, the "same" sequence of whole cube rotations (tl, in this case) will have the same effect on the same corner cubie. Here, we are using the lower case letters t and l to denote whole cube quarter-turns and the upper case letters T and L to denote the face quarter-turns. The converse is also true if we are careful. That is, each whole cube quarter-turn may be denoted in two ways. For example, t is the same as d'. To convert from whole cube rotations back to quarter-turn face turns, we would convert t to T or to D' depending on whether the cubie in question were on the Top face or the Down face at the time. The same trick does not work for the edges. The problem is that face turns and whole cube turns are not fully interchangeable. For instance, T and t are interchangeable for the Top edge cubies, as are D and d for the Down edge cubies. But there is no equivalent interchange for the "equator" of edge cubies fl, lb, br, and rf. (Well, maybe you could do it if you allowed slice moves, but we are not working with slice moves.) I am always interested in symmetry, usually as represented by conjugacy. For whole cube rotations, there are five conjugacy classes. (Again, see my note of 14 November 1995.) For individual cubies, we define conjugacy as follows. Let X and Y be functions (not permutations) which are the restriction of normal permutations to the cubie in question. Then X and Y are conjugate if m'Xm=Y for some m in M, the set of 48 rotations and reflections of the cube. m' must be restricted to the pre-image of the domain of X, and m must be restricted to the range of X. With the various permutations thus restricted to functions on the single flt cubie, the conjugacy classes are as follows: 1. I 2. F, F', L, L', T, T' 3. FF, LL, TT 4. TL', TB, FT', FR, LF', LD 5. TL, L'T' 6. FRR, LDD, TBB 7. FTT, LFF, TLL Note that if we treat all the moves as whole cube permutations rather than as functions on the flt cubie, then #4 and #5 are collapsed down into a single conjugacy class, as are #6 and #7. Then, the conjugacy classes are the same as the ones for the 1x1x1 cube. When I first started working on this little problem, I thought the conjugacy classes for a single cubie might provide a non-arbitrary frame of reference for defining twist. They almost do, but not quite. a. When the cubie is in its home cubicle, its twist is obvious. However, we can observe that I, TL, and L'T' place the flt cubie in the flt cubicle. TL and L'T' are conjugate, but they are not conjugate to I. Hence, it is natural to take I as the untwisted state. b. When the cubie is immediately adjacent to its home cubicle (there are three such cubicles), the conjugacy classes can be used to define twist. For example, the flt cubie is placed into the ftr cubicle by F, T', and by LFF. F and T' are conjugate, but they are not conjugate to LFF. Hence, we can take LFF as the untwisted state. c. When the cubie is immediately adjacent to the diametrically opposed cubicle (there are three such cubicles), the conjugacy classes can be used to define twist. For example, the flt cubie is placed into the frd cubicle by FF, LD, and by LF'. LD and LF' are conjugate, but they are not conjugate to FF. Hence, we can take FF as the untwisted state. d. When the cubie is in the diametrically opposed cubicle (there is only one such cubicle), I don't see any way to use the conjugacy classes to define twist. All three twists are conjugate, and hence none is inherently different from the other two. For example, FRR, LDD, and TBB are all conjugate. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7127 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@curry.epilogue.com Tue Oct 1 14:32:11 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA14238; Tue, 1 Oct 1996 14:32:11 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Tue, 1 Oct 1996 17:19:48 +0100 From: Riccardo Distasi Message-Id: <9610011619.AA00774@irsip.na.cnr.it> To: Cube-Lovers Subject: Intro to cube group theory? Dear mathematical cubologists, I am creeping on this list since a few months, but I have to admit that most of the more advanced mathematical technicalities are beyond my understanding, mainly because I lack knowledge of the basic facts and terminology about groups. All I studied about groups was a part of Birkhoff/McLaine's "Algebra" some 10 years ago. Is there any good reference on groups where I can educate myself? I would prefer freeware papers over costly and hard-to-find (at least in Italy) books. Does anybody have a hint for me? The aim of my training is that of learning about M-conjugacy and the Shamir algorithm, and to be able to follow the technical discussions about the Rubik cube that appear on this list. Riccardo -- Riccardo Distasi, ric@irsip.na.cnr.it From cube-lovers-errors@curry.epilogue.com Tue Oct 1 19:36:41 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA14676; Tue, 1 Oct 1996 19:36:40 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Authentication-Warning: coronado.nadn.navy.mil: wdj owned process doing -bs Date: Tue, 1 Oct 1996 18:43:19 -0400 (EDT) From: Assoc Prof W David Joyner X-Sender: wdj@coronado To: Riccardo Distasi cc: Cube-Lovers Subject: Re: Intro to cube group theory? In-Reply-To: <9610011619.AA00774@irsip.na.cnr.it> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Tue, 1 Oct 1996, Riccardo Distasi wrote: > Dear mathematical cubologists, > I am creeping on this list since a few months, but I have to admit > that most of the more advanced mathematical technicalities are beyond > my understanding, mainly because I lack knowledge of the basic > facts and terminology about groups. > > All I studied about groups was a part of Birkhoff/McLaine's "Algebra" > some 10 years ago. Is there any good reference on groups where I can > educate myself? I would prefer freeware papers over costly and > hard-to-find (at least in Italy) books. Does anybody have a hint for > me? The aim of my training is that of learning about M-conjugacy and > the Shamir algorithm, and to be able to follow the technical > discussions about the Rubik cube that appear on this list. > I think the best book is Bandelow's Inside Rubik's cube and beyond, which might be in a local library. I have lecture notes for a course I'm teaching on the Rubik's cube which I can send you for free. Also, a group-theorist friend of mine has several hundred copies of an elementary group theory book (printed by the US government and I think free) available - you can email me or him (Prof Gaglione, amg@nadn.navy.mil) if you're interested. Finally, Puzzletts is still selling Singmaster's Notes on the Rubik's cube, though they are also out of print. - David Joyner > Riccardo > -- > Riccardo Distasi, ric@irsip.na.cnr.it > > From cube-lovers-errors@curry.epilogue.com Wed Oct 2 14:38:26 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA01653; Wed, 2 Oct 1996 14:38:25 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 2 Oct 1996 19:39:28 +0300 (EET DST) From: Timo Berry X-Sender: taberry@kyberias To: Cube-lovers@ai.mit.edu Subject: An amateur humbly approaches Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Dear Sir(s)! I'm a student of graphic design in the University of Industrial Arts in Helsinki. I'm working on a school project and I need some information on Rubik's Cube. I'm making a piece that takes advantage of the visual language of the cube, but I need to know more. I'm familiar with the game from years back but I never really learned how the actual mechanism worked, nor anything on the history of the game. I would appreciate a few hints on the vast amount of home pages on the subject. What I'm mostly looking for is a basic, no-nonsense explanation of the history and the philosophy and especially the mechanism of the cube (I'd hate to take my only cube apart!). Any illustrations of the mechanism would be great. Are there any working pictures or plans available? Sincerely Yours, Timo Berry taberry@uiah.fi From cube-lovers-errors@curry.epilogue.com Wed Oct 2 14:55:19 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA01686; Wed, 2 Oct 1996 14:55:19 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199610020614.BAA01730@mail.utexas.edu> Date: Wed, 02 Oct 96 01:13:22 -0700 From: C-Money Organization: University of Texas at Austin X-Mailer: Mozilla 1.1N (Windows; I; 16bit) MIME-Version: 1.0 To: Cube-Lovers@ai.mit.edu Subject: http://sdg.ncsa.uiuc.edu/~mag/Misc/CubeLoversInfo.txt Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii I am located in Austin, Tx. I was wondering where I can purchase a rubik's cube. If you could help me out I would greatly appreciate it. From cube-lovers-errors@curry.epilogue.com Wed Oct 2 16:52:14 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA02017; Wed, 2 Oct 1996 16:52:14 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199610021956.OAA67514@opus.cs.utexas.edu> X-Mailer: exmh version 1.6.2 7/18/95 To: C-Money cc: Cube-Lovers@ai.mit.edu Subject: cubes in austin In-reply-to: Your message of "Wed, 02 Oct 1996 01:13:22 CDT." <199610020614.BAA01730@mail.utexas.