From raymond@cps.msu.edu Sat Mar 20 13:24:09 1993 Return-Path: Received: from atlantic.cps.msu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA24769; Sat, 20 Mar 93 13:24:09 EST Received: from pacific.cps.msu.edu by atlantic.cps.msu.edu (4.1/rpj-5.0); id AA20390; Sat, 20 Mar 93 13:24:02 EST Received: by pacific.cps.msu.edu (4.1/4.1) id AA02491; Sat, 20 Mar 93 13:24:02 EST Date: Sat, 20 Mar 93 13:24:02 EST From: raymond@cps.msu.edu Message-Id: <9303201824.AA02491@pacific.cps.msu.edu> To: cube-lovers@ai.mit.edu Subject: Seeking magic dodecahedron Hello cube lovers, I haven't seen any activity on this mailing list in a long time! Cubing is not dead, is it? Anyway, I'm trying to find a good quality magic dodecahedron. Does anyone know where I can get one? Thanks, Carl raymond@cps.msu.edu From news@nntp-server.caltech.edu Sun Mar 21 14:11:53 1993 Return-Path: Received: from punisher.caltech.edu by life.ai.mit.edu (4.1/AI-4.10) id AA11179; Sun, 21 Mar 93 14:11:53 EST Received: from gap.cco.caltech.edu by punisher.caltech.edu (4.1/DEI:4.41) id AA00501; Sun, 21 Mar 93 11:12:38 PST Received: by gap.cco.caltech.edu (4.1/DEI:4.41) id AA28694; Sun, 21 Mar 93 11:10:18 PST To: mlist-cube-lovers@nntp-server.caltech.edu Path: joelong From: joelong@cco.caltech.edu (Joseph Louis Long) Newsgroups: mlist.cube-lovers Subject: Re: Seeking magic dodecahedron Date: 21 Mar 1993 19:10:17 GMT Organization: California Institute of Technology, Pasadena Lines: 13 Message-Id: <1oieipINNs0k@gap.caltech.edu> References: <9303201824.AA02491@pacific.cps.msu.edu> Nntp-Posting-Host: punisher.caltech.edu raymond@cps.msu.edu writes: >Hello cube lovers, > I haven't seen any activity on this mailing list in a long time! I'll say... when I saw this post it reminded me that I've been watching this group for about two months hoping to see some mention of Square-1, but have been left disapointed.. (wimper wimper whine.) :) So let me ask... Does anyone have a solution to Square-1? Is there a simple ``operator'' based method, like there is for the cube? If it is simple enough to explain in text, could someone please post it? Has there been a ``solutions book'' published? obviously in the dark on recent cubic developments, joe From pbeck@pica.army.mil Mon Mar 22 07:59:36 1993 Return-Path: Received: from COR4.PICA.ARMY.MIL ([129.139.68.9]) by life.ai.mit.edu (4.1/AI-4.10) id AA23749; Mon, 22 Mar 93 07:59:36 EST Date: Mon, 22 Mar 93 7:58:16 EST From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Cc: pbeck@pica.army.mil Subject: cubing Message-Id: <9303220758.aa08917@COR4.PICA.ARMY.MIL> NO cubing isn't dead. CFF will hold its annual meeting the day after IPP, which is in amsterdam this year. their newsletter is still published. There is a renewed interest as indicated by sales of magic solids. In particular ISHI PRESS, san jose is comerrcially selling: 5xs super novas - hungarian made regular dodecahedrons I believe that there are 2 books on square 1 in the editing stage. I don't know when/if they will go to print. there was discussion and a solution to square 1 posted to this list, in addition the test issue of a puzzling magazine from ISHI press featured square 1. check the literature. pete beck THE FUTURE IS PUZZLING, BUT CUBING IS FOREVER !! From cosell@world.std.com Mon Mar 22 15:05:44 1993 Return-Path: Received: from world.std.com by life.ai.mit.edu (4.1/AI-4.10) id AA04703; Mon, 22 Mar 93 15:05:44 EST Received: by world.std.com (5.65c/Spike-2.0) id AA25131; Mon, 22 Mar 1993 15:05:38 -0500 Date: Mon, 22 Mar 1993 15:05:38 -0500 Message-Id: <199303222005.AA25131@world.std.com> From: cosell@world.std.com (Bernie Cosell) In-Reply-To: <1oieipINNs0k@gap.caltech.edu> (from joelong@cco.caltech.edu (Joseph Louis Long)) (at 21 Mar 1993 19:10:17 GMT) X-Mailer: //\\miga Electronic Mail (AmiElm 1.18) Reply-To: cosell@world.std.com Path: world.std.com!cosell Organization: Fantasy Farm Fibers To: joelong@cco.caltech.edu (Joseph Louis Long) Subject: Re: Seeking magic dodecahedron Cc: cube-lovers@life.ai.mit.edu Content-Length: 774 In <1oieipINNs0k@gap.caltech.edu> on Mar 21, Joseph Louis Long wrote: } So let me ask... Does anyone have a solution to Square-1? Is there } a simple ``operator'' based method, like there is for the cube? } If it is simple enough to explain in text, could someone please } post it? Has there been a ``solutions book'' published? Dunno about the former, but the answer to the latter is 'yes'. In the April GAMES magazine there is an ad: BAFFLED BY SQUARE 1 Now you can solve the world's most challenging cube puzzle. Clear, easy to unerstand book shows you how. Send $5 to Turn to Square 1 PO Box 1451 Westford, MA 01886 /Bernie\ -- Bernie Cosell cosell@world.std.com Fantasy Farm Fibers, Pearisburg, VA (703) 921-2358 From pbeck@pica.army.mil Thu Mar 25 08:40:16 1993 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA19762; Thu, 25 Mar 93 08:40:16 EST Date: Thu, 25 Mar 93 8:27:15 EST From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Subject: square-1 Message-Id: <9303250827.aa21198@COR4.PICA.ARMY.MIL> PRODUCT ANNOUNCEMENT: SQUARE 1 SOLUTION BOOK & PROPOSED NEWSLETTER RICHARD SNYDER POB 1451 WESTFORD, MA 01886 617-246-0700 VOICE 617-246-1167 FAX has written a book. He says it is do back from printing on april 10. If you are interested CONTACT him directly. THe world is probably waiting for the difinitive book review. From myrberger@e.kth.se Fri Apr 16 13:08:19 1993 Return-Path: Received: from elmer.e.kth.se by life.ai.mit.edu (4.1/AI-4.10) id AA09441; Fri, 16 Apr 93 13:08:19 EDT Received: by e.kth.se (MX V3.2) id 31997; Fri, 16 Apr 1993 19:08:18 +0200 Date: Fri, 16 Apr 1993 19:07:26 +0100 From: myrberger@e.kth.se To: Cube-Lovers@ai.mit.edu Message-Id: <0096B212.D5231500.31997@e.kth.se> Subject: 3x3x3 puzzles and other lists? Hi, I have recently found this mailing list and have just finished reading through the earlier postings. Perhaps this don't really belong to this list, but I have a list of different puzzles that in their final state is a 3x3x3 cube. Among them are, of course, Rubik's cube. The Soma cube and related puzzles are also there. If you'd like a copy of the list, please MAIL me. I also wonder if you know about other mailing lists or such which deals with puzzles (preferrable mechanical). (I know about USENET/NEWS group rec.puzzles.) Thanks Johan MAIL: myrberger@e.kth.se From cosell@world.std.com Wed May 26 22:37:04 1993 Return-Path: Received: from world.std.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24908; Wed, 26 May 93 22:37:04 EDT Received: by world.std.com (5.65c/Spike-2.0) id AA27291; Wed, 26 May 1993 22:37:02 -0400 Date: Wed, 26 May 1993 22:37:02 -0400 Message-Id: <199305270237.AA27291@world.std.com> From: cosell@world.std.com (Bernie Cosell) X-Mailer: //\\miga Electronic Mail (AmiElm 2.43) Reply-To: cosell@world.std.com Organization: Fantasy Farm Fibers To: cube-lovers@life.ai.mit.edu Subject: Ishi Intternational Puzzles. NEW! (fwd) Content-Length: 2127 On May 20, Anton Dovydaitis wrote: [-------------------- text of forwarded message follows --------------------] Ishi Press International is now directly accessable via e-mail: our on-line InterNet address is 'ishius@ishius.com' and for European customers it's 'ishi@cix.compulink.co.uk'. Ishi Press International sells a wide variety of puzzles from simple glass, wood and metal puzzles, to collector's items. Our line consists of several hundred puzzles, including: Toyo Puzzle City Glass, Hikimi Puzzland Wood and Cast Iron puzzles by Nob Arjeu wood puzzles from France Wood, string and wire disentanglement puzzles by Jean-Claude Constantin Magic Bottle puzzles Handcrafted English Puzzles, including handblown glass Klein bottles, Single and Double Hourglass Paradoxes, and Wooden Trench Puzzles Wooden puzzles by Bill Cutler and Stuart Coffin Puzzle books by Slocum and Nob Yamanaka Kumiki Works wooden burr puzzles Traditional inlaid wood Trick Puzzle Boxes by Okiyama and Ninomiya Yamanaka Kumiki Works wooden burr puzzles Kamei puzzle boxes, including the Top Box, Die Box, Book, Cup and Saucer and the Fan Bolt puzzles by Strijbos Puzzling People Puzzles, wooden 3-D jigsaw puzzles from England, including the Flummox Wire Puzzle Sculptures by Rick Irby, including a 3' Dragon, The Hong Kong Horror and puzzle earrings and more. Our puzzle line is always increasing: this summer we will be carrying wood puzzles from Pentangle. To receive our puzzle catalog with color photographs, please e-mail your real mailing address to 'ishius@ishius.com'. To receive regular e-mail on our latest offerings, e-mail us at 'ishius@ishius.com'. Please be sure to put the word PUZZLE in the subject header, as many of our customers are interested in GO, not puzzles. Please write me if you have any questions. ================================================ Anton Dovydaitis Ishi Press International ishius@ishius.com 76 Bonaventura Drive Tel: 800/859-2086 San Jose, CA 95134 FAX: 408/944-9110 [------------------------- end of forwarded message ------------------------] From dik@cwi.nl Sun Jun 13 19:53:20 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19146; Sun, 13 Jun 93 19:53:20 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA25903 (5.65b/3.8/CWI-Amsterdam); Mon, 14 Jun 1993 01:53:19 +0200 Received: by boring.cwi.nl id AA19557 (4.1/2.10/CWI-Amsterdam); Mon, 14 Jun 93 01:53:16 +0200 Date: Mon, 14 Jun 93 01:53:16 +0200 From: Dik.Winter@cwi.nl Message-Id: <9306132353.AA19557.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Contents of CFF31 Last Friday I received issue #31 of Cubism For Fun. A short summary of the contents: 1. Short articles by Jan de Geus and Frans de Vreugd about Cube Day 1992. 2. An article by Herbert Kociemba about a classification of pretty patterns on the cube. 3. Reflections by Tom Verhoeff about puzzles and computers. 4. Announcement by Koos Verhoef and Tom Verhoeff of a contest *. 5. A short article by Jaques Haubrich about Rubik's Tangle and how to position 24 parts in a cube like way (four on a side). 6. An article by Jan Verbakel about the creation of castles with solid pentominoes. 7. A short article by Trevor Wood on the pecking of octacubes. 8. A short article by Jaques Haubrich about a difficult packing problem. 9. An article by David Singmaster about a gathering in Atlanta in honor of Martin Gardner. (Nearly the whole puzzling world appears to have been there.) 10. A new contest by Anton Hanegraaf. 11. Announcement of the 13th Dutch cube day on August 22 in Amsterdam. This day is next to the 13th International Puzzle Party. * This is an interesting puzzle indeed. Consider the densest sphere packing in 3D. This is the packing where you start with a lattice of spheres based on a triangular lattice, and put on top of it another, similar, lattice such that each sphere of the new layer fits in a hole in the lower layer. Add more layers. Pick from that all possible configurations of 4 connected spheres. There are 25 such configurations. The puzzle is to create from these 25 pieces a pyramid with a side of 8 spheres (which contains 120 spheres), with a hole at the center that consists of a pyramid with a side of 4 spheres (remember those sums of triangular numbers!). It is not known whether there is a solution. The authors tell how they have a TRS-80 now running 5 years on this problem, using backtracking techniques. Until now the first 6 pieces did not move. The could fit 24 pieces already 521,010 times. The puzzle was first announced at the previous Cube Day. CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch Cubists Club). It appears a bit irregular, but a few times a year. Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to approximately $ 15.-. Information: Anton Hanegraaf Heemskerkstraat 9 6662 AL Elst The Netherlands (sorry, there is no e-mail address). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From @cunyvm.cuny.edu:Matt_Drobel@Novell.COM Tue Jun 29 11:02:15 1993 Received: from CUNYVM.CUNY.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01717; Tue, 29 Jun 93 11:02:15 EDT Received: from ns.Novell.COM by CUNYVM.CUNY.EDU (IBM VM SMTP V2R2) with TCP; Mon, 28 Jun 93 15:58:19 EDT Received: by ns.Novell.COM (4.1/SMI-4.1) id AA12073; Mon, 28 Jun 93 13:58:30 MDT Received: by MHS.Novell.COM id 259C7D7D810A02D0; Mon, 28 Jun 93 13:58:29 MDT Return-Path: Return-Receipt-To: Matt_Drobel@novell.com Precedence: special-delivery To: cube-lovers%ai.mit.edu@cunyvm.cuny.edu Message-Id: <4E917D7D010A02D0@MHS.Novell.COM> Subject: From: Matt_Drobel@novell.com (Drobel, Matthias) Date: Mon, 28 Jun 93 13:55:57 MDT Total-To: cube-lovers%ai.mit.edu@cunyvm.cuny.edu signoff cube-lovers From @cunyvm.cuny.edu:rbm8p@darwin.clas.virginia.edu Tue Jun 29 23:15:45 1993 Return-Path: <@cunyvm.cuny.edu:rbm8p@darwin.clas.virginia.edu> Received: from CUNYVM.CUNY.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29385; Tue, 29 Jun 93 23:15:45 EDT Received: from virginia.edu by CUNYVM.CUNY.EDU (IBM VM SMTP V2R2) with TCP; Tue, 29 Jun 93 13:36:04 EDT Received: from darwin.clas.virginia.edu by uvaarpa.virginia.edu id aa19787; 29 Jun 93 13:36 EDT Received: by darwin.clas.Virginia.EDU (5.65c/1.34) id AA23767; Tue, 29 Jun 1993 17:36:13 GMT Date: Tue, 29 Jun 1993 17:36:13 GMT From: Richard Burd Macdonald Message-Id: <199306291736.AA23767@darwin.clas.Virginia.EDU> X-Mailer: Mail User's Shell (7.2.3 5/22/91) To: cube-lovers%ai.mit.edu@cunyvm.cuny.edu unsubscribe From hoey@aic.nrl.navy.mil Tue Jul 6 17:29:01 1993 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14021; Tue, 6 Jul 93 17:29:01 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA28843; Tue, 6 Jul 93 17:28:56 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Tue, 6 Jul 93 17:28:55 EDT Date: Tue, 6 Jul 93 17:28:55 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9307062128.AA04699@sun13.aic.nrl.navy.mil> To: Cube-Lovers@life.ai.mit.edu Subject: Rubik's Shrimp I hear on Usenet that there's a show on BBC1 called Wildlife 100, where people saw a Mantis Shrimp playing with a Rubik's Cube. Reports are inconclusive as to whether it was able to actually turn faces, or whether it just waved it around, or even just took it apart. Now if they could get it to turn faces, presumably they could film it and play it back in reverse.... Dan Hoey Hoey@AIC.NRL.Navy.Mil From CPELLEY@delphi.com Tue Jul 20 21:30:11 1993 Return-Path: Received: from bos3a.delphi.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA02226; Tue, 20 Jul 93 21:30:11 EDT Received: from delphi.com by delphi.com (PMDF V4.2-11 #4520) id <01H0S38KLDRA91WX1L@delphi.com>; Tue, 20 Jul 1993 21:29:06 EDT Date: Tue, 20 Jul 1993 21:29:06 -0400 (EDT) From: CPELLEY@delphi.com Subject: New idea for a puzzle To: cube-lovers@life.ai.mit.edu Message-Id: <01H0S38KLNEW91WX1L@delphi.com> X-Vms-To: INTERNET"cube-lovers@ai.ai.mit.edu" Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII Content-Transfer-Encoding: 7BIT I have a great concept for a new variant on Rubik's Cube. Where can I contact the people who manufacture these puzzles today? I understand Jean-Claude Constantin and Uwe Meffert are still around. The idea is for a dodecahedral puzzle that is sliced up differently than a Skewb or Megaminx. From ronnie@cisco.com Wed Jul 28 19:37:15 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00159; Wed, 28 Jul 93 19:37:15 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA06770 (5.67a/IDA-1.5 for ); Wed, 28 Jul 1993 16:37:08 -0700 Message-Id: <199307282337.AA06770@lager.cisco.com> To: Cube-Lovers@life.ai.mit.edu Subject: Hint wanted for 4x4x4 Date: Wed, 28 Jul 1993 16:37:08 -0700 From: "Ronnie B. Kon" I've been beating my head against the order 4 Rubik's cube for long enough, and I want a hint. (Not a solution--I have a solution book if I wanted to use it). My problem is I cannot flip a pair of adjacent edges (this is equivalent to not being able to exchange a pair of knights-move separated edges). All my other transformations have no side effects, so I can solve the edges first. But I can't see how to just affect two of them. I tend to solve using commutators, but I don't see a way here. The move I use on the top moves the marked pieces clockwise (this pattern . . 0 . . . . . . . . 0 . 0 . . rotates and reflects, of course). There is no way to combine these into a pair exchange (after doing the move, you still have two pieces out of place--nothing changed from the original). I tried to find a move that would exchange three pieces, the third being the correctly placed piece next to one of the incorrectly placed pieces (ie., treat a right edge cubie as if it should be a left edge cubie) but this can easily be shown as impossible: Define the parity of a piece as being left if it is a left edge cubie when the red facelet is up, right if it is a right edge cubie when the red facelet is up. The parity is undefined if there is no red facelet. There are only three moves available that affect an edge cubie--none of them alter the parity. QED So, what am I missing? As I said before, I really just want a hint here. Ronnie From diamond@jit081.enet.dec.com Wed Jul 28 20:13:19 1993 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01523; Wed, 28 Jul 93 20:13:19 EDT Received: by enet-gw.pa.dec.com; id AA04792; Wed, 28 Jul 93 17:13:17 -0700 Message-Id: <9307290013.AA04792@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Wed, 28 Jul 93 17:13:18 PDT Date: Wed, 28 Jul 93 17:13:18 PDT From: 29-Jul-1993 0914 To: cube-lovers@life.ai.mit.edu Apparently-To: cube-lovers@life.ai.mit.edu Subject: Re: Hint wanted for 4x4x4 Ronnie B. Kon asked for a hint but not a solution. So here is a hint. If I understood correctly your descriptions of two transforms which you asserted to be equivalent, then in fact they are not equivalent. (Of course, if I didn't understand your descriptions correctly, then this isn't a hint.) -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From dik@cwi.nl Wed Jul 28 20:17:38 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01767; Wed, 28 Jul 93 20:17:38 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA06051 (5.65b/3.8/CWI-Amsterdam); Thu, 29 Jul 1993 02:17:27 +0200 Received: by boring.cwi.nl id AA08740 (4.1/2.10/CWI-Amsterdam); Thu, 29 Jul 93 02:17:25 +0200 Date: Thu, 29 Jul 93 02:17:25 +0200 From: Dik.Winter@cwi.nl Message-Id: <9307290017.AA08740.dik@boring.cwi.nl> To: Cube-Lovers@life.ai.mit.edu, ronnie@cisco.com Subject: Re: Hint wanted for 4x4x4 > I've been beating my head against the order 4 Rubik's cube for long > enough, and I want a hint. (Not a solution--I have a solution book if > I wanted to use it). Yes, it is not really simple. > ... > I tend to solve using commutators, but I don't see a way here. The > move I use on the top moves the marked pieces clockwise (this pattern > . . 0 . > . . . . > . . . 0 > . 0 . . > rotates and reflects, of course). There is no way to combine these > into a pair exchange (after doing the move, you still have two pieces > out of place--nothing changed from the original). Still you are halfway there if you are willing to forgo the pattern of the centers (which can always be done later). Turn the front face (at the bottom in the frawing), the right face and the back face one quarter turn before your turn, and back after. Observe that that constitutes a cycle of three edge cubies in a single middle slice. Combine with a quarter turn of that middle slice. > So, what am I missing? As I said before, I really just want a hint > here. I hope that is enough of a hint and not enough of a giveaway. (I thought there was a shorter sequence, but I disremember at the moment.) My solution for the 4x4x4 always was: first corners, next edges and finally centers. Because there are many identical pieces for the centers those are reasonably simple. It would be much more difficult if each center had its own place. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From hoey@aic.nrl.navy.mil Thu Jul 29 08:36:32 1993 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20298; Thu, 29 Jul 93 08:36:32 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA08504; Thu, 29 Jul 93 08:36:14 EDT Date: Thu, 29 Jul 93 08:36:14 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9307291236.AA08504@Sun0.AIC.NRL.Navy.Mil> To: Cube-Lovers@life.ai.mit.edu Cc: Dik.Winter@cwi.nl, ronnie@cisco.com (Ronnie B. Kon) Subject: Re: Hint wanted for 4x4x4 Newsgroups: ml.cube-lovers In-Reply-To: <9307290017.AA08740.dik@boring.cwi.nl> Organization: Navy Center for Applied Research in AI Cc: ronnie@cisco.com (Ronnie B. Kon) asks for hints for exchanging a pair of edges: > > I tend to solve using commutators, but I don't see a way here.... The key is that commutators are always odd permutations. So do the move that is an odd permutation of the edges, then use commutators. Dik.Winter@cwi.nl (dik t. winter) shows a neat way of moving most of the cubies affected by the odd permutation into the top slice, where they can be cycled using Ronnie's commutator, which cycles the TB(R), TR(F), and TF(L) cubies: > > . . 