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Date: Wed, 02 Oct 1996 14:56:54 -0500 From: Norman Richards > I am located in Austin, Tx. I was wondering where I can purchase a rubik's > cube. If you could help me out I would greatly appreciate it. I just bought a new one a few weeks ago at the Kay-bee toy store in Highland Mall. They had about 6 cubes last friday when I was there. They also have some triamids and a couple snakes left. No magic or mini-cube's though. :( The Imaginarium in Higland Mall had the Rubik's C4 cube a while back (the one where you have to align the centers of 4 faces also), but I was quite surprised to find the store closed when I went to the mall last week. :( All the Rubik's stuff goes for $10 a pop. If you find a better price somewhere else, let me know. Also, if you happen to see any stores here that have the mini-cube, please let me know! ______________________________________________________________________________ orb@cs.utexas.edu soli deo gloria From cube-lovers-errors@curry.epilogue.com Wed Oct 2 21:06:02 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA02598; Wed, 2 Oct 1996 21:06:02 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 2 Oct 1996 20:14:55 -0400 From: der Mouse Message-Id: <199610030014.UAA03565@Collatz.McRCIM.McGill.EDU> To: cube-lovers@ai.mit.edu Subject: Re: cube programs, etc > Date: Mon, 2 Sep 1996 22:50:07 +0300 (IDT) Guess who's going through backed-up mail... :-) > I have a computer program that learn to solve the 2X2X2 cube. I mean > that after several hours of 'learning' the program knows to solve any > legal start position of this cube. Before learning the program solve > the cube after about 15 minuets, after learning it takes about 5 > seconds. Interesting. Is the program available? > The letter from Prof Joyner made me think about the following things: > 1. Does anyone have a program (in C) that can take a move (a string > or a line from a file) and show it on the display. Well, I have something of the sort, though it's for the 3-Cube. For example, here's first turning the R face once, then illustrating the Spratt wrench, first defining a slice turn (and checking it), then using it to write the wrench more simply than it would be if done directly with the primitives. % twist > R Cube: u u f u u f u u f l l l f f d r r r u b b l l l f f d r r r u b b l l l f f d r r r u b b d d b d d b d d b Cycles: (ur,br,dr,fr) (ubr,bdr,dfr,fur) [4] Already centered > .set SLICER CUBER R' L `SLICER' defined > SLICER Cube: u f u u f u u f u l l l f d f r r r b u b l l l f d f r r r b u b l l l f d f r r r b u b d b d d b d d b d Cycles: (u,b,d,f) (ub,bd,df,fu) [4] Centred: (ul,fl,dl,bl) (ur,fr,dr,br) (ulb,flu,dlf,bld) (ubr,fur,dfr,bdr) [4] > (SLICER U) 4 Cube: u b u l u u u u u l u l f f f r r r b u b l l l f f f r r r b b b l l l f d f r r r b d b d f d d d d d b d Cycles: (ub)+ (ul)+ (fd)+ (bd)+ [2] Already centered > > 2. Does anyone know about similar programs to my. Program that > 'learn' to solve the cube by themselves. Someone I know once wrote such a program in Lisp. (Incidentally, this was also one of the most stunning examples of hot-spot hand-tuning I ever saw. It represented the cube as a bunch of conses pointing to one another, no leaves at all. The "apply a rotation" call worked by juggling links with rplaca and rplacd. I rewrote this one call in assembly (approximately the same number of lines of code, incidentally) and got three orders of magnitude, a factor of a thousand, speed improvement in the overall program.) The program was somewhat interesting in that it solved the cube by experimenting and discovering macros, somewhat akin to the way humans tend to. I don't know whether this program still exists anywhere. If anyone cares I can try to find out. der Mouse mouse@rodents.montreal.qc.ca 01 EE 31 F6 BB 0C 34 36 00 F3 7C 5A C1 A0 67 1D From cube-lovers-errors@curry.epilogue.com Thu Oct 3 01:05:43 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id BAA03062; Thu, 3 Oct 1996 01:05:43 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 3 Oct 1996 07:05:35 +0200 Message-Id: <1.5.4.16.19961003070403.41ef5388@mailsvr.pt.lu> X-Sender: geohelm@mailsvr.pt.lu X-Mailer: Windows Eudora Light Version 1.5.4 (16) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Assoc Prof W David Joyner From: Georges Helm Subject: Re: Intro to cube group theory? Cc: Cube-Lovers At 18:43 01/10/1996 -0400, you wrote: >I think the best book is Bandelow's Inside Rubik's cube and beyond, >which might be in a local library. I have lecture notes for a >course I'm teaching on the Rubik's cube which I can send you >for free. Also, a group-theorist friend of mine has several >hundred copies of an elementary group theory book (printed >by the US government and I think free) available - you can >email me or him (Prof Gaglione, amg@nadn.navy.mil) if you're >interested. Finally, Puzzletts is still selling Singmaster's >Notes on the Rubik's cube, though they are also out of >print. - David Joyner > I think a very helpful book is Handbook of Cubic Math by Alexander H. Frey, Jr. + David Singmaster Contents: Preface 1.Introduction 2.A Cubik Orientation 3.Restoring the Cube 4.The What, Why, and How of Cube Movements 5.Improved Restoration Processes 6.The Cube Group and Subgroups 7.Permutation Structures and the Order of Groups 8.Advanced Restoration Methods 9.Epilogue A. A Small Catalogue of Processes B. Solutions to Exercises Index It was published by Enslow publishers Georges geohelm@pt.lu http://www.geocities.com/Athens/2715 http://ourworld.compuserve.com/homepages/Georges_Helm From cube-lovers-errors@curry.epilogue.com Thu Oct 3 14:20:38 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA04396; Thu, 3 Oct 1996 14:20:37 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 03 Oct 1996 13:31:03 -0500 (EST) From: Jerry Bryan Subject: Re: Intro to cube theory? To: Cube-Lovers Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII Content-transfer-encoding: 7BIT > On Tue, 1 Oct 1996, Riccardo Distasi wrote: > Does anybody have a hint for > me? The aim of my training is that of learning about M-conjugacy and > the Shamir algorithm, and to be able to follow the technical > discussions about the Rubik cube that appear on this list. Several good books have already been mentioned here, so I thought I would try briefly to answer your specific questions rather than listing books again. I doubt you are going to find any references to M-conjugacy in any Group Theory Books, nor even in any books that are specific to the Cube. What you will find is discussions of conjugacy. The conjugate of X by Y is defined either as Y'XY or as YXY'. Here I am following the E-mail convention that Y' means Y^(-1) or "Y inverse". We use Y' because it is hard to write a proper superscript 1 on E-mail. One reason for the different definitions for conjugacy (Y'XY vs. YXY') may be that some authors use a right-to-left definition for group operators and some use a left-to-right. (Cube-Lovers uses left-to-right almost exclusively). But I think that even with a consistent left-to-right convention, you fill find differences between authors in their definition of conjugacy. I think I remember a discussion in Singmaster about why some authors do it one way and others do it the other. The best I recall, both ways of doing it make sense in the proper context. I will try to chase down the reference and post a followup. I don't think it makes much difference which convention you use as long as you are consistent. If Y'XY is a conjugate, then YXY' is also. That is, if Y'XY is the conjugate of X by Y, then YXY' is the conjugate of X by Y'. Frey and Singmaster use the YXY' convention. Cube-Lovers (including the things I have posted) primarily uses the Y'XY convention. I actually think the YXY' convention makes more sense. Roughly speaking, it means to do one thing, then to do a second thing, and finally to undo the first thing. The effect is essentially to do the second thing, but to do it shifted by the first thing. For example, suppose you know how to do something to the Top layer of the cube but you don't know how (or find it awkward) to do the same thing to the Down layer of the cube. What you could do is turn the cube upside down, perform your operation on the Top layer, and then turn the cube right side up. You will have performed your operation on the Down layer. In Cube-Lovers, we would probably write this as cXc'. We call the set of twenty-four rotations of the cube C, and c would be one of the elements of C that turns the cube upside down. So c would turn the cube upside down, X would be your operation, and c' would restore the cube to right side up. Except that we would really write it as c'Xc, which in some ways makes no sense. I read it as undo the first thing, then do the second thing, and finally do the first thing. I really do have to chase down Singmaster's explanation of why this makes sense. I confess I struggle with the real geometric significance of Y'XY. That is, if we have Z=Y'XY, then what is the relationship between X and Z? They have the same cycle structure, but that is about as far as I get in a geometric interpretation. Here I am assuming that each of X, Y, and Z are in the cube group. But I find c'Xc or m'Xm easy to interpret. In Cube-Lovers convention, M is the set of forty-eight rotations and reflections of the cube to go along with C as the set of twenty-four rotations of the cube. So C is a subset of M and C-conjugacy is a subset of M-conjugacy. But we nearly always talk about M-conjugacy. But C and M are not really in the cube group G as we usually define it. That is, the standard model for G is a fixed face center model where we do not rotate the whole cube. To use Group Theory properly with M-conjugacy, we have to deal with M-conjugacy in terms of a larger group which is sometimes called MG or G+M. MG includes all the face turns, rotations, and reflections of the cube. However, it is the case that if X is in G, then so too is m'Xm. So if we want to, we can treat M-conjugacy as a function on G without having to expand our group to MG. Many Group Theory books will talk about symmetry. A symmetry is just a special kind of permutation which preserves some kind of property, usually a geometric property. For example, there are eight symmetries of a square. A square can be rotated in four different ways and still look the same, and each of the four rotations can be turned inside out. You can also think of the "turned inside out" versions as being mirror images, so they are called reflections. Similarly, a cube has twenty-four rotations and twenty-four reflections as symmetries. This was true long before Rubik's cube was invented, and you will find discussions of the symmetries of the cube in books that were written before Rubik's cube was invented. Cube-Lovers simply calls the set of forty-eight symmetries of the cube M on a fairly consistent basis, and so M-conjugacy is born. It is really just conjugacy by the symmetries of the cube. M-conjugacy is important because it identifies positions which are "really the