0 . > > . . . . > > . . . 0 > > . 0 . . (I'm naming them by their edge and their near side.) I suspect Ronnie is using something like (F Ti F') T (F Ti' F) T (F Ti F') T^2 (F Ti' F). (For this I'm using "i" to mark inside slabs). But you can cycle the FL(T), FR(T), RB(T) cubies directly, using a different commutator. With more effort, there is a commutator that doesn't mess up face centers. We are getting to the part where it's hard to distinguish between the hintable and the obvious, but if people send me email about not being able to figure out what commutators I'm talking about I'll answer, and post them if such nobility is common. >My solution for the 4x4x4 always was: first corners, next edges and finally >centers. Because there are many identical pieces for the centers those are >reasonably simple. It would be much more difficult if each center had its >own place. As I mentioned years ago, I've made places for mine by cutting corners of to clusters of face centers and their neighboring edges on each face. +----+----+----+----+ | | | | | | | | | | +----+---( )---+----+ | | | | | | | | | | +---( )---+----+----+ | | | | | | | | | | +----+----+----+----+ | | | | | | | | | | +----+----+----+----+ It's not that hard to fix the face centers, just time-consuming. It's a good thing we do the edges first, though, or it would be hard to figure where the cuts go. Dan Hoey Hoey@AIC.NRL.Navy.Mil ( So much discussion on this quiescent list will probably flush out someone who wants to unsubscribe. Remember to send your note to cube-lovers-request@ai.ai.mit.edu to avoid annoyance.) From ncramer@bbn.com Thu Jul 29 09:40:49 1993 Return-Path: Received: from LABS-N.BBN.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22477; Thu, 29 Jul 93 09:40:49 EDT Message-Id: <9307291340.AA22477@life.ai.mit.edu> Date: Thu, 29 Jul 93 8:59:44 EDT From: Nichael Cramer To: "Ronnie B. Kon" Cc: Cube-Lovers@life.ai.mit.edu Subject: Re: Hint wanted for 4x4x4 Hi Ronnie Let me try to restate the problem slightly to make sure we are talking about the same problem. Basically, you can solve the entire cube, except that the pieces "1" and "2" in the diagram are flipped/exchanged: XXXX XXXX XXXX X12X Assuming this is the problem, the hint is as follows: The problem here is that some of the face pieces are not really in their right places. In short, one of the center slices is 1/4 turn out of phase. The simplest way to proceed (at least for me) is to move to the following state: XX2X (i.e. rotate one of the central slices 1/4 turn) XXOX XXOX X1OX ^ | Now, you can solve for pieces "1" and "2" and --using these pieces as a landmark-- proceed from there. Hint #2: You can help avoid this problem by solving the face pieces last. Extra Credit: Actually, the state as shown in the first diagram above is pretty interesting in that the analogous position on a 3X3X3 cube (i.e. a single flipped edge cube) is of course impossible. From this state it is relatively easy to get to another, very interesting state: namely the 4X4X4 appears to be completely solved except that two opposite corners are exchanged. (Again, this is obviously impossible on the 3X3X3.) Left as an exercise for the reader. ;) Nichael ncramer@bbn.com Dr Pepper: It's not just for breakfast any more. From hoey@aic.nrl.navy.mil Thu Jul 29 10:11:15 1993 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23418; Thu, 29 Jul 93 10:11:15 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA09096; Thu, 29 Jul 93 10:11:13 EDT Date: Thu, 29 Jul 93 10:11:13 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9307291411.AA09096@Sun0.AIC.NRL.Navy.Mil> To: Cube-Lovers@life.ai.mit.edu Subject: Oops... Re: Hint wanted for 4x4x4 Cc: Dik.Winter@cwi.nl, ronnie@cisco.com (Ronnie B. Kon) I wrote: ? The key is that commutators are always odd permutations. So do the ? move that is an odd permutation of the edges, then use commutators. But I *Meant*: ! The key observation is that commutators are always even ! permutations. So you to perform an odd permutation on edge, you ! should do the move that is an odd permutation of the edges, then use ! commutators. Dan From tomgm@physics.purdue.edu Thu Jul 29 11:02:30 1993 Return-Path: Received: from bohr.physics.purdue.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26047; Thu, 29 Jul 93 11:02:30 EDT Received: by bohr.physics.purdue.edu (5.65/2.7) id AA21974; Thu, 29 Jul 93 10:05:22 -0500 Message-Id: <9307291505.AA21974@bohr.physics.purdue.edu> From: Tom G. Miller Subject: Re: Hint wanted for 4x4x4 To: ronnie@cisco.com (Ronnie B. Kon) Date: Thu, 29 Jul 93 10:05:22 EST Cc: Cube-Lovers@life.ai.mit.edu In-Reply-To: <199307282337.AA06770@lager.cisco.com>; from "Ronnie B. Kon" at Jul 28, 93 4:37 pm X-Mailer: ELM [version 2.3 PL11] Ronnie, It's been so long since I've messed around with my 4x4x4 that I can't answer your question directly, however when I see descriptions for the method in which people solve the 4x4x4 it is usually different from the way I first solved it: What I did was to pair up the middle two edgies, and the four central face cubes. There are few enough restraints that this is not too hard to do for someone who has never touched a 4x4x4 cube. One then has a cube like with faces similar to the following: r b b g y o o b y o o b r y y w One can then "pretend" it is a 3x3x3 cube and then solve it. Unfortunately you will occasionally end up in an orbit of the "pseudo-3x3x3" that is impossible to solve. Oh well... scramble it and try it again. Using this technique I was able to solve a scrambled 4x4x4 cube within an hour or so of when I set my hands on one. Needless to say, this is NOT a good technique for solving a 4x4x4 cube if one is interested only in the 4x4x4. In fact I suspect it is a pretty awful algorithm, especially since you frequently end up in an unsolveable orbit using your standard 3x3x3 techniques. But it is a useful trick for maximizing the hard work one used in learning the 3x3x3. As most people who have a 4x4x4 realize, if you never make any twists of a solved 4x4x4 cube except along the center, you of course have a 2x2x2. And as I described, if you only make moves 1-deep, it is equivalent to a 3x3x3, and of course it is also a 4x4x4. Tom Miller tomgm@physics.purdue.edu From ronnie@cisco.com Fri Jul 30 00:46:55 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01754; Fri, 30 Jul 93 00:46:55 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA12561 (5.67a/IDA-1.5 for ); Thu, 29 Jul 1993 21:46:49 -0700 Message-Id: <199307300446.AA12561@lager.cisco.com> To: Cube-Lovers@life.ai.mit.edu Subject: Re: Hint wanted for 4x4x4 Date: Thu, 29 Jul 1993 21:46:48 -0700 From: "Ronnie B. Kon" Thanks to all who responded. I haven't yet got what I consider a solution for my problem (shift a slice and resolve is my current method which is slow and ugly) but at least I understand my problem slightly better. A few questions: 1. What is the definition of parity by which commutators are even, but slice turns are odd? I haven't been able to come up with a cube-wide parity. (I know no group theory). 2. How many orbits does the order 4 cube have? I can only think of three (twirling a corner cubie). Then again, I haven't painted the facelets yet, so there could be orbits I haven't begun to see involving them. 3. Would an order 6 cube have any challenge beyond the order 4? I think the answer is no--if you are able to solve the 3-cube and the 4-cube you can solve any cube. From dik@cwi.nl Mon Aug 2 20:52:23 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05478; Mon, 2 Aug 93 20:52:23 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA19947 (5.65b/3.8/CWI-Amsterdam); Tue, 3 Aug 1993 02:52:21 +0200 Received: by boring.cwi.nl id AA23253 (4.1/2.10/CWI-Amsterdam); Tue, 3 Aug 93 02:52:19 +0200 Date: Tue, 3 Aug 93 02:52:19 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308030052.AA23253.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Fitting puzzle solved Some time ago I posted an article (c.q. mailed a message) describing the contents of Cubism For Fun, the newsletter published by the Dutch Cubists Club (NKC). In that article (message) I gave a more elaborate description about a problem involving fitting pieces. Briefly: The base problem is as follows. Build a tetrahedron consisting of balls, 8 balls on an edge. When you look at the lattice induced by this tetrahedron after some thinking you will find there are 25 ways to pick 4 connected balls. Now take those 25 ways and make "pieces" from it. Again, go back to the tetrahedron and inside it create a hollow tetrahedron with 4 balls on an edge. The remainder requires 100 balls to fill. Try to do that with the 25 "pieces" you just created. This has been a fairly long-standing problem but it is now (partly) solved. I just had word that Jan de Ruiter from Purmerend (the Netherlands) found a number of solutions. Details will likely be presented in a forthcoming issue of CFF. An amusing side-note. Between the 25 pieces there are two that can be created interlocked. It is not clear whether it is possible to separate those two pieces by hand when interlocked, so it is not clear whether a solution that has those two pieces interlocked really is a solution. The first solutions Jan de Ruiter found *had* those two pieces interlocked. But after some time he found a solution with those two pieces far away from each other, so there is really a true solution. Remaining questions: How many solutions are there? How many do not have those two pieces interlocked? Is it possible to separate those two pieces when interlocked? (The last puzzle resembles one of those chinese metal separation puzzles.) (Information about CFF can be obtained from Anton Hanegraaf, Heemskerkstraat 9, 6662 AL Elst, The Netherlands. E-mail is now also possible: gm@phys.uva.nl.) dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From dik@cwi.nl Mon Aug 2 21:10:32 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06234; Mon, 2 Aug 93 21:10:32 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA20559 (5.65b/3.8/CWI-Amsterdam); Tue, 3 Aug 1993 03:10:23 +0200 Received: by boring.cwi.nl id AA23300 (4.1/2.10/CWI-Amsterdam); Tue, 3 Aug 93 03:10:22 +0200 Date: Tue, 3 Aug 93 03:10:22 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308030110.AA23300.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Diameter of cube group? I have now running (for about 60 days already) a program that implements Kociemba's algorithm to solve the cube. It tries to solve random configurations and stops when a solution of 20 turns or less is found. The random configurations are created by doing 100 random turns. Until now, with 9000 configurations tried, all proved to be solvable in 20 turns or less. This strongly suggests that the diameter of the cube group is at most 21, or perhaps 22; but not more. The figure of 9000 configurations in 60 days indicates that solution of one configuration takes slightly less than 10 minutes. This is contrary to what I thought was possible. Whenever I tried configurations they were mostly solved within 2 or 3 minutes. This suggests that the random configurations are more difficult to solve than what I and many others brought up as possible difficult patterns. But I still need to do some analysis on the ouput (now 3 Mb of data). Continuing and waiting for a config that requires 21 turns, dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From W.Taylor@math.canterbury.ac.nz Mon Aug 2 22:20:16 1993 Return-Path: Received: from CANTVA.CANTERBURY.AC.NZ by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08764; Mon, 2 Aug 93 22:20:16 EDT Received: from math.canterbury.ac.nz by csc.canterbury.ac.nz (PMDF V4.2-13 #2553) id <01H1B8FCMWR4AYP5CF@csc.canterbury.ac.nz>; Tue, 3 Aug 1993 14:19:57 +1200 Received: from sss330.math.canterbury.ac.nz by math.canterbury.ac.nz (4.1/SMI-4.1) id AA23489; Tue, 3 Aug 93 14:19:52 NZS Date: Tue, 3 Aug 93 14:19:52 NZS From: W.Taylor@math.canterbury.ac.nz (Bill Taylor) Subject: re: Diameter of cube group? To: Cube-Lovers@life.ai.mit.edu Message-Id: <9308030219.AA23489@math.canterbury.ac.nz> X-Envelope-To: Cube-Lovers@AI.AI.MIT.EDU Content-Transfer-Encoding: 7BIT Fascinating news from Dik Winter about the solving of 9000 configurations. Can someone please remind us exactly what Kociemba's algorithm is; or at least a breif outline of how it works. I know I've heard the name before, but can't remember anything about it. Thanks, Bill Taylor. wft@math.canterbury.ac.nz From dik@cwi.nl Tue Aug 3 18:44:13 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18203; Tue, 3 Aug 93 18:44:13 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA19810 (5.65b/3.8/CWI-Amsterdam); Wed, 4 Aug 1993 00:44:10 +0200 Received: by boring.cwi.nl id AA26535 (4.1/2.10/CWI-Amsterdam); Wed, 4 Aug 93 00:44:09 +0200 Date: Wed, 4 Aug 93 00:44:09 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308032244.AA26535.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Kociemba's algorithm I have received a few requests for information about the algorithm and for the program. I have put the program available for ftp. In the set of files there is also a Description I edited from a number of messages I mailed a long time ago to this mailing list. They give a reasonable description of the algorithm. Ftp to ftp.cwi.nl. File is: /pub/dik/cube.tar.Z. Do not forget to set binary mode. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From acw@bronze.lcs.mit.edu Thu Aug 5 17:03:17 1993 Return-Path: Received: from bronze.lcs.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16999; Thu, 5 Aug 93 17:03:17 EDT Received: by bronze.lcs.mit.edu id AA22841; Thu, 5 Aug 93 17:02:45 EDT Date: Thu, 5 Aug 93 17:02:45 EDT From: acw@bronze.lcs.mit.edu (Allan C. Wechsler) Message-Id: <9308052102.AA22841@bronze.lcs.mit.edu> To: Dik.Winter@cwi.nl Cc: cube-lovers@life.ai.mit.edu In-Reply-To: Dik.Winter@cwi.nl's message of Tue, 3 Aug 93 03:10:22 +0200 <9308030110.AA23300.dik@boring.cwi.nl> Subject: Diameter of cube group? I wonder about the validity of your Monte Carlo analysis. It seems to be based on an intuition about how fast the number of configurations falls off with the distance from SOLVED. I share the intuition, but I'm not sure I can rigorize it, and that makes me cautious. What prevents a group from having a "pointy tail", that is, a "corridor" of elements at increasing distances from the identity? In fact, does the number of elements as a function of distance have to be unimodal? Could this function have a "waist"? Intuitively, this sounds impossible, but I am wondering what constraints on such functions are known. From ronnie@cisco.com Thu Aug 5 19:55:48 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23537; Thu, 5 Aug 93 19:55:48 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA23583 (5.67a/IDA-1.5 for ); Thu, 5 Aug 1993 16:55:37 -0700 Message-Id: <199308052355.AA23583@lager.cisco.com> To: acw@bronze.lcs.mit.edu (Allan C. Wechsler) Cc: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu Subject: Re: Diameter of cube group? In-Reply-To: Your message of "Thu, 05 Aug 1993 17:02:45 EDT." <9308052102.AA22841@bronze.lcs.mit.edu> Date: Thu, 05 Aug 1993 16:55:36 -0700 From: "Ronnie B. Kon" Disclaimer: this sounds more authoritative than is intended--I really don't know what I'm talking about. It couldn't be very pointy. From the most distant configuration, there are 6 positions immediately before it. There are 6^2 two steps away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually. This is necessarily so, as if any of the configurations reachable with two twists (for example) are closer in than (max - 2) steps then you could reach the maximum configuration by getting there and then doing the two steps. This gives me the feeling that Monte Carlo is fairly valid. (How's that for rigor?) Ronnie > I wonder about the validity of your Monte Carlo analysis. It seems > to be based on an intuition about how fast the number of configurations > falls off with the distance from SOLVED. I share the intuition, but > I'm not sure I can rigorize it, and that makes me cautious. > > What prevents a group from having a "pointy tail", that is, a "corridor" > of elements at increasing distances from the identity? In fact, does > the number of elements as a function of distance have to be unimodal? > Could this function have a "waist"? Intuitively, this sounds > impossible, but I am wondering what constraints on such functions are known. > From dik@cwi.nl Thu Aug 5 19:55:54 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23538; Thu, 5 Aug 93 19:55:54 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA05077 (5.65b/3.9/CWI-Amsterdam); Fri, 6 Aug 1993 01:55:52 +0200 Received: by boring.cwi.nl id AA05297 (4.1/2.10/CWI-Amsterdam); Fri, 6 Aug 93 01:55:50 +0200 Date: Fri, 6 Aug 93 01:55:50 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308052355.AA05297.dik@boring.cwi.nl> To: acw@bronze.lcs.mit.edu Subject: Re: Diameter of cube group? Cc: cube-lovers@life.ai.mit.edu > I wonder about the validity of your Monte Carlo analysis. It seems > to be based on an intuition about how fast the number of configurations > falls off with the distance from SOLVED. I share the intuition, but > I'm not sure I can rigorize it, and that makes me cautious. I am not sure (that is obvious). But when looking at other (similar) puzzles I did I think it is a save guess. > What prevents a group from having a "pointy tail", that is, a "corridor" > of elements at increasing distances from the identity? The groups I did calculate in full do *not* have a pointy tail. This holds for 2x2x2, 3x3x3 corners only, magic domino. I think it would be a big surprise if there is a pointy tail. Obviously we can say a priory that there is not a single configuration opposite from start, so the tail is not very pointy, if it is at all. For instance for the magic domino the tail of the list of number of configuration a certain distance from start is: 14: 508704668 15: 232904952 16: 14508468 17: 129376 18: 112 With the maximum at 14 turns. (Here I took the table where only a single solution is allowed; i.e. no full rotations of the domino.) 1 in 2 (approx.) configurations requires 14 turns or more. 1 in 100 requires 16 turns or more. Of course the number of configurations of the cube is quite a bit more. Still after doing about 9000 configurations not a single one is found that requires more than 20 turns. If we assume a picture similar to the domino (which in my opinion is a save guess), there might be configurations that retuire 21 or perhaps 22 turns, but more is extremely unlikely. However, there is a remaining question; is the random choice of configuration unbiased? I think it is. To create a random configuration I do 100 random turns chosen from 18 possible turns (quarter turns, half turns and reverse turns). The random number generator is (as far as I know) quite good (Berkeley Unix's random). dik From dik@cwi.nl Thu Aug 5 20:01:34 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23600; Thu, 5 Aug 93 20:01:34 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA05147 (5.65b/3.9/CWI-Amsterdam); Fri, 6 Aug 1993 02:01:27 +0200 Received: by boring.cwi.nl id AA05312 (4.1/2.10/CWI-Amsterdam); Fri, 6 Aug 93 02:01:26 +0200 Date: Fri, 6 Aug 93 02:01:26 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308060001.AA05312.dik@boring.cwi.nl> To: ronnie@cisco.com Subject: Re: Diameter of cube group? Cc: cube-lovers@life.ai.mit.edu The last remark first: > This gives me the feeling that Monte Carlo is fairly valid. (How's > that for rigor?) Not very ;-). > It couldn't be very pointy. From the most distant configuration, > there are 6 positions immediately before it. There are 6^2 two steps > away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually. This still can create a pointy tail; just as pointy as the front. My experience is that the tail is much more blunt than the front. That there are already more than a single configuration at maximum distance makes that reasonable. From CPELLEY@delphi.com Fri Aug 6 02:47:53 1993 Return-Path: Received: from bos3a.delphi.