same", even if they may look different superficially. That is, if Y=m'Xm for some m in M, then X and Y look the same except that they may be rotated or recolored with respect to each other. In particular, X and Y may be solved in the "same way", and each will require the same number of moves for solution. My view of Shamir's method is that it really has nothing to do with Group Theory. Rather, it has to do with data structures and information theory. There are several components of Shamir's method, but the most important is addressing the following problem. Suppose you have a collection of objects in a computer program in some arbitrary (possibly "random" order), and suppose you want to eliminate any duplicate objects to make the collection into a true set in a mathematical sense. Almost any algorithm you come up with is equivalent to sorting the objects to place the duplicate occurences adjacent to each other, and then scanning the collection front to back to identify the duplicates. Now you may not literally sort. You may build trees, hash tables, or any of a number of interesting and efficient structures, but they all reduce to sorting at the conceptual level. A variation on this theme is suppose you have two (or more) such collections, and you want to eliminate all duplicate objects. At the conceptual level, almost any algorithm you come up with is equivalent to sorting each collection, and then merging and matching the sorted collections. Shamir's method provides a very efficient way to accomplish this "sorting". Given a collection of objects which is sorted already, it lets you create a second collection which is sorted in a totally different way, without any of the objects moving in memory -- by simply traversing a search tree in a clever way. The issue arises in search programs for Rubik's cube because you often have a set of cube positions which you need to compose with another position or set of positions. When you are done, you need to "sort and match" or "merge and match" the results. Literally sorting and merging can take ridiculous amounts of time and memory. If the first set of positions is already sorted, Shamir's method tells us how to compose the first set of positions with other positions in such a way that the newly generated sets of positions come out automagically in the right order, with no additional sorting required. Much more detail than this is available the the Cube-Lovers archives. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7127 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@curry.epilogue.com Fri Oct 4 17:33:19 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id RAA06958; Fri, 4 Oct 1996 17:33:19 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <1.5.4.16.19961004231924.2eef8b5e@pop3.redestb.es> X-Sender: estelada@pop3.redestb.es (Unverified) X-Mailer: Windows Eudora Light Version 1.5.4 (16) Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable To: Cube-Lovers-Request@ai.mit.edu From: Joaquim Folch Subject: Rubik revenge (4x4) Date: Fri, 4 Oct 1996 12:03:40 +0100 > >>Dear Sir: >> >>I=B4m Joaquin Folch (Barcelona-Spain). I have a blind friend who needs an >>Rubik cube, large size whith 16 squares per side (Revenge type, 4x4= lines), >>because his actual cube is broken. He played very well and fast whith his >>marked cube. >>Please tell me how much I have to pay for it.=20 >>Please answer me. Many thanks to all, Joaquin. >>My adress: EMail: estelada@redestb.es >> >>Joaquin Folch >>Espigol 6 >>08328 Alella (Barcelona) >>Spain=20 >> > > > > > >> > From cube-lovers-errors@curry.epilogue.com Wed Oct 16 14:01:16 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA01742; Wed, 16 Oct 1996 14:01:15 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 16 Oct 1996 08:54:11 -0500 To: cube-lovers@ai.mit.edu From: Peter Beck Subject: largest cube HI folks, I would like to revisit the question of what is the largest (number of slices) size cube that can be made. As I recollect the center spindle mechanism has been analyzed and the conclusion was a 5x5x5. There is a new mechanism used in the MOZAIKA puzzle (info below). I wonder if anybody has analyzed it to determine what configurations of cubes it could be used for. PS This mechanism also seems to answer the question of whether or not the cube is a sliding block puzzle on a spherical surface or a solid rotating puzzle. ******************************** * "MOZAIKA" is a spherical sliding block * puzzle like Rubik's cube with a new * mechanism. I have only seen the 3x3x3 * version. It has 2 types of pieces * (the third, a sphere in the center appears to be * unnecessary): a triangular piece analogous * to the cubes corner and a rectangular piece * analogous to the center piece , using 2 of * these to make an edge piece. The puzzle * thus has 3 orthogonal equators made up of * the rectangular pieces and the corners. * * These pieces interlock to form a spherical * surface - the center is hollow. The interlock * method is that the corner pieces have a rail * that the rectangular pieces ride on. The * corner pieces are held in space by the * rectangular pieces (sorry for poor description). ******************************** * FROM: J&R DESIGNS * 1126 SOUTH STREET * POB 315 * NILES, MICHIGAN 49120 * COST: US $15 * + $3 POSTAGE USA OR $5 OVERSEAS * ******************************** THE FUTURE IS PUZZLING, but CUBING IS FOREVER !!! Peter Beck,aka, Just Puzzles, 201-625-4191 answering machine a cube WEB site;2/27/96 - ...................................................... my career site - updated 5/31/96 ...................................................... last modified 31 May 1996 From cube-lovers-errors@curry.epilogue.com Wed Oct 16 22:51:41 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA02682; Wed, 16 Oct 1996 22:51:41 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 17 Oct 1996 01:49:55 +0200 From: Dik.Winter@cwi.nl Message-Id: <9610162349.AA04173=dik@bever.cwi.nl> To: cube-lovers@ai.mit.edu, pbeck@pica.army.mil Subject: Re: largest cube > I would like to revisit the question of > what is the largest (number of slices) > size cube that can be made. As I > recollect the center spindle mechanism > has been analyzed and the conclusion > was a 5x5x5. If I remember well the limit was not order 5 but 6, and not due to the mechanism but only because during turning the corner cubes will extend so much outside the cube that they are held by only 2 neighbours. *But* this holds only if your requirement is that all cubelets have the same size. When you allow cubelets to grow when going from the center you can get larger (although I think even in that case there will be a limit, right now I am too lazy to think about it even further). dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ From cube-lovers-errors@curry.epilogue.com Wed Oct 23 14:06:47 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA09246; Wed, 23 Oct 1996 14:06:46 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Organization: Centro di Calcolo - Dipartimento di Informatica di Pisa - Italy From: Mario Velucchi Message-Id: <199610231533.RAA02193@helen.cli.di.unipi.it> Subject: Re: DEAR TANOFF <<<<<<<<<<<<<<<< (fwd) To: cube Date: Wed, 23 Oct 1996 17:33:50 +0200 (MET DST) X-Mailer: ELM [version 2.4 PL24] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Forwarded message: > From velucchi@CLI.DI.Unipi.IT Wed Oct 23 17:32:34 1996 > Subject: Re: DEAR TANOFF <<<<<<<<<<<<<<<< > To: TANOFF%SMOOKE@BIOMED.MED.YALE.EDU > Date: Wed, 23 Oct 1996 17:32:20 +0200 (MET DST) > > > > > What is the Siamese Cube? > > > > Two (usual/normal) Rubik Cubes in One ... > > > ------ > | | > | | > -----+----- > | | > | | > ------ > > The goal is equal to normal cube but the moves are differents ... > because the two cubes are "uniti" .... > Do You understand? let me know! > > > Sorry for my English and my Picture! > > > -- > Best Regards, MV > \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// > Mario Velucchi University of PISA > Via Emilia, 106 Department of Computer Science > I-56121 Pisa e-mail:velucchi@cli.di.unipi.it > ITALY talk:velucchi@helen.cli.di.unipi.it > http://www.cli.di.unipi.it/~velucchi/intro.html > \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// > From cube-lovers-errors@curry.epilogue.com Thu Oct 24 16:27:10 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA12157; Thu, 24 Oct 1996 16:27:09 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 24 Oct 1996 10:29:40 +0100 Message-Id: <1.5.4.32.19961024102451.002cf550@mentda.me.ic.ac.uk> X-Sender: ars2@mentda.me.ic.ac.uk X-Mailer: Windows Eudora Light Version 1.5.4 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Mario Velucchi From: "The Official Thermo-Fluids Fan Club of the UK. (Andy Southern)" Subject: Re: DEAR TANOFF <<<<<<<<<<<<<<<< (fwd) Cc: Cube-Lovers@ai.mit.edu At 17:33 23/10/96 +0200, you wrote: >Forwarded message: >> From velucchi@CLI.DI.Unipi.IT Wed Oct 23 17:32:34 1996 >> Subject: Re: DEAR TANOFF <<<<<<<<<<<<<<<< >> To: TANOFF%SMOOKE@BIOMED.MED.YALE.EDU >> Date: Wed, 23 Oct 1996 17:32:20 +0200 (MET DST) >> >> > >> > What is the Siamese Cube? >> > >> >> Two (usual/normal) Rubik Cubes in One ... >> >> >> ------ >> | | >> | | >> -----+----- >> | | >> | | >> ------ >> >> The goal is equal to normal cube but the moves are differents ... >> because the two cubes are "uniti" .... >> Do You understand? let me know! >> >> >> Sorry for my English and my Picture! >> >> >> -- >> Best Regards, MV >> \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// >> Mario Velucchi University of PISA >> Via Emilia, 106 Department of Computer Science >> I-56121 Pisa e-mail:velucchi@cli.di.unipi.it >> ITALY talk:velucchi@helen.cli.di.unipi.it >> http://www.cli.di.unipi.it/~velucchi/intro.html >> \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\//////////////////////////////// >> > > > I think I understand. There are two cubes, orientated the same, which share a common corner piece. The shared corner piece has no stickers on it, but is a "Double Inside" corner piece. The effect is that they share the same line from corner to corner, passing through the dead centre of the cube. The appearence would be like a (5x5x5) which had been cut away. There would be a cubie at the locations: (1,1,1),(1,1,2),(1,1,3),(1,2,1),(1,2,2),(1,2,3),(1,3,1),(1,3,2),(1,3,3), (2,1,1),(2,1,2),(2,1,3),(2,2,1),(2,2,2),(2,2,3),(2,3,1),(2,3,2),(2,3,3), (3,1,1),(3,1,2),(3,1,3),(3,2,1),(3,2,2),(3,2,3),(3,3,1),(3,3,2),(3,3,3),(3,3 ,4),(3,3,5),(3,4,3),(3,4,4),(3,4,5),(3,5,3),(3,5,4),(3,5,5), (4,3,3),(4,3,4),(4,3,5),(4,4,3),(4,4,4),(4,4,5),(4,5,3),(4,5,4),(4,5,5), (5,3,3),(5,3,4),(5,3,5),(5,4,3),(5,4,4),(5,4,5),(5,5,3),(5,5,4),(5,5,5), These cubes would *not* rotate about the apparent centre (3,3,3), but about the two real centres (4,4,4) and (2,2,2). I could see there being a few perceptual problems. The conecting cubie at (3,3,3) would have no colour stickers on it, hence position and rotation must be determined from the other corners. The cube would also appear to the operator to turn only the outer slice and middle slice of each cube because the operator would always use the centre of mass as his/her frame of referance. That is different to the standard (3x3x3) because the operator feels the outer slices move. sorry if this is either wrong or nothing new, I just thought I'd share my thoughts with you. Andrew Southern From cube-lovers-errors@curry.epilogue.com Fri Oct 25 01:26:18 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id BAA00762; Fri, 25 Oct 1996 01:26:17 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <9610250500.AA13321@jrdmax.jrd.dec.com> Date: Fri, 25 Oct 96 14:00:53 +0900 From: Norman Diamond 25-Oct-1996 1355 To: cube-lovers@ai.mit.edu Apparently-To: cube-lovers@ai.mit.edu Subject: Siamese Rubik's Cubes (was Re: DEAR TANOFF <<<<<<<<<<<<<<<< (fwd)) Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=ISO-2022-JP A. Southern misinterpreted M. Velucchi's picture: >> ------ >> | | >> | | >> -----+----- >> | | >> | | >> ------ Siamese Rubik's cubes share an entire column of cubies, i.e. in the case of two 3x3x3's they share an edge cubie and two corner cubies. Cubies cannot move from one cube to the other. The shared column of cubies cannot be separated or rearranged. The effect is like bandaging an edge column on one 3x3x3 cube and bandaging an edge column on another 3x3x3 cube and having two identical puzzles. The idea of bandaging has been extended further by Dieter Gebhardt (publications in CFF) and others. Most variations of bandaging cannot be constructed by joining another cube onto it; they just have to be done in a simpler and straightforward manner :-) And even when a collector wants duplicates of some version, there's no need for two duplicates to be stuck to each other :-) So there is no real demand for Siamese cubes any more. But bandaged cubes, yeah some variations are really really difficult. -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital.] From cube-lovers-errors@curry.epilogue.com Fri Oct 25 16:08:00 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA02317; Fri, 25 Oct 1996 16:08:00 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com To: Cube-Lovers@AI.MIT.EDU From: Wei-Hwa Huang Subject: Re: Siamese Rubik's Cubes (was Re: DEAR TANOFF <<<<<<<<<<<<<<<< (fwd)) Date: 25 Oct 1996 14:02:32 GMT Organization: California Institute of Technology, Pasadena Message-ID: <54qh9o$4tu@gap.cco.caltech.edu> References: NNTP-Posting-Host: off.ugcs.caltech.edu X-Newsreader: NN version 6.5.0 #2 (NOV) Norman Diamond 25-Oct-1996 1355 writes: >A. Southern misinterpreted M. Velucchi's picture: >>> ------ >>> | | >>> | | >>> -----+----- >>> | | >>> | | >>> ------ >Siamese Rubik's cubes share an entire column of cubies, i.e. in the >case of two 3x3x3's they share an edge cubie and two corner cubies. >Cubies cannot move from one cube to the other. The shared column >of cubies cannot be separated or rearranged. The effect is like >bandaging an edge column on one 3x3x3 cube and bandaging an edge >column on another 3x3x3 cube and having two identical puzzles. A "creative" question: Suppose we want to be able to rotate the 17-cubie faces 180 degrees. Can anyone think of a mechanical structure that could achieve this? From cube-lovers-errors@curry.epilogue.com Fri Oct 25 22:41:42 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA03041; Fri, 25 Oct 1996 22:41:42 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Fri, 25 Oct 96 18:51:55 EDT Message-Id: <9610252251.AA14688@sun34.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Subject: Re: Siamese Rubik's Cubes Norman Diamond wrote: ... > Siamese Rubik's cubes share an entire column of cubies, i.e. in the > case of two 3x3x3's they share an edge cubie and two corner cubies. ... > The idea of bandaging has been extended further by Dieter Gebhardt > (publications in CFF) and others.... Most cases of bandaging create a puzzle whose transition graph is not the Cayley graph of a group. For instance, if two disjoint edge-corner pairs were taped together, you would have some positions with ten possible QT neighbors and some with eight. But the corner-edge-corner bandaging does create a group: Fix the position of the bandaged part, and permute the other 46 facelets (six corners, eleven edges, and six face centers) with two face moves and two slice moves. The resulting group can have at most 5! corner permutations, as in the two-generator group (see Singmaster or the archives (21 July 1981, 31 Aug 1994)). There are at most 11! edge permutations, and the face center permutations represent the rotation group of the cube, with 24 elements. There can be at most 3^5 corner orientations and 2^10 edge orientations. Finally, the total permutation parity (corner, edge, and face center) must be even. Gap tells me the group has 14302911135744000 = 5! 3^5 11! 2^10 24/2 elements, so all such positions are achievable. I haven't run the Supergroup through Gap, so I'm not sure whether it 2048 times as many positions. Of course the regular Siamese cube has the square of this many positions, because there are two cubes. A different kind of Siamese cube would be one in which the three 17-cube slabs can rotate 180 degrees with respect to each other. It would certainly be difficult to build. I think the interaction between the slab moves and the Lucky Six group would make it hard to solve, as well. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@curry.epilogue.com Sat Oct 26 00:18:08 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id AAA03251; Sat, 26 Oct 1996 00:18:08 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Fri, 25 Oct 1996 23:47:50 -0400 (EDT) From: Nicholas Bodley To: Wei-Hwa Huang cc: Cube-Lovers@ai.mit.edu Subject: Re: Siamese Rubik's Cubes (was Re: DEAR TANOFF <(fwd)) In-Reply-To: <54qh9o$4tu@gap.cco.caltech.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On 25 Oct 1996, Wei-Hwa Huang wrote: }Norman Diamond 25-Oct-1996 1355 writes: }>A. Southern misinterpreted M. Velucchi's picture: {Snips} }A "creative" question: } }Suppose we want to be able to rotate the 17-cubie faces 180 degrees. }Can anyone think of a mechanical structure that could achieve this? Here's hoping that this "stream of consciousness with revisions" style is acceptable!: I can conceive of such a structure, but whether it could be made to work decently is quite open to doubt. It would have a great many pieces; the whole top layer would have to consist of cubies with two physical parts, one that would travel to its new location, and the other which would remain behind. Holding the whole works together while rotating it is difficult enough, but reliably reattaching the two parts of each cubie once the rotation was complete is borderline crazy! Of course, all edge and corner cubies would need to be two-part. If someone is ambitious enough to attempt such a design, it would be very costly and out of the question for mass production. It might help if a tool (such as a Torx (TM) wrench) were provided to insert into both "face-center" cubies (or the common corner cubie) to unlock the top layer from its underlying parts and to lock the top-layer cubies together. However, just a clamping frame to hold the top layer together would make sense, IMO. A strictly-mechanical solution is at least borderline impractical, but shrewd design with rare-earth magnets might help. Dismantle a regular Cube to see what would be involved. An edge cubie has a "foot" that extends below the top layer, as does a corner cubie. These "feet" would have to be left behind once a move began. It's really nice to have all the unlocking and reattaching taken care of "automatically" by just the twisting shear force created by gripping the Siamese Cube, but for such a move as this, that's a formidable luxury. If I were an experienced mechanical engineer, I'd say it just isn't practical. However, it is great fun to think of how it could be done. (If e-mail had a universal graphics format, illustrations would be nice, but I honestly don't feel that ambitious!) I also suspect that when it came time to design in detail, new conceptual problems would arise which might be extremely difficult to overcome. Consider, for instance, that if you don't use a clamping frame, the mere act of locking the top layer together has to hold the corner cubies in place. The locking pieces need to be operated by sliding members passing through the neighboring edge cubies, and that's not all, by far. My regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 0. -------------------------------------------------------------------------- From cube-lovers-errors@curry.epilogue.com Mon Oct 28 23:06:18 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id XAA04616; Mon, 28 Oct 1996 23:06:18 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-ID: <327582D5.71A3@host1.dia.net> Date: Mon, 28 Oct 1996 22:06:45 -0600 From: Scott Crawford Reply-To: scrawfor@host1.dia.net X-Mailer: Mozilla 3.0Gold (Win95; I) MIME-Version: 1.0 To: Cube List Subject: Rubik's Revenge Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit I am looking for anyone who has a Rubik's Revenge they'd like to part