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04343; Fri, 6 Aug 93 02:47:53 EDT Received: from delphi.com by delphi.com (PMDF V4.2-11 #4520) id <01H1EMUYYXKW91XFD4@delphi.com>; Fri, 6 Aug 1993 00:53:23 EDT Date: Fri, 06 Aug 1993 00:53:23 -0400 (EDT) From: CPELLEY@delphi.com Subject: Square-1 Puzzle Party To: cube-lovers@life.ai.mit.edu Message-Id: <01H1EMUZ0T1E91XFD4@delphi.com> X-Vms-To: INTERNET"cube-lovers@ai.ai.mit.edu" Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII Content-Transfer-Encoding: 7BIT Richard Snyder's book on Square-1 is now being published. He sent me the following press release about a forthcoming Square-1 Puzzle Party: Square-1 is the most challenging puzzle since Rubik's Cube! When you turn it, it forms many unfamiliar shapes, and it seems impossible to get it back to the cube shape! And if you do somehow manage to turn it into a cube, it is scrambled and needs to be solved Rubik-style. It's really two puzzles in one! Harder than Rubik's, it's so hard that only 5 people in the whole world have ever been able to come up with a complete solution to it! Richard Snyder of Dracut is the only person in the USA who has written a book which shows how to solve Square-1! His book is clear and easy to follow, leading you step by step from any scrambled state to the completely solved cube! Then he gives formulas for over 100 colored patterns which you can make on Square-1's symmetrical shapes, and teaches you how to make your own symmetrical patterns! There's no other book quite like it in the world! Richard will be presenting his new book, Turn to Square-1, to Boston in a great puzzle party, which will be held at 1PM on Sat., Aug. 7, 1993, at The Games People Play, 1105 Massachusetts Ave., Cambridge, MA (near Harvard Square) Richard will demonstrate solving Square-1, making Square-1 patterns, and he will also demonstrate solving Rubik's Cube, the Skewb, and other cube puzzles. He will be autographing copies of his new book, and presenting many other fine puzzles and books that are carried by The Games People Play. The Press and the Media will be there, and you are invited to come. Bring your Square-1, your Rubik's Cube, and your other Rubik's puzzles that you haven't been able to solve! Richard will solve them, and show you his solution to Rubik's Cube, the world's best, fastest, and most concise Rubik's Cube solution! But most of all, be prepared to be astounded as Richard shows you how you too can Turn to Square-1! From ccw@eql12.caltech.edu Fri Aug 6 22:37:43 1993 Return-Path: Received: from EQL12.Caltech.Edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09504; Fri, 6 Aug 93 22:37:43 EDT Date: Fri, 6 Aug 93 12:50:32 PDT From: ccw@eql12.caltech.edu (Chris Worrell) Message-Id: <930806124838.23c011ac@EQL12.Caltech.Edu> Subject: Re: Square-1 Puzzle Party In-Reply-To: Your message <01H1EMUZ0T1E91XFD4@delphi.com> dated 6-Aug-1993 To: CPELLEY@delphi.com Cc: ccw@eql12.caltech.edu, cube-lovers@life.ai.mit.edu Sorry. I can't let this one pass by without comment. CPELLEY@delphi.com says > Richard Snyder's book on Square-1 is now being published. He sent me the > following press release about a forthcoming Square-1 Puzzle Party: > It's really two puzzles in one! Harder than Rubik's, it's so hard that > only 5 people in the whole world have ever been able to come up with a > complete solution to it! Unless Snyder or his agent is talking about a God's Algorithm for Square-1, this statement is ridiculous. I doubt that this number includes myself, as I have only told a few family members and friends that I have solved this. (I expect many of you can say the same thing.) Harder than Rubik's? This is a matter of opinion and definition. Do they mean conceptually harder, harder to derive a solution method, harder to prove a solution method, or harder to achieve an individual solution attempt? Or does harder just mean more time? More time to derive a solution method, more time to prove a solution method, or more time to achieve an individual solution attempt? I don't really doubt the last. Except for the Pyraminx (and the 2-Cube), all of the puzzles of this type take me longer to solve than the Cube. I think that the Rubik's cube still holds the record as the puzzle that took me longest to derive a solution method. (Of course all of the others borrowed substantially from the cube.) >Bring your Square-1, your Rubik's Cube, and your other Rubik's puzzles that >you haven't been able to solve! Sorry, I don't have any. Except the 10x10 Rubik's Tangle. Chris Worrell ccw@eql.caltech.edu From dn1l+@andrew.cmu.edu Fri Aug 6 23:27:13 1993 Return-Path: Received: from po3.andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10683; Fri, 6 Aug 93 23:27:13 EDT Received: from localhost (postman@localhost) by po3.andrew.cmu.edu (8.5/8.5) id XAA27259; Fri, 6 Aug 1993 23:27:10 -0400 Received: via switchmail; Fri, 6 Aug 1993 23:27:10 -0400 (EDT) Received: from niobe.weh.andrew.cmu.edu via qmail ID ; Fri, 6 Aug 1993 23:22:39 -0400 (EDT) Received: from niobe.weh.andrew.cmu.edu via qmail ID ; Fri, 6 Aug 1993 23:22:30 -0400 (EDT) Received: from mms.0.1.23.EzMail.2.0.CUILIB.3.45.SNAP.NOT.LINKED.niobe.weh.andrew.cmu.edu.pmax.ul4 via MS.5.6.niobe.weh.andrew.cmu.edu.pmax_ul4; Fri, 6 Aug 1993 23:22:28 -0400 (EDT) Message-Id: Date: Fri, 6 Aug 1993 23:22:28 -0400 (EDT) From: "Dale I. Newfield" To: cube-lovers@life.ai.mit.edu Subject: Tangle (Was: Re: Square-1 Puzzle Party) In-Reply-To: <930806124838.23c011ac@EQL12.Caltech.Edu> Excerpts from mail: 6-Aug-93 Re: Square-1 Puzzle Party by Chris Worrell@eql12.calt >>Bring your Square-1, your Rubik's Cube, and your other Rubik's puzzles that >>you haven't been able to solve! >Sorry, I don't have any. Except the 10x10 Rubik's Tangle. I only have one quarter of that puzzle...(section 4). I worked on it for a considerable amount of time, and concluded that the only solution method was trial and error. So I wrote a program to do it for me. I know all 16 solutions (2 unique)*(2 identical exchanged pieces)*(4 orientations). Has anyone come up with a method, besides trial and error, that solves this thing? (or the 10x10?) (hmmm--I wonder how much the other 3 would cost?) -Dale Newfield dn1l@{cs,andrew}.cmu.edu From @uccvma.ucop.edu:MJTOL@UCCMVSA.BITNET Sat Aug 7 04:25:18 1993 Return-Path: <@uccvma.ucop.edu:MJTOL@UCCMVSA.BITNET> Received: from uccvma.ucop.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14988; Sat, 7 Aug 93 04:25:18 EDT Message-Id: <9308070825.AA14988@life.ai.mit.edu> Received: from UCCVMA.UCOP.EDU by uccvma.ucop.edu (IBM VM SMTP V2R2) with BSMTP id 1516; Thu, 05 Aug 93 16:24:06 PDT Received: from UCCMVSA.BITNET (NJE origin MJT$OL@UCCMVSA) by UCCVMA.UCOP.EDU (LMail V1.1d/1.7f) with BSMTP id 8037; Thu, 5 Aug 1993 16:24:06 -0700 Received: by UCCMVSA.BITNET Thu, 05 Aug 93 16:23:29 PST Date: Thu, 05 Aug 93 16:23:29 PST From: "Michael Thwaites" To: cube-lovers@life.ai.mit.edu Subject: cube tail? > What prevents a group from having a "pointy tail", that is, > a "corridor" of elements at increasing distances from the > identity? In fact, does the number of elements as a > function of distance have to be unimodal? Could this > function have a "waist"? Intuitively, this sounds > impossible, but I am wondering what constraints on such > functions are known. > It seems to me it can't be too pointy. Working backwards, the number of arrangements working from the end has to explode (probably in symetry) with the number of arrangements form the start. From weber@src.dec.com Sat Aug 7 17:33:08 1993 Return-Path: Received: from inet-gw-2.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04351; Sat, 7 Aug 93 17:33:08 EDT Received: by inet-gw-2.pa.dec.com; id AA19111; Sat, 7 Aug 93 14:33:02 -0700 Received: by chaucer; id AA01481; Sat, 7 Aug 93 14:32:53 -0700 Message-Id: <9308072132.AA01481@chaucer> To: "Dale I. Newfield" Cc: cube-lovers@life.ai.mit.edu Subject: Tangle (Was: Re: Square-1 Puzzle Party) In-Reply-To: Message of Fri, 6 Aug 1993 23:22:28 -0400 (EDT) from "Dale I. Newfield" Date: Sat, 07 Aug 93 14:32:53 -0700 From: weber@src.dec.com X-Mts: smtp >>>Bring your Square-1, your Rubik's Cube, and your other Rubik's puzzles that >>>you haven't been able to solve! >>Sorry, I don't have any. Except the 10x10 Rubik's Tangle. > >I only have one quarter of that puzzle...(section 4). > >I worked on it for a considerable amount of time, and concluded that the only >solution method was trial and error. I was thinking about the Rubik's Tangle, and what was puzzling me was WHY there should be only one solution (apart from the obvious symmetries). After all, all pieces are identical except for coloring, and a set consists of all 24 possible coloring, and 1 duplicate, and this doesn't sound like an artificial construction. Is there any mathematical reason for the uniqueness of the solution? What possible "Tangle-like" puzzles have unique solutions? -Sam From dn1l+@andrew.cmu.edu Sun Aug 8 00:21:54 1993 Return-Path: Received: from andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16550; Sun, 8 Aug 93 00:21:54 EDT Received: from localhost (postman@localhost) by andrew.cmu.edu (8.5/8.5) id AAA06744; Sun, 8 Aug 1993 00:21:45 -0400 Received: via switchmail; Sun, 8 Aug 1993 00:21:37 -0400 (EDT) Received: from dollar.mg.andrew.cmu.edu via qmail ID ; Sat, 7 Aug 1993 19:36:59 -0400 (EDT) Received: from dollar.mg.andrew.cmu.edu via qmail ID ; Sat, 7 Aug 1993 19:36:48 -0400 (EDT) Received: from mms.0.1.23.EzMail.2.0.CUILIB.3.45.SNAP.NOT.LINKED.dollar.mg.andrew.cmu.edu.pmax.ul4 via MS.5.6.dollar.mg.andrew.cmu.edu.pmax_ul4; Sat, 7 Aug 1993 19:36:39 -0400 (EDT) Message-Id: Date: Sat, 7 Aug 1993 19:36:39 -0400 (EDT) From: "Dale I. Newfield" To: cube-lovers@life.ai.mit.edu Subject: Re: Tangle (Was: Re: Square-1 Puzzle Party) Cc: In-Reply-To: <9308072132.AA01481@chaucer> Excerpts from mail: 7-Aug-93 Tangle (Was: Re: Square-1 P.. by weber@src.dec.com > I was thinking about the Rubik's Tangle, and what was puzzling me was > WHY there should be only one solution (apart from the obvious symmetries). > After all, all pieces are identical except for coloring, and a set consists > of all 24 possible coloring, and 1 duplicate, and this doesn't sound like > an artificial construction. Is there any mathematical reason for the > uniqueness of the solution? What possible "Tangle-like" puzzles have > unique solutions? The section I had (4) had 2 distinct solutions (apart from the exchange of the 2 identical pieces, and the 4 orientations). In fact, the box that the puzzle came in said it should have 2 solutions. -Dale From @mail.uunet.ca:mark.longridge@dosgate.canrem.COM Mon Aug 9 12:02:01 1993 Return-Path: <@mail.uunet.ca:mark.longridge@dosgate.canrem.COM> Received: from ghost.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA02937; Mon, 9 Aug 93 12:02:01 EDT Received: from canrem.COM ([198.133.42.251]) by mail.uunet.ca with SMTP id <100821(1)>; Mon, 9 Aug 1993 12:01:49 -0400 Received: from dosgate by unixbox.canrem.COM id aa21348; Mon, 9 Aug 93 12:01:39 EDT Received: by dosgate (PCB-UUCP 1.1c) id 180A7E; Mon, 9 Aug 93 08:19:43 -0400 To: CUBE-LOVERS@ai.mit.edu Subject: SQUARE'S GROUP ANALYSIS From: Mark Longridge Message-Id: <60.250317.104.0C180A7E@canrem.com> Date: Sun, 8 Aug 1993 15:40:00 -0400 Organization: CRS Online (Toronto, Ontario) After reading Dik's post I figured I'd add my 2 cents worth: Mark's Notes on the Squares Group --------------------------------- On studying the squares group I have found 16 antipodal cases requiring the maximum 15 moves. Two of these cases cycle all 8 corners and leave the edges in place. A third case "2 DOT/Inverted T's" is pleasingly symmetric. Also I have noted that cycling only the 4 edges in the U or D layer requires 1 move less that cycling only the 4 corners in U or D when using only moves in the square's group, 12 moves for edges and 13 moves for corners. If we define "symmetry level" as the number of distinct patterns generated by rotating the cube through it's 24 different orientations in space then most known antipodes are symmetry level 6. Thus the lower the number the higher the level of symmetry. The least symmetric positions have level 24, and this is very common. The most symmetric positions have level 1, the two positions START and 6 X order 2. I have also found positions with levels 3, 8 and 12. Given the fact that 8 antipodal cases have symmetry level 6 and 8 cases have symmetry level 12 we can now account for ALL 8 * 6 + 8 * 12 = 144 of the 144 cases! Cases with symmetry level 6: p66 Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 D2 (15) p67 Antipode 2 R2 B2 D2 F2 D2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 (15) p80 2 DOT, Invert T's R2 B2 D2 R2 B2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 T2 (15) p99 2 DOT, 4 ARM R2 B2 D2 L2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 (15) p100 2 Cross, 4 ARCH 1 R2 B2 T2 R2 F2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 (15) p130 2 Cross, 4 ARCH 2 L2 B2 D2 B2 L2 D2 F2 L2 (T2 D2 F2 L2) F2 L2 T2 (15) p135 2 X, 4 T L2 B2 D2 F2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 D2 F2 (15) p137 2 X, 4 ARM L2 F2 T2 B2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 D2 F2 (15) Cases with symmetry level 12: p108 2 DOT, 2 T, 2 ARM L2 F2 T2 R2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 (15) p128 2 H, 2 T, 2 CRN L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 T2 (15) p129 2 H, 2 T, 2 ARCH R2 F2 L2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 T2 (15) p131 2 H, 2 ARM, 2 ARCH L2 F2 R2 D2 B2 L2 D2 L2 (T2 D2 F2 T2) F2 L2 F2 (15) p132 2 Cross,2 ARCH,2CRN L2 F2 D2 R2 F2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 D2 (15) p133 2 Cross, 2 T, 2 ARM L2 F2 D2 F2 D2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 (15) p134 2 CRN, 2 X, 2 ARCH L2 F2 D2 B2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 (15) p136 2 H, 2 ARM, 2 CRN R2 F2 L2 T2 B2 L2 T2 L2 (T2 D2 F2 T2) F2 L2 B2 (15) 5 of the 16 known antipodes are within 4 and 2 face turns (or 2 and 1 slice turns) of each other: p66 + L2 R2 T2 D2 = p80 (allowing for whole cube rotations) p66 + F2 B2 = p100 p80 + T2 D2 = p99 P66 + T2 D2 = P128 Using full group moves these antipodes can be reduced to: P66a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1 (13) p67a alternate method F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2 (13) p80a alternate method U1 F2 R2 L2 U2 D2 F2 U2 D3 (9) p99a alternate method U1 R2 F2 B2 U2 D2 R2 D1 (8) P100a alternate method F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1 (12) p108a alternate method R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1 (12) p130a alternate method F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3 (13) p133a alternate method R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2 (11) Both p80a and p99a are surprisingly compact, p99a being a full 7 turns less than it's square's group equivalent. Note that in p99a a square's group sequence is sandwiched between 2 turns on opposite faces. It is the final turn D1 which brings it back into a sq group state! In general U1 (sq group sequence) D1 does not lead to a sq group sequence. Another interesting discovery was comparing the full group sequences: L1 R1 D2 L3 R3 (antislice, 5 moves) L1 R3 D2 L3 R1 (slice , 5 moves) F1 B1 D2 F1 B1 (clockwise, 5 moves) ... to their square's group equivalents: R2 F2 T2 L2 B2 L2 T2 R2 F2 (9 moves) R2 B2 L2 D2 R2 F2 L2 T2 F2 (9 moves) R2 B2 T2 F2 L2 F2 T2 R2 F2 (9 moves) Also it was found possible to permute 3 edges only using: L2 T2 R2 B2 R2 T2 L2 F2 (8 moves) or L3 R1 F2 L1 R3 D2 (6 moves) In general any sequence L1 R1 (any squares group moves) L3 R3 will always result in a squares group position, for example: L1 R1 (D2 F2 B2) R3 L3 F1 B1 (T2 B2 F2 L2) F3 B3 p66a (F2 R2 U2 F2 R2) U3 D3 (B2 L2 F2 B2) U1 D1 (13 moves) The longest irreducible square's group sequence discovered so far, which is an embedded part of longest Phase 2 sequence (p94): (Thus it can't be reduced by using full group moves using current techniques) R2 B2 U2 B2 L2 D2 L2 F2 (8) Later on I discovered this irreducible sequence by chance: T2 B2 T2 B2 D2 F2 R2 T2 L2 F2 (10) Edges only (with corners in place) can be 14 moves at most, e.g. D2 L2 F2 D2 F2 L2 T2 L2 T2 D2 F2 R2 F2 R2 (14) This answers the question David Singmaster posed in "Notes on Rubik's Magic Cube" on Thistlethwaite's last stage. That is: "Are there any positions in the square's group with corners fixed of length 14 or can they be done in less moves?" A few observations... - It is not possible to swap just 1 pair of edges and corners - It is only possible to have 4, 6 or 8 corners out of place - Known antipodal cases can be solved in <=13 moves using full group - In reaching an antipode one may start with any of the 6 turns (since antipodes are global maxima, any turn will get you one move closer) - If the corners are fixed, the position is NOT an antipode - Longest order appears to be 12 - All known (probably all!) antipodes have symmetry level 6 or 12 - Although only conjectural, it is now believed that one turn of a face MUST lead to a new state which is either 1 move closer or 1 move farther from START Question: Are there any irreducible square's group sequences that are longer then 10 moves? Are these truly irreducible or only irreducible under Dik Winter's Kociemba inspired program? Oh well, the full group beckons. I still want to try and come up with my own algorithm though. -> Mark <- From @mail.uunet.ca:mark.longridge@dosgate.canrem.COM Mon Aug 9 12:39:08 1993 Return-Path: <@mail.uunet.ca:mark.longridge@dosgate.canrem.COM> Received: from ghost.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04795; Mon, 9 Aug 93 12:39:08 EDT Received: from canrem.COM ([198.133.42.251]) by mail.uunet.ca with SMTP id <100967(5)>; Mon, 9 Aug 1993 12:01:48 -0400 Received: from dosgate by unixbox.canrem.COM id aa21334; Mon, 9 Aug 93 12:01:34 EDT Received: by dosgate (PCB-UUCP 1.1c) id 180A7C; Mon, 9 Aug 93 08:19:42 -0400 To: CUBE-LOVERS@ai.mit.edu Subject: CUBES (OF COURSE!) From: Mark Longridge Message-Id: <60.250315.104.0C180A7C@canrem.com> Date: Sun, 8 Aug 1993 15:34:00 -0400 Organization: CRS Online (Toronto, Ontario) Well, I finally know what all the square's group antipodes look like. Next message I'll post a detailed summary on these patterns. I wrote a colour printer driver for my cube program today, tested it and it turned out pretty slick. I'm using a Star NX2420 rainbow printer and I'm (out of necessity) using FACE Star NX2420 Code Real Cube ---- ----------- ---- --------- TOP/DOWN BLACK/CYAN 0/2 WHITE/BLUE LEFT/RIGHT VIOLET/ORANGE 3/5 RED/ORANGE FRONT/BACK YELLOW/GREEN 4/6 YELLOW/GREEN ...which works pretty good except violet is more like blue. I could have also used magenta (sort of a pink) for red but it does not contrast well with orange. Should make the DOTC (Domain of the Cube) Newsletter look a lot better if I can ever finish the damn thing! More notes to follow.... -> Mark Longridge, still cubing after all of these years <- From @mail.uunet.ca:mark.longridge@dosgate.canrem.COM Tue Aug 10 11:07:52 1993 Return-Path: <@mail.uunet.ca:mark.longridge@dosgate.canrem.COM> Received: from ghost.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16327; Tue, 10 Aug 93 11:07:52 EDT Received: from canrem.COM ([198.133.42.251]) by mail.uunet.ca with SMTP id <101001(4)>; Tue, 10 Aug 1993 11:07:26 -0400 Received: from dosgate by unixbox.canrem.COM id aa01045; Tue, 10 Aug 93 11:06:06 EDT Received: by dosgate (PCB-UUCP 1.1c) id 180C04; Tue, 10 Aug 93 01:23:31 -0400 To: cube-lovers@life.ai.mit.edu Subject: Cubes (of course) From: Mark Longridge Message-Id: <60.251050.104.0C180C04@canrem.com> Date: Tue, 10 Aug 1993 01:22:00 -0400 Organization: CRS Online (Toronto, Ontario) Well, I finally know what all the square's group antipodes look like. Next message I'll post a detailed summary on these patterns. I wrote a colour printer driver for my cube program today, tested it and it turned out pretty slick. I'm using a Star NX2420 rainbow printer and I'm (out of necessity) using FACE Star NX2420 Code Real Cube ---- ----------- ---- --------- TOP/DOWN BLACK/CYAN 0/2 WHITE/BLUE LEFT/RIGHT VIOLET/ORANGE 3/5 RED/ORANGE FRONT/BACK YELLOW/GREEN 4/6 YELLOW/GREEN ...which works pretty good except violet is more like blue. I could have also used magenta (sort of a pink) for red but it does not (contrast well with orange. Should make the DOTC (Domain of the Cube) Newsletter look a lot better if I can ever finish the damn thing! More notes to follow.... -> Mark Longridge, still cubing after all of these years <- From @mail.uunet.ca:mark.longridge@dosgate.canrem.com Tue Aug 10 11:11:29 1993 Return-Path: <@mail.uunet.ca:mark.longridge@dosgate.canrem.com> Received: from ghost.