with. Scott Crawford From cube-lovers-errors@curry.epilogue.com Sun Nov 3 21:05:08 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA11798; Sun, 3 Nov 1996 21:05:07 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com From: Pet Milk Organization: Arkansas School for Math & Science To: Cube-Lovers@ai.mit.edu Date: Sun, 3 Nov 1996 17:41:33 CDT Subject: Greetings Priority: normal X-mailer: Pegasus Mail for Windows (v2.42a) Message-ID: <319C4F363C9@ASMS3.DSC.K12.AR.US> Hi My name is Nathan, and, obviously, I'm a newcomer to the list. I have need of some help. I'm forced to write a paper concerning some famous individual that has contributed to mathematics in some way. After looking carefully, I picked Mr. Rubik. I searched the Net forever, but only came up with an interview that was conducted with Rubik. However, I need more information. Here's where I need you. Does anyone have any further information concerning the work of Mr. Rubik in any way? Anything would be of help: Net sites, books, lists, anything. I'm not in any bug hurry, however I would like the information in time to sort it, etc. Thank you for your help... Nathan I spy a boy I spy a girl I spy a chance to change the world From cube-lovers-errors@curry.epilogue.com Sun Nov 3 21:44:47 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA11881; Sun, 3 Nov 1996 21:44:46 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com From: Stan Isaacs Message-Id: <199611040235.AA111104943@hpcc01.corp.hp.com> Subject: Book on Bandaged Cubes To: cube-lovers@ai.mit.edu Date: Sun, 3 Nov 96 18:35:43 PST Mailer: Elm [revision: 70.85.2.1] I just got a book that might be very interesting to cube lovers. It's called "Bandaged Cubes", by Dieter Gebhardt. Some cube lovers may already know Dieter and about bandaged cubes, from articles in the CFF magazine; this is all about bandaging in one place. He presents notation and classification, and discusses many types, the group-theory of them, and how to solve them. It even has color pictures of some of the variations. One type is the C-block cube (also called "Rigit Edge Cube"), which is just half of the Siamese cube recently discussed here. For those who haven't seen articles on this, bandaged cubes are regular Rubik's cubes with some edges taped together. If you tape 2 cubies, one corner and one edge, that is an "A-block". If you tape an edge and a center, that's a "B-block". 2 corners and an edge (3 cubies) is a "C-block. And so on - he has notation for all the bandage possibilities, and discusses (as far as I can tell) all the interesting variations in a 3x3x3. (He leaves 4x4x4 and 5x5x5 bandaged cubes for a later time.) Anyway, if you get tired of Rubik's cube itself, these offer dozens of variations, each with its own quirks and limitations, and many chances for new discoveries. According to CFF, the booklet can be bought from Dieter for $24 (DM 36) (including postage) at: Dieter Gebhardt Norikerstrasse, 23, D-90402 Nurnberg, GERMANY Its 100 pages, with 74 figures and 4 color plates. I highly recommend it. Every Cube-lover should have a copy. (Of course, now I need a cheap source of blank cubes to tape.) -- Stan Isaacs From cube-lovers-errors@curry.epilogue.com Mon Nov 4 14:13:43 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA13736; Mon, 4 Nov 1996 14:13:42 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com To: mlist-cube-lovers@nntp-server.caltech.edu From: Wei-Hwa Huang To: Cube-Lovers@AI.MIT.EDU Subject: Re: Book on Bandaged Cubes Date: 4 Nov 1996 16:37:13 GMT Organization: California Institute of Technology, Pasadena Lines: 17 Message-ID: <55l63p$kuh@gap.cco.caltech.edu> References: NNTP-Posting-Host: accord.cco.caltech.edu X-Newsreader: NN version 6.5.0 #12 (NOV) Stan Isaacs writes: >For those who haven't seen articles on this, bandaged cubes are regular >Rubik's cubes with some edges taped together. If you tape 2 cubies, >one corner and one edge, that is an "A-block". If you tape an edge and a >center, that's a "B-block". 2 corners and an edge (3 cubies) is a "C-block. >And so on - he has notation for all the bandage possibilities, and >discusses (as far as I can tell) all the interesting variations in a >3x3x3. (He leaves 4x4x4 and 5x5x5 bandaged cubes for a later time.) Does he cover non-adjacent bandages; for example, two corner cubies? -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ ------------------------------------------------------------------------------- Not technically an "evil alum". From cube-lovers-errors@curry.epilogue.com Mon Nov 4 14:12:38 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA13732; Mon, 4 Nov 1996 14:12:38 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Authentication-Warning: coronado.nadn.navy.mil: wdj owned process doing -bs Date: Mon, 4 Nov 1996 06:58:58 -0500 (EST) From: Assoc Prof W David Joyner X-Sender: wdj@coronado Reply-To: Assoc Prof W David Joyner To: Pet Milk cc: Cube-Lovers@ai.mit.edu Subject: Re: Greetings In-Reply-To: <319C4F363C9@ASMS3.DSC.K12.AR.US> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Sun, 3 Nov 1996, Pet Milk wrote: > Hi > > > My name is Nathan, and, obviously, I'm a newcomer to the list. I have > need of some help. > > I'm forced to write a paper concerning some famous individual that > has contributed to mathematics in some way. After looking carefully, > I picked Mr. Rubik. I searched the Net forever, but only came up > with an interview that was conducted with Rubik. However, I need > more information. Here's where I need you. > > Does anyone have any further information concerning the work of Mr. > Rubik in any way? Anything would be of help: Net sites, books, > lists, anything. I'm not in any bug hurry, however I would like the > information in time to sort it, etc. Have you seen the book "Rubik's cubic compendium", by Rubik, et al? It has an article by Rubik which is interesting and is still in print (published by Oxford Univ Press I think). - David Joyner > > Thank you for your help... > > > Nathan > > I spy a boy > I spy a girl > I spy a chance to change the world > > From cube-lovers-errors@curry.epilogue.com Mon Nov 4 15:29:43 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id PAA13904; Mon, 4 Nov 1996 15:29:43 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Mon, 4 Nov 1996 15:12:06 -0500 (EST) Message-Id: <199611042012.PAA24488@itchy.mindspring.com> X-Sender: gammet@mindspring.com X-Mailer: Windows Eudora Light Version 1.5.2 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: cube-lovers@ai.mit.edu From: Aben Gentry Subject: Rubik's Clock... Hey Guys, Have any of you figured out how to solve Rubik's clock yet? Also, what is best source for cubes (and cube-like puzzles) that you know of? ...I normally shop at Puzzletts. Aben Gentry abeng@mindspring.com From cube-lovers-errors@curry.epilogue.com Mon Nov 4 21:53:51 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA15107; Mon, 4 Nov 1996 21:53:51 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com X-Authentication-Warning: arthur.st.nepean.uws.edu.au: lrylands owned process doing -bs Date: Tue, 5 Nov 1996 13:22:02 +1100 (EST) From: Leanne Rylands X-Sender: lrylands@arthur To: Aben Gentry cc: cube-lovers@ai.mit.edu Subject: Re: Rubik's Clock... In-Reply-To: <199611042012.PAA24488@itchy.mindspring.com> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII > > Have any of you figured out how to solve Rubik's clock yet? Also, what is > best source for cubes (and cube-like puzzles) that you know of? ...I > normally shop at Puzzletts. Don Taylor and I wrote a book ``Mastering Rubik's Clock''. Published in 1988 by Simon and Schuster which gives the solution. The clock is very easy to solve (hence the book is very thin, only 16 pages). Leanne Rylands From cube-lovers-errors@curry.epilogue.com Tue Nov 5 23:16:36 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id XAA17832; Tue, 5 Nov 1996 23:16:35 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com From: AirWong@aol.com Date: Tue, 5 Nov 1996 18:43:57 -0500 Message-ID: <961105184355_222918906@emout11.mail.aol.com> To: CUBE-LOVERS@ai.mit.edu Subject: Re: Rubik's Clock... > Have any of you figured out how to solve Rubik's clock yet? What exactly is the Rubik's clock? I've only heard of the Rubik's cube, dice, pyramid, tangle, fifteen... how many puzzles have the Rubik's name on them, anyway? Aaron Wong From cube-lovers-errors@curry.epilogue.com Wed Nov 6 14:32:49 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA19590; Wed, 6 Nov 1996 14:32:49 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com To: Cube-Lovers@AI.MIT.EDU From: Wei-Hwa Huang Subject: Re: Rubik's Clock... Date: 6 Nov 1996 16:10:50 GMT Organization: California Institute of Technology, Pasadena Lines: 37 Message-ID: <55qdaa$jig@gap.cco.caltech.edu> References: NNTP-Posting-Host: accord.cco.caltech.edu X-Newsreader: NN version 6.5.0 #12 (NOV) AirWong@aol.com writes: >> Have any of you figured out how to solve Rubik's clock yet? >What exactly is the Rubik's clock? I've only heard of the Rubik's cube, dice, >pyramid, tangle, fifteen... how many puzzles have the Rubik's name on them, >anyway? Hum de hum... Cube (Several releases) Mini Cube Revenge 4th Dimension (A cube with pictures) Race Game Snake (Many colors, three sizes) Magic (Link the Rings) Magic (Make the Cube) Magic (Unlink the Rings) Magic Game Magic Puzzle Clock Fifteen Rabbits Dice Triamid Tangle (4 versions) Maze (The Pyraminx, the Octagon, the 5x5x5, and the Missing Link have never been labeled with Rubik's name, AFAIK...) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ ------------------------------------------------------------------------------- Not technically an "evil alum". From cube-lovers-errors@curry.epilogue.com Thu Nov 7 16:07:58 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA22332; Thu, 7 Nov 1996 16:07:58 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 7 Nov 1996 08:15:38 -0500 (EST) From: Nicholas Bodley To: AirWong@aol.com cc: CUBE-LOVERS@ai.mit.edu Subject: Re: Rubik's Clock... In-Reply-To: <961105184355_222918906@emout11.mail.aol.com> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Aaron is probably not the only person wondering! By any chance, was this a temporary lapse, with the calendar cube in mind? Regards to all, NB * * * On Tue, 5 Nov 1996 AirWong@aol.com wrote: {Snips} }> Have any of you figured out how to solve Rubik's clock yet? } }What exactly is the Rubik's clock? I've only heard of the Rubik's cube, dice, }pyramid, tangle, fifteen... how many puzzles have the Rubik's name on them, }anyway? From cube-lovers-errors@curry.epilogue.com Thu Nov 7 16:09:46 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA22336; Thu, 7 Nov 1996 16:09:46 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <1.5.4.32.19961107155402.002b5d48@mentda.me.ic.ac.uk> X-Sender: ars2@mentda.me.ic.ac.uk X-Mailer: Windows Eudora Light Version 1.5.4 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 07 Nov 1996 15:54:02 +0000 To: Cube-Lovers@ai.mit.edu From: "The Unofficial Thermofluids Fan Club of the UK." Subject: Re: Rubik's Clock... I wrote this a few days ago but sent it to the wrong address......durrrrrrrrrrrrrrrrrrrrrr! here it is with an amendment. At 13:22 05/11/96 +1100, you wrote: >> >> Have any of you figured out how to solve Rubik's clock yet? Also, what is >> best source for cubes (and cube-like puzzles) that you know of? ...I >> normally shop at Puzzletts. > >Don Taylor and I wrote a book ``Mastering Rubik's Clock''. >Published in 1988 by Simon and Schuster which gives the >solution. >The clock is very easy to solve (hence the book is very >thin, only 16 pages). > >Leanne Rylands > > > > I once set about building a 5x5 rubiks clock, but I never got round to finishing it as I realised that I already knew how to solve the general NxN clock. I did get around to building a 32x2 Rubik's Magic, that's eight Rubiks Magics built into one array. It required a couple of customised inlays (i.e. I cut and pasted), and is still prone to misalignment and deligamentation (i.e. it falls apart a bit), but it works and the solution is just an extended version of the 4x2. It takes me about half an hour to solve, and is the equivelent of a good work out, that's why I haven't used it since I was about 14! I only used it to get one above the guys at school that could do the magic (4x2) in about one second, because I could never get any faster than 2 seconds! The smallest "Rubik's Magic" I've ever custom built was a 2x1, most people didn't have any problem with that one! Has anyone else ever extrapulated a puzzle to form a "Custom Master Edition"? I'd be interested to hear. from Wei-Hwa Huang Cube (Several releases) Mini Cube Revenge 4th Dimension (A cube with pictures) Race Game Snake (Many colors, three sizes) Magic (Link the Rings) Magic (Make the Cube) Magic (Unlink the Rings) Magic Game Magic Puzzle Clock Fifteen Rabbits Dice Triamid Tangle (4 versions) Maze I think there was also Rubik's Illusion, which was a game of chess using some sort of complex mapping. Cheers! Andrew R. Southern, The unofficial Thermo-Fluids Fan Club of the UK. From cube-lovers-errors@curry.epilogue.com Fri Nov 8 17:23:27 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id RAA25136; Fri, 8 Nov 1996 17:23:27 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Fri, 8 Nov 1996 12:02:10 GMT Message-Id: <96110812021006@glam.ac.uk> From: Vanessa Paradis WANTS me!! To: CUBE-LOVERS@ai.mit.edu Subject: HELLO AGAIN! X-VMS-To: CUBE-LOVERS@AI.MIT.EDU I have solved the Rubiks Clock. I also have the Rubiks Illusion. It is like 4-in-a-row game, but you need 5-in-a-row, using a mirror (which is connected to the back of the board) as another part of the board. There are 3 types of pieces. RED, YELLOW and RED/YELLOW. These can be used to make a row of 5 in any direction From cube-lovers-errors@curry.epilogue.com Tue Nov 12 16:10:33 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA02082; Tue, 12 Nov 1996 16:10:33 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199611121842.NAA17030@life.ai.mit.edu> From: Pete Beck To: cube-lovers@ai.mit.edu Subject: Fw: [Dan Galvin: Thought for Tuesday, Nov 12, 1996] Date: Tue, 12 Nov 1996 13:32:50 -0500 X-Msmail-Priority: Normal X-Priority: 3 X-Mailer: Microsoft Internet Mail 4.70.1155 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit ---------- > To: pbeck@qa.pica.army.mil > Subject: [Dan Galvin: Thought for Tuesday, Nov 12, 1996] > Date: Tuesday, November 12, 1996 12:40 PM > > > ----- Forwarded message # 1: > > Received: from postal.tamu.edu by COR6.PICA.ARMY.MIL id ab27608; > 12 Nov 96 9:44 EST > Received: from postal (postal.tamu.edu [128.194.103.24]) by postal.tamu.edu (8.7.5/8.7.5) with SMTP id IAA13954; Tue, 12 Nov 1996 08:35:44 -0600 (CST) > Received: from TAMVM1.TAMU.EDU by TAMVM1.TAMU.EDU (LISTSERV-TCP/IP release > 1.8b) with spool id 9135 for TFTD-L@TAMVM1.TAMU.EDU; Tue, 12 Nov 1996 > 08:33:52 -0600 > Received: from TAMVM1 (NJE origin SMTPH@TAMVM1) by TAMVM1.TAMU.EDU (LMail > V1.2a/1.8a) with BSMTP id 5159; Tue, 12 Nov 1996 04:02:02 -0600 > Received: from tam2000.tamu.edu by tamvm1.tamu.edu (IBM VM SMTP V2R2) with TCP; > Tue, 12 Nov 96 04:02:01 CST > Received: (from galvin@localhost) by tam2000.tamu.edu (8.8.2/8.8.2) id EAA02104 > for TFTD-L@TAMVM1.TAMU.EDU; Tue, 12 Nov 1996 04:02:02 -0600 (CST) > Approved-By: Dan Galvin > Message-ID: <199611121002.EAA02104@tam2000.tamu.edu> > Date: Tue, 12 Nov 1996 04:02:02 -0600 > Reply-To: Dan Galvin > Sender: THOUGHT FOR THE DAY > From: Dan Galvin > Subject: Thought for Tuesday, Nov 12, 1996 > To: Multiple recipients of list TFTD-L > > * > Easiest Color to Solve on a Rubik's Cube: > Black. Simply remove all the little colored stickers on the > cube, and each of side of the cube will now be the original > color of the plastic underneath -- black. According to the > instructions, this means the puzzle is solved. > -- Steve Rubenstein > > ----- End of forwarded messages From cube-lovers-errors@curry.epilogue.com Tue Nov 12 16:12:10 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA02086; Tue, 12 Nov 1996 16:12:10 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <9611121909.AA31489@milo.cfw.com> From: Carey To: Cube-Lovers@ai.mit.edu Subject: Square 1 Date: Tue, 12 Nov 1996 13:59:31 -0500 X-Msmail-Priority: Normal X-Priority: 3 X-Mailer: Microsoft Internet Mail 4.70.1155 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Hello, I'm working on a solution to the Square 1 puzzle. Does anyone know the maximum number of moves required? Also I'm looking for the minimum number of moves required to solve it if you have three consecutive edge wedges. Pete Carey g-carey@cfw.com From cube-lovers-errors@curry.epilogue.com Wed Nov 13 16:27:44 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA04550; Wed, 13 Nov 1996 16:27:44 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 13 Nov 1996 16:13:57 -0500 (EST) From: Michael C Masonjones X-Sender: mcmj@world.std.com To: Cube-Lovers@ai.mit.edu Subject: Re: Square 1 In-Reply-To: <9611121909.AA31489@milo.cfw.com> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Tue, 12 Nov 1996, Carey wrote: > Hello, > I'm working on a solution to the Square 1 puzzle. Does anyone know the > maximum number of moves required? Also I'm looking for the minimum number > of moves required to solve it if you have three consecutive edge wedges. > > Pete Carey > g-carey@cfw.com > I assume this means a permutation of three edge wedges. I can do it in 8 flips through the center divisor, the most convenient way I've found to count moves on Square-1. Start with the permutation on top (in the square/square configuration, of course). Position top and bottom squares so that the left side of the top edge wedge facing you lies above the central turning slot and the right side of the bottom edge wedge lies below the same slot. (If you flip through the center, you still have square configurations, top and bottom). T+n = rotate top n/12 of a turn counterclockwise, as seen from top.. T-n = ..........................clockwise............ B+n, B-n are the same for the bottom face when looking at it from the bottom. F = flip through center slot. Try this: F T+3 F T-1 B-1 F T-2 B+1 F T-3 F T+3 F T-1 B-1 F T-2 B+1 F T+3 Notice that half of this produces two 2-permutations. I'm curious if anyone speed cubes Square 1. My average is about 1:35, with a best time of 1:15 for partial fluke. I can always do it in under 2 minutes for the worst parity situation. How does this compare? There have got to be faster people out there, because I can only do the regular Rubik's cube in 55 seconds on average, which is pretty slow by this group's standards. Mike Masonjones. mcmj@blazetech.com From cube-lovers-errors@curry.epilogue.com Wed Nov 13 22:28:12 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id WAA05354; Wed, 13 Nov 1996 22:28:12 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199611140236.AA29041@world.std.com> To: "cube-lovers@ai.mit.edu" Subject: Re: Square 1 Date: Wed, 13 Nov 96 22:37:52 -0500 From: michael X-Mailer: E-Mail Connection v2.5.03 -- [ From: michael * EMC.Ver #2.5.02 ] -- Whoops! the flip is done with the right hand keeping the left side of the puzzle stationary, if that was not already clear. Doing it with a left hand twist does a double double switch of edge-wedge and corner-wedge pairs. Sorry for the errata post on this. Maybe I should add something else to avoid the complete wast of bandwidth. Since he asked... The only thing I know about confirmed maximal moves for Square 1 is that any possible shape can be put back to two squares with at most 7 flips, and only one configuration requires that many moves. That's the one with a square on one side (CECECECE) and the CEECECCE shape (C= corner wedge, E=edge-wedge) on the other. Satisfyingly symmetric antipode. Corrected permutation of 3 edge wedges: >T+n = rotate top n/12 of a turn counterclockwise, as seen from top.. >T-n = ..........................clockwise............ >B+n, B-n are the same for the bottom face when looking at it from the bottom. > >F = flip through center slot. **** with right hand.