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16446; Tue, 10 Aug 93 11:11:29 EDT Received: from canrem.COM ([198.133.42.251]) by mail.uunet.ca with SMTP id <101002(1)>; Tue, 10 Aug 1993 11:07:29 -0400 Received: from dosgate by unixbox.canrem.COM id aa01048; Tue, 10 Aug 93 11:06:06 EDT Received: by dosgate (PCB-UUCP 1.1c) id 180C07; Tue, 10 Aug 93 01:30:17 -0400 To: cube-lovers@life.ai.mit.edu Subject: Square's group From: Mark Longridge Message-Id: <60.251051.104.0C180C07@canrem.com> Date: Tue, 10 Aug 1993 01:29:00 -0400 Organization: CRS Online (Toronto, Ontario) After reading Dik's post I figured I'd add my 2 cents worth: Mark's Notes on the Squares Group --------------------------------- On studying the squares group I have found 16 antipodal cases requiring the maximum 15 moves. Two of these cases cycle all 8 corners and leave the edges in place. A third case "2 DOT/Inverted T's" is pleasingly symmetric. Also I have noted that cycling only the 4 edges in the U or D layer requires 1 move less that cycling only the 4 corners in U or D when using only moves in the square's group, 12 moves for edges and 13 moves for corners. If we define "symmetry level" as the number of distinct patterns generated by rotating the cube through it's 24 different orientations in space then most known antipodes are symmetry level 6. Thus the lower the number the higher the level of symmetry. The least symmetric positions have level 24, and this is very common. The most symmetric positions have level 1, the two positions START and 6 X order 2. I have also found positions with levels 3, 8 and 12. Given the fact that 8 antipodal cases have symmetry level 6 and 8 cases have symmetry level 12 we can now account for ALL 8 * 6 + 8 * 12 = 144 of the 144 cases! Cases with symmetry level 6: p66 Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 D2 p67 Antipode 2 R2 B2 D2 F2 D2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 p80 2 DOT, Invert T's R2 B2 D2 R2 B2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 T2 p99 2 DOT, 4 ARM R2 B2 D2 L2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 p100 2 Cross, 4 ARCH 1 R2 B2 T2 R2 F2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 p130 2 Cross, 4 ARCH 2 L2 B2 D2 B2 L2 D2 F2 L2 (T2 D2 F2 L2) F2 L2 T2 p135 2 X, 4 T L2 B2 D2 F2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 D2 F2 p137 2 X, 4 ARM L2 F2 T2 B2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 D2 F2 Cases with symmetry level 12: p108 2 DOT, 2 T, 2 ARM L2 F2 T2 R2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 p128 2 H, 2 T, 2 CRN L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 T2 p129 2 H, 2 T, 2 ARCH R2 F2 L2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 T2 p131 2 H, 2 ARM, 2 ARCH L2 F2 R2 D2 B2 L2 D2 L2 (T2 D2 F2 T2) F2 L2 F2 p132 2 Cross,2 ARCH,2CRN L2 F2 D2 R2 F2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 D2 p133 2 Cross, 2 T, 2 ARM L2 F2 D2 F2 D2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 p134 2 CRN, 2 X, 2 ARCH L2 F2 D2 B2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 T2 B2 p136 2 H, 2 ARM, 2 CRN R2 F2 L2 T2 B2 L2 T2 L2 (T2 D2 F2 T2) F2 L2 B2 5 of the 16 known antipodes are within 4 and 2 face turns (or 2 and 1 slice turns) of each other: p66 + L2 R2 T2 D2 = p80 (allowing for whole cube rotations) p66 + F2 B2 = p100 p80 + T2 D2 = p99 P66 + T2 D2 = P128 Using full group moves these antipodes can be reduced to: P66a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1 p67a alternate method F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2 p80a alternate method U1 F2 R2 L2 U2 D2 F2 U2 D3 p99a alternate method U1 R2 F2 B2 U2 D2 R2 D1 P100a alternate method F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1 p108a alternate method R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1 p130a alternate method F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3 p133a alternate method R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2 Both p80a and p99a are surprisingly compact, p99a being a full 7 turns less than it's square's group equivalent. Note that in p99a a square's group sequence is sandwiched between 2 turns on opposite faces. It is the final turn D1 which brings it back into a sq group state! In general U1 (sq group sequence) D1 does not lead to a sq group sequence. Another interesting discovery was comparing the full group sequences: L1 R1 D2 L3 R3 (antislice, 5 moves) L1 R3 D2 L3 R1 (slice , 5 moves) F1 B1 D2 F1 B1 (clockwise, 5 moves) ... to their square's group equivalents: R2 F2 T2 L2 B2 L2 T2 R2 F2 (9 moves) R2 B2 L2 D2 R2 F2 L2 T2 F2 (9 moves) R2 B2 T2 F2 L2 F2 T2 R2 F2 (9 moves) Also it was found possible to permute 3 edges only using: L2 T2 R2 B2 R2 T2 L2 F2 (8 moves) or L3 R1 F2 L1 R3 D2 (6 moves) In general any sequence L1 R1 (any squares group moves) L3 R3 will always result in a squares group position, for example: L1 R1 (D2 F2 B2) R3 L3 F1 B1 (T2 B2 F2 L2) F3 B3 p66a (F2 R2 U2 F2 R2) U3 D3 (B2 L2 F2 B2) U1 D1 (13 moves) The longest irreducible square's group sequence discovered so far, which is an embedded part of longest Phase 2 sequence (p94): (Thus it can't be reduced by using full group moves using current techniques) R2 B2 U2 B2 L2 D2 L2 F2 (8) Later on I discovered this irreducible sequence by chance: T2 B2 T2 B2 D2 F2 R2 T2 L2 F2 (10) Edges only (with corners in place) can be 14 moves at most, e.g. D2 L2 F2 D2 F2 L2 T2 L2 T2 D2 F2 R2 F2 R2 (14) This answers the question David Singmaster posed in "Notes on Rubik's Magic Cube" on Thistlethwaite's last stage. That is: "Are there any positions in the square's group with corners fixed of length 14 or can they be done in less moves?" A few observations... - It is not possible to swap just 1 pair of edges and corners - It is only possible to have 4, 6 or 8 corners out of place - Known antipodal cases can be solved in <=13 moves using full group - In reaching an antipode one may start with any of the 6 turns (since antipodes are global maxima, any turn will get you one move closer) - If the corners are fixed, the position is NOT an antipode - Longest order appears to be 12 - All known (probably all!) antipodes have symmetry level 6 or 12 - Although only conjectural, it is now believed that one turn of a face MUST lead to a new state which is either 1 move closer or 1 move farther from START Question: Are there any irreducible square's group sequences that are longer then 10 moves? Are these truly irreducible or only irreducible under Dik Winter's Kociemba inspired program? The full group beckons.... -> Mark <- From hoey@aic.nrl.navy.mil Fri Aug 13 19:19:41 1993 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04412; Fri, 13 Aug 93 19:19:41 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA05894; Fri, 13 Aug 93 18:26:10 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Fri, 13 Aug 93 18:26:09 EDT Date: Fri, 13 Aug 93 18:26:09 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9308132226.AA28300@sun13.aic.nrl.navy.mil> To: Mark Longridge , cube-lovers@life.ai.mit.edu Subject: Re: Squares group In-Reply-To: <60.251051.104.0C180C07@canrem.com> Mark Longridge has some interesting things to say about the antipodes of the group generated by half-turns: > If we define "symmetry level" as the number of distinct patterns > generated by rotating the cube through it's 24 different > orientations in space then most known antipodes are symmetry level > 6. Thus the lower the number the higher the level of symmetry. The > least symmetric positions have level 24, and this is very common. This approach is somewhat unfortunate in two ways. First, it would be better to use the full 48-element symmetry group M of the cube, because some patterns are not recognized as transformed images of each other if you only use the 24-element group C of rotations. For instance, the positions reached by processes F2R2T2 and F2T2R2 cannot be related with C, so you would see four classes of positions at distance three rather than three. But the antipodes you give are all mirror-symmetric, so there is no new coalescence there. Relating processes that are conjugates by a reflection is usually somewhat tricky, since the moves of the process must be changed in direction (replacing clockwise by counterclockwise) but in the squares group this is a nonproblem. The second deficiency of your approach is that you lose information by specifying only the index of the symmetry subgroup (the ``number of distinct patterns generated ...''). It makes sense to find out exactly which subgroup of M is the symmetry group of your positions. I've done that, below. Each of these symmetry groups comes in three conjugates, so I've transformed some of the processes (marked x) so they all use the same particular symmetry group(s). The group elements are given as cycles of the cube faces, so (TD)(FRBL) means to reflect T<->D and rotate F->R->B->L->F. > Cases with symmetry level 6: These are cases where the symmetry group has order 8. > p66x Double 4 corner sw L2 D2 R2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 B2 > p80 2 DOT, Invert T's R2 B2 D2 R2 B2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 T2 > p99 2 DOT, 4 ARM R2 B2 D2 L2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 > p100 2 Cross, 4 ARCH 1 R2 B2 T2 R2 F2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 p66x, p80, p99, and p100 have symmetry group P=<(TD)(FRBL),(FB),(LR)>. > p67x Antipode 2 R2 D2 B2 T2 B2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 > p130 2 Cross, 4 ARCH 2 L2 B2 D2 B2 L2 D2 F2 L2 (T2 D2 F2 L2) F2 L2 T2 p67x and p130 have symmetry group Q=<(TD),(FRBL)>. > p135x 2 X, 4 T D2 B2 L2 F2 R2 F2 R2 D2 (R2 L2 F2 R2) D2 L2 F2 > p137 2 X, 4 ARM L2 F2 T2 B2 T2 F2 T2 L2 (T2 D2 F2 T2) L2 D2 F2 p135x and p137 have symmetry group S=<(TD),(FB)(LR),(FR)(BL)>. > Cases with symmetry level 12: These have 4-element symmetry groups. > p108 2 DOT, 2 T, 2 ARM L2 F2 T2 R2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 > p128x 2 H, 2 T, 2 CRN L2 D2 R2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 F2 > p129x 2 H, 2 T, 2 ARCH R2 T2 L2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 F2 > p131x 2 H, 2 ARM, 2 ARCH L2 T2 R2 B2 D2 L2 B2 L2 (F2 B2 T2 F2) T2 L2 T2 > p132 2 Cross,2 ARCH,2CRN L2 F2 D2 R2 F2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 D2 > p136x 2 H, 2 ARM, 2 CRN R2 T2 L2 F2 D2 L2 F2 L2 (F2 D2 T2 F2) T2 L2 D2 p108, p128x, p129x, p131x, p132, and p136x have symmetry group HP=<(FB),(LR)>. > p133x 2 