**** > >Try this: >F T+3 F T-1 B-1 F T-2 B+1 F T-3 >F T+3 F T-1 B-1 F T-2 B+1 F T+3 > Mike Masonjones. mcmj@blazetech.com From cube-lovers-errors@curry.epilogue.com Wed Nov 13 23:44:48 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id XAA05475; Wed, 13 Nov 1996 23:44:47 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 13 Nov 1996 23:36:47 -0500 Message-Id: <13Nov1996.162951.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: Cube-Lovers@ai.mit.edu In-reply-to: Jerry Slocum's message of 11 Nov 96 18:29:11 EST <961111232911_70410.1050_JHD113-1@CompuServe.COM> Subject: Directory of Puzzlers I received the following note from Jerry Slocum, which he asked me to pass on to Cube-Lovers if I though it was appropriate. I see nothing wrong with it, so here it is. But let me take this opportunity to request that people -not- send things to Cube-Lovers-Request and ask that I forward them on to Cube-Lovers -- that just makes more work for me. Please just send what you want to go to Cube-Lovers to Cube-Lovers. If you feel the need to make some explanation to Cube-Lovers-Request, send a separate message to Cube-Lovers-Request. Thanks! - Alan ------- Begin Forwarded Message ------- Date: 11 Nov 96 18:29:11 EST From: Jerry Slocum <70410.1050@compuserve.com> To: Alan Bawden Subject: Directory of Puzzlers Message-ID: <961111232911_70410.1050_JHD113-1@CompuServe.COM> Dear Alan, In 1994 the Slocum Puzzle Foundation published the Second Edition of the "Directory of Puzzle Collectors and Sellers". It includes a list of 232 puzzle collectors and designers and described in detail what puzzles interest them. More than 120 of them collect, and are interested in, combinatorial Rubik-type puzzles. It also includes 96 mail order puzzle sellers and 147 retail stores that sell puzzles (and were recommended by collectors). I will be sending out letters of inquiry to all collectors & sellers included in the Directory within the next week for updated information for a new Third Edition that will be published early in 1997. Some, but not all, of the Cube Lovers subscribers will receive mailings. I would be glad to invite Cube Lovers who are puzzle collectors and/or sellers to email me if they wish to be in the Third Edition and are not in the current Directory. I will add a section on puzzle related WWW Internet pages and sites and expand the coverage of email in the new Directory. I will send a letter of inquiry to all that request one and provide their mailing address. I am asking for the replies to be returned to me 3 weeks after they are received. Let me know if you have any questions or you would like more details. I would be glad to have this notice posted for Cube Lovers if you think it is appropriate.. Regards, Jerry Slocum 257 South Palm Drive, Beverly Hills, CA 90212 USA Fax 310-274-3644 email:70410.1050@compuserve.com ------- End Forwarded Message ------- From cube-lovers-errors@curry.epilogue.com Thu Nov 14 14:06:14 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA06978; Thu, 14 Nov 1996 14:06:13 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 14 Nov 1996 08:33:29 -0500 (EST) From: Jerry Bryan Subject: y'xy vs. yxy' To: Cube-Lovers Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII Content-transfer-encoding: 7BIT As promised, here is my followup on why the conjugate of x by y is y'xy rather than yxy'. Recall that y'xy informally means undo y, then do x, and finally do y. It seems strange to undo something before you do it, but nonetheless y'xy is the conventional definition of congugacy rather than yxy'. My first reference is Singmaster, Notes on Rubik's 'Magic Cube', Fifth Edition, pp. 57-58. We adopt left to right notation so that (a)xy=y(x(a)). a is the argument, and x and y are permutations which are composed left to right. I paraphrase slightly, but here is what Singmaster says. We desire the conjugate of x by y to be x shifted by y. By "x shifted by y", we mean the following. Suppose in cycle notation we have x=(...,a,b,c...). Then, x shifted by y is z, where z=(...,(a)y,(b)y,(c)y,...). I will defer the presentation of Singmaster's proof, but the final conclusion is that z=y'xy. So our definition of the conjugate of x by y becomes y'xy. By contrast, we have yxy'=(...,(a)y',(b)y',(c)y',....), or x shifted by y'. While I was chasing down this reference in Singmaster, a message arrived from Dan Hoey giving an alternative justification for the y'xy definition. I will quote Dan's message extensively. Dan first credits Jim Saxe with the explanation, and then goes on to say the following. > Suppose we are conjugating elements of a group X >by elements of a group G. Congugation by an element g induces a >permutation on X. This is a very old idea in Cube-Lovers. I believe the first occurrence is in Symmetry and Local Maxima. Elements of the standard cube group G were conjugated by elements of the set M of rotations and reflections of the cube. Conjugation of all the elements of G by a fixed element m of M were viewed as a permutation on G. We denote m'gm by g^m for fixed g in G and fixed m in M. We then denote {m'gm | m in M} as g^M and {m'gm | g in G} as G^m. I normally tend to think of M-conjugation in terms of g^M -- that is, take one fixed element g and calculate its 48 M-conjugates. By contrast, G^m means take each g in G and calculate m'gm using the same fixed m for each g. It is G^m which is a permutation on G. Dan continues: > It is useful to have the mapping from g to its >conjugation permutation be a homomorphism into S[X]. Suppose f is the >mapping > > f: a -> {x -> a' x a}. > >To make this a homomorphism, we must have > > f:a.b -> f(a).f(b) > >so {x -> (a.b)' x (a.b)} = {x -> a' x a} . {x -> b' x b}. > >The right hand side is the product of two permutations. Indeed. It's probably obvious to everybody else how to form the indicated composition of the two permutations, but I was bumfuzzled for a while. Once I figured it out, I just kicked myself for being so dense. Let me explain. My day job is as a bureaucrat, but most semesters I am also adjunct faculty teaching elementary algebra and calculus. As such, I end up teaching simple funcions -- e.g., f(x) = x^2 + 1. You teach students to calculate such things as f(2) or f(3). Then, you teach them such things as f(a) and try to explain that "x is a variable" but "a is a constant that you just don't know the value of". Finally, you get into such things as f(a+b) or f(x^2 + 1). The latter is the one that really confuses most of my students. They can handle "replace x with 2" or "replace x with a". But they have great conceptual difficulty with "replace x with x^2 + 1". The truth is, it is a bit of a different concept because it is really function composition in disguise, although most elementary math books don't teach function composition for several chapters after introducing functions. Anyway, with Dan's equation we really just have a function composition where in the end we replace x with a'xa. So x->b'xb becomes a'xa->b'(a'xa)b. I kick myself because I couldn't quickly figure out the same concept that I am forever emphasizing with my students. Dan continues: > If we are >writing them left to right, as in f.g (x) = g(f(x)), then it is > > {x->b'(a' x a)b} which corresponds to the left hand side. > >>But we write permutation composition from right to left, >f.g(x)=f(g(x)) we would get > > {x->a'(b' x b)a} ? > >for the right hand side, and that is wrong, since a'b' is not (ab)'. > >>People who write right to left define conjugation by a as >f:a->{x->axa'} for this reason. > It seems to me that we could rescue the homomorphism and the yxy' definition, but it would be awkward. We would have to have the mapping from g' to its conjugation permutation be the homomorphism, rather than the mapping from g. Now for Singmaster's proof: given the cycle in our definition of x, we have x:a->b. We need y'xy:(a)y->(b)y. But (a)yy'xy=(a)xy=(b)y. So y'xy carries (a)y to (b)y, and we are done. Let me finish by talking a little more about the equivalence between conjugacy and cycle structure. Again, this is from Singmaster. It is the case in Sn that two elements x and z are conjugate if and only if they have identical cycle structure. Any finite permutation group may be viewed as a subgroup of Sn for suitable choice of n. The theorem may or may not be true in any particular subgroup of Sn. The part about conjugates having identical cycle structure is always true. But the converse may or may not be true. To say that x and z are conjugate means that there exists some y such that z=y'xy. It's easy to see that if x and z have the same cycle structure, then such a y must exist in Sn (e.g., line up the cycles of x with the cycles of z, see what goes to what, and that is a y which will work). The problem in the general case is that a subgroup of Sn might contain x and z which have the same cycle structure, without also containing an appropriate y which would make them conjugate. Singmaster shows that the converse of the theorem is true for the constructable group of the cube, but that it is not true for the standard cube group G. The counter-example is as follows. Let x be a 7-cycle on the corners and an 11-cycle on the edges -- e.g., x=(C1,C2,C3,C4,C5,C6,C7)(E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11). Let z be only slightly different (reversing two corners) -- e.g., z=(C2,C1,C3,C4,C5,C6,C7),(E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11) The obvious conjugating element is y=(C1,C2), which is in the constructable group but which is not in G. There are other conjugating elements, but they are all of the form (C1,C2) (C2,C1,C3,C4,C5,C6,C7)^i (E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11)^i, which also are in the constructable group but not in G. Hence, x and z have the same cycle structure, but are not conjugate in G. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7127 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@curry.epilogue.com Sat Nov 16 21:51:20 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA00394; Sat, 16 Nov 1996 21:51:19 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Sat, 16 Nov 1996 07:24:20 -0500 (EST) From: Nicholas Bodley To: Cube-Lovers@ai.mit.edu Subject: Non-cubical Rubik cousins; physical realizability In-Reply-To: <13Nov1996.162951.Alan@LCS.MIT.EDU> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII (My subject line is a spur-of-the-moment phrase; not deeply considered.) I just got to wondering whether some people have considered theoretical larger analogs of the Magic Domino (btw, would somebody please manufacture some Magic Dominoes? Binary Arts?). To get back on topic, these would be Cube-like puzzles with such "cubie counts" ("Dimensions") as 3X3X4, 3X4X4, etc. Whether these are trivial, I haven't yet thought out; making real, physical ones might not be simple. If this topic is covered in the archives, I apologize; in such a case, could someone recommend non-obvious keywords or names? Is there an agreed-upon concise way of defining the "size/count/dimensions" of a Cube; i.e., a 2X2X2 is a Pocket Cube, a 4... is Rubik's Revenge, etc.? How about "order-3" for a regular Rubik's, or simply (given proper context) "n", so that "2" signifies Pocket, "4" Revenge, etc.? Perhaps it's just a personal reaction, but I find it cumbersome to type "5X5X5" more than a few times, for instance. Thinking about this brings up another topic, and probably a difficult one to completely characterize. Given any arbitrary puzzle composed of cubies, is it always possible to create a mechanism to realize that specific puzzle physically? As far as I know (and here I stick my neck waaaaay out!), there is no theory of mechanisms in the general case that would, for instance, say whether an order-2 is realizable (as we know, it can be made, and has been); the Magic Domino is more of a challenge, imho, in that it isn't as easy to say whether such a structure can be made. Some matters affecting realizability are relatively easy to anticipate, such as the matter of holding the corner cubies in place in a "7" (with all cubies of equal size) when one plane is rotated with respect to the other six. Other matters are a question of what's reasonable to design mechanically; while theoretically possible, some structures might not be at all practical, because of such problems as cumulative friction, lack of rigidity, and dimensional tolerances. Such real-world considerations (unfortunately!) muddy the waters until a really good mind comes along to settle the mud. A preliminary guess at an answer to the question is that probably all "low-order" collections of cubies are realizable, but we are far from having a theory of mechanisms that tells us how to design the innards. I maintain that the mechanism of the ordinary Rubik's Cube is the most ingenious simple one ever invented; I have studied mechanisms to a fair degree. (A good competitor is the programmable pushbutton combination lock that has five buttons in a row. This is mechanical, digital, programmable, combinatorial, and sequential.) Hope and trust this hasn't been a waste of bitspace! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 0. -------------------------------------------------------------------------- From cube-lovers-errors@curry.epilogue.com Sun Nov 17 19:42:27 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id TAA02462; Sun, 17 Nov 1996 19:42:27 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <9611180041.AA06743@jrdmax.jrd.dec.com> Date: Mon, 18 Nov 96 09:41:55 +0900 From: Norman Diamond 18-Nov-1996 0937 To: cube-lovers@ai.mit.edu Subject: Re: Non-cubical Rubik cousins; physical realizability Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=ISO-2022-JP By bandaging a 4x4x4, you can make several variations: 3x3x4, 3x4x4, 2x4x4, and 2x3x3. The 4x4x4 is no longer made or sold through ordinary distribution channels any more, but probably still available from Puzzletts at a high price. At the IPP a few months ago someone was offering a real 2x3x3 for around US$40 I think, which is cheaper than a 4x4x4 is now but still rarer. -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital.] From cube-lovers-errors@curry.epilogue.com Tue Nov 26 20:05:10 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id UAA13447; Tue, 26 Nov 1996 20:05:09 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-Id: <199611270059.TAA07056@dns.city-net.com> To: "Cube-Lovers@AI.MIT.EDU" Subject: Rubic's Revenge Date: Tue, 26 Nov 96 19:56:00 -0500 From: Bill Edwards X-Mailer: E-Mail Connection v2.5.03 -- [ From: Bill Edwards * EMC.Ver #2.5.02 ] -- Anybody know where I can get some more Rubic's Revenges? I wore out my last one more than a year ago. Does a 5x5 matrix Rubic's cube exist? I think I know the general solution, as an extrapolation from the solution to a 4x4. Hope to hear from somebody soon. Bill From cube-lovers-errors@curry.epilogue.com Wed Nov 27 14:23:19 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA15812; Wed, 27 Nov 1996 14:23:19 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com From: Tim Botham To: cube-lovers@ai.mit.edu, Bill Edwards Subject: Re: Rubic's Revenge Date: Tue, 26 Nov 1996 21:34:10 -0800 X-MSMail-Priority: Normal X-Priority: 3 X-Mailer: Microsoft Internet Mail 4.70.1155 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Message-Id: Puzzletts (on the web at http://www.puzzletts.com/) has a good selection of 4x4x4, 5x5x5, and other variations available by mail-order. Ordering can be done through their web page. Tim ---------- > From: Bill Edwards > To: Cube-Lovers@AI.MIT.EDU > Subject: Rubic's Revenge > Date: November 26, 1996 4:56 PM > > -- [ From: Bill Edwards * EMC.Ver #2.5.02 ] -- > > Anybody know where I can get some more Rubic's Revenges? I wore out my last > one more than a year ago. > > Does a 5x5 matrix Rubic's cube exist? I think I know the general solution, > as an extrapolation from the solution to a 4x4. > > Hope to hear from somebody soon. > > Bill From cube-lovers-errors@curry.epilogue.com Wed Nov 27 14:24:05 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA15816; Wed, 27 Nov 1996 14:24:04 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 27 Nov 1996 10:18:53 -0800 From: Aaron Coles Subject: Rubik's Tangle Puzzles To: cube-lovers@ai.mit.edu Reply-to: acoles@fec.gov Message-id: <329C860D.60E0@fec.gov> MIME-version: 1.0 X-Mailer: Mozilla 3.0 (Win16; U) Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7bit Anyone know where I can purchase Rubik Tangle #1 from?? I already have 2-4. I lent it to someone and never got it back. Also has anyone created the 10x10 grid with these puzzles yet?? From cube-lovers-errors@curry.epilogue.com Wed Nov 27 16:05:13 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id QAA16084; Wed, 27 Nov 1996 16:05:12 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Wed, 27 Nov 1996 12:56:20 -0800 From: Don Woods Message-Id: <199611272056.MAA21078@altum.com> To: cube-lovers@ai.mit.edu Subject: Re: Rubik's Tangles Someone (I've deleted the message) recently asked about the Tangles. My recollection is that the four puzzles are all the same except for permutations of the colors. That is, each Tangle consists of the 24 possible distinct pieces, plus one duplicated piece. Which piece is duplicated varies, but the resulting puzzles are the same. (Oh yeah, the pattern of the crossing ropes on each tile is also the same for each puzzle.) Really disappointing, especially since I think there were two distinct solutions, and if they'd varied the mix a bit more they could've had unique solutions as well as having four truly different puzzles. Also, another fellow and I independently did some analyses about three years ago that proved that you cannot make a 10x10 using the four combined puzzles. Presumably the marketing blurb that suggests doing so was written by someone who had no clue whether it was possible or not. Again, if they'd varied the puzzles a bit I have no doubt they could've made the 10x10 achievable as well. -- Don. From cube-lovers-errors@curry.epilogue.com Thu Nov 28 13:10:28 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id NAA18263; Thu, 28 Nov 1996 13:10:28 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Thu, 28 Nov 1996 11:51:48 -0500 (EST) From: Nicholas Bodley To: cube-lovers@ai.mit.edu Subject: Lubricants for puzzles Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII My apologies in advance if this is a repeated topic; it hasn't appeared recently, at least. Question is: What is a really good lubricant for plastic puzzles (such as the classic Cube) with moving parts? What's good for metal machinery isn't necessarily good for plastics; there is even a risk that some additives in metal lubricants would dissolve or etch some plastics. A liquid, probably with a benign solvent to distribute it, would be desirable. Powdered (or colloidal?) PTFE ("Teflon", a Du Pont TM in the USA) particles or flakes should help a good bit. A "carrier" grease (which might as well be a lubricant) would keep any particles in place. Molybdenum disulfide might be good, but might also tend to stain hands and clothing. Some waxes might work. Powdered graphite would probably work loose and make a mess. Lubricants that stain clothing aren't welcome, either! The lubricant must also be benign toward metal, because the Cube is held together with metal screws and tensioned by springs. I suspect that someone, somewhere, knows about a commercial (proprietary) formulation that meets most or all of these criteria. (I would not recommend WD-40, by the way; I expect it would evaporate after some months. It has its place, but I don't think it's a good plastic lubricant.) About 10 years ago, I found such a product, and lubricated some of my puzzles with it, with good success, but then a major personal crisis came, and I lost track of what it was... I'll try to summarize, if any significant number of replies comes by... Thanks in advance, and best regards! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@curry.epilogue.com Fri Nov 29 03:45:13 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id DAA19651; Fri, 29 Nov 1996 03:45:13 -0500 Precedence: bulk Errors-To: cube-lovers-e