Cross, 2 T, 2 ARM L2 T2 B2 T2 B2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 > p134x 2 CRN, 2 X, 2 ARCH L2 T2 B2 D2 F2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 p133x and p134x have symmetry group HS=<(TD),(FB)(LR)>. In case you have trouble forming the closure of these groups: P = {I, (FB)(LR), (TD)(FRBL), (TD)(FLBR), (FB), (LR), (TD)(FR)(BL), (TD)(FL)(BR)} Q = {I, (FB)(LR), (TD), (TD)(FB)(LR), (TD)(FRBL), (TD)(FLBR), (FLBR), (FRBL)} S = {I, (FB)(LR), (TD), (TD)(FB)(LR), (TD)(FR)(BL), (TD)(FL)(BR), (FR)(BL), (FL)(BR)} HP = {I, (FB)(LR), (FB), (LR)} HS = {I, (FB)(LR), (TD), (TD)(FB)(LR)}. I should note that the subgroup names M, C, P, Q, S, HP, and HS are part of a general classification of subgroups of M that I worked out some time ago. I have a chart of them I can send; just ask by email. > A few observations... > - It is not possible to swap just 1 pair of edges and corners Certainly, all the generators are even permutations on the edges and on the corners. > - It is only possible to have 4, 6 or 8 corners out of place That is a nice, concise way of putting it. To elaborate, if you permute one of the corner orbits in a 3-cycle, the other will also be permuted in a 3-cycle; otherwise, any pair of cycle structures of the same parity is possible. > - In reaching an antipode one may start with any of the 6 turns > (since antipodes are global maxima, any turn will get you one move > closer) Careful! This also relies on the fact you call a conjecture, below. Otherwise you could have two neighboring global maxima, and their inverses would be antipodes that do not have this property. For instance, consider the corner group as generated by the 24 pairs of neighboring squares (F2R2, etc). This is a 48-element group with diameter 2, trivial enough to be analyzed by hand. Antipodes (L2B2)(D2R2) and (D2R2)(T2F2) are neighbors, because (L2B2)(D2R2)(F2T2)=(D2R2)(T2F2). So there is no length-2 process equivalent to (F2T2)(R2D2) that starts with T2F2. > - If the corners are fixed, the position is NOT an antipode > - All known (probably all!) antipodes have symmetry level 6 or 12 I presume these comments are left over from before you found them all. > - Longest order appears to be 12 Appears? The orbits are all of size 4 (two orbits of corners, three orbits of edges), so 12=LCM(2,3,4) is an easy upper bound. Finding one is easy given the processes Singmaster lists. > - Although only conjectural, it is now believed that one turn of a > face MUST lead to a new state which is either 1 move closer or 1 > move farther from START Conjectural? It's immediate from the fact that each generator is an odd permutation of the corner orbit {FTR,FDL,BTL,BDR}. > Question: Are there any irreducible square's group sequences that > are longer then 10 moves? Are these truly irreducible or only > irreducible under Dik Winter's Kociemba inspired program? Well, that could be searched for; a matter of checking 600K positions for each of the 15K or so pattern representatives. I hope I can find the time to hack it up. Dan Hoey Hoey@AIC.NRL.Navy.Mil From hoey@aic.nrl.navy.mil Mon Aug 16 20:20:25 1993 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06633; Mon, 16 Aug 93 20:20:25 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA29078; Mon, 16 Aug 93 18:05:48 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Mon, 16 Aug 93 18:05:47 EDT Date: Mon, 16 Aug 93 18:05:47 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9308162205.AA12648@sun13.aic.nrl.navy.mil> To: Mark Longridge , cube-lovers@life.ai.mit.edu Subject: Squares group, correction I should proofread these things better. I got the processes for p130, p135x, p137, and p136x wrong in my last message. Here is the corrected list of squares-group antipodes and their symmetry groups. SG Pos Name Process P p66x Double 4 corner sw L2 D2 R2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 B2 P p80 2 DOT, Invert T's R2 B2 D2 R2 B2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 T2 P p99 2 DOT, 4 ARM R2 B2 D2 L2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 P p100 2 Cross, 4 ARCH 1 R2 B2 T2 R2 F2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 Q p67x Antipode 2 R2 D2 B2 T2 B2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 Q p130x 2 Cross, 4 ARCH 2 T2 B2 R2 B2 T2 R2 F2 T2 (L2 R2 F2 T2) F2 T2 L2 S p135x 2 X, 4 T L2 D2 B2 T2 F2 T2 F2 L2 (F2 B2 T2 F2) L2 B2 T2 S p137x 2 X, 4 ARM L2 T2 F2 D2 F2 T2 F2 L2 (F2 B2 T2 F2) L2 B2 T2 HP p108 2 DOT, 2 T, 2 ARM L2 F2 T2 R2 B2 L2 F2 L2 (T2 D2 F2 T2) F2 L2 T2 HP p128x 2 H, 2 T, 2 CRN L2 D2 R2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 F2 HP p129x 2 H, 2 T, 2 ARCH R2 T2 L2 T2 L2 T2 F2 R2 (F2 B2 T2 F2) T2 L2 F2 HP p131x 2 H, 2 ARM, 2 ARCH L2 T2 R2 B2 D2 L2 B2 L2 (F2 B2 T2 F2) T2 L2 T2 HP p132 2 Cross,2 ARCH,2CRN L2 F2 D2 R2 F2 L2 B2 L2 (T2 D2 F2 T2) F2 L2 D2 HP p136x 2 H, 2 ARM, 2 CRN R2 T2 L2 F2 D2 L2 F2 L2 (F2 B2 T2 F2) T2 L2 D2 HS p133x 2 Cross, 2 T, 2 ARM L2 T2 B2 T2 B2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 HS p134x 2 CRN, 2 X, 2 ARCH L2 T2 B2 D2 F2 T2 F2 L2 (F2 B2 T2 F2) L2 F2 D2 Sorry if anyone was led astray. Dan From acw@bronze.lcs.mit.edu Thu Aug 19 15:06:28 1993 Return-Path: Received: from bronze.lcs.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15753; Thu, 19 Aug 93 15:06:28 EDT Received: by bronze.lcs.mit.edu id AA27266; Thu, 19 Aug 93 15:04:18 EDT Date: Thu, 19 Aug 93 15:04:18 EDT From: acw@bronze.lcs.mit.edu (Allan C. Wechsler) Message-Id: <9308191904.AA27266@bronze.lcs.mit.edu> To: ronnie@cisco.com Cc: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu In-Reply-To: "Ronnie B. Kon"'s message of Thu, 05 Aug 1993 16:55:36 -0700 <199308052355.AA23583@lager.cisco.com> Subject: Diameter of cube group? Date: Thu, 05 Aug 1993 16:55:36 -0700 From: "Ronnie B. Kon" Disclaimer: this sounds more authoritative than is intended--I really don't know what I'm talking about. Don't worry. Mathematical reasoning stands on its own merits. It couldn't be very pointy. From the most distant configuration, there are 6 positions immediately before it. There are 6^2 two steps away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually. Very good. This is a necessary insight, regardless of the exact numerical details. (For example, you mean 12, not 6.) But the possible flaw is that there might be more than one maximally distant state; if their sets of neighbors overlap viciously enough, this effect could make the tail pointier. You can make valence-12 graphs (not of groups, just arbitrary graphs) that have fairly bumpy distance-vs-population functions. Any argument that rigorously constrains N(d) must somehow appeal to the fact that the cube graph is a Cayley graph, that is, the graph of a group. [...] This gives me the feeling that Monte Carlo is fairly valid. (How's that for rigor?) It's a start. But we have to use groupness somehow. From @mitvma.mit.edu:DWR2560@TAMZEUS.BITNET Fri Aug 20 09:47:26 1993 Return-Path: <@mitvma.mit.edu:DWR2560@TAMZEUS.BITNET> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19046; Fri, 20 Aug 93 09:47:26 EDT Message-Id: <9308201347.AA19046@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 3102; Fri, 20 Aug 93 06:56:16 EDT Received: from TAMZEUS.BITNET (DWR2560) by MITVMA.MIT.EDU (Mailer R2.10 ptf000) with BSMTP id 4997; Fri, 20 Aug 93 06:56:15 EDT Date: Fri, 20 Aug 93 05:56 CST From: Subject: pointy tails To: cube-lovers@life.ai.mit.edu X-Original-To: cube-lovers@life.ai.mit.edu, DWR2560 Allan C. Wechsler writes: > It couldn't be very pointy. From the most distant configuration, > there are 6 positions immediately before it. There are 6^2 two steps > away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually. > >Very good. This is a necessary insight, regardless of the exact >numerical details. (For example, you mean 12, not 6.) But the >possible flaw is that there might be more than one maximally distant >state; if their sets of neighbors overlap viciously enough, this >effect could make the tail pointier. You can make valence-12 graphs [deletia] All this misses the point (so to speak) which is that 12^N is _exceedingly_ pointy for our purposes. If one samples only 1000 positions out of ~10E19, then one could very well miss a 12^N tail of length 14 moves! The estimate of 22 as an upper limit relies on the intuition that the distribution is MUCH blunter than this. Dave Ring dwr2560@zeus.tamu.edu From reid@math.berkeley.edu Mon Aug 23 04:10:58 1993 Return-Path: Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14953; Mon, 23 Aug 93 04:10:58 EDT Received: from jacobi.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math)) id AA19047; Mon, 23 Aug 93 01:10:56 PDT Date: Mon, 23 Aug 93 01:10:56 PDT From: reid@math.berkeley.edu (michael reid) Message-Id: <9308230810.AA19047@math.berkeley.edu> To: cube-lovers@life.ai.mit.edu Subject: Re: Diameter of cube group? > Continuing and waiting for a config that requires 21 turns, dik here's a pattern to try: first do 6 checkerboards of order 2 (F2 B2 R2 L2 U2 D2) and then do superfliptwist. in other words, the group product of these two elements. From dik@cwi.nl Tue Aug 24 20:43:14 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05145; Tue, 24 Aug 93 20:43:14 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA02014 (5.65b/3.10/CWI-Amsterdam); Wed, 25 Aug 1993 02:43:01 +0200 Received: by boring.cwi.nl id AA11725 (4.1/2.10/CWI-Amsterdam); Wed, 25 Aug 93 02:42:58 +0200 Date: Wed, 25 Aug 93 02:42:58 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308250042.AA11725.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu, reid@math.berkeley.edu Subject: Re: Diameter of cube group? > > Continuing and waiting for a config that requires 21 turns, dik > here's a pattern to try: > first do 6 checkerboards of order 2 (F2 B2 R2 L2 U2 D2) and then do > superfliptwist. in other words, the group product of these two elements. As they commute I did it the other way around. But I am highly suspicious that you tried it yourself. 10 minutes and only down to 22 turns. But continuing, possibly for weeks/months. On another machine I am trying to prove that 20 is minimal for superfliptwist. 90 hours gone, still nothing. Most of the time is not spend with phase 1 set to 16 turns. Phase 1 to 13 got it doen to 20. Nothing new with phase 1 to 14 or 15. 16 turns in phase 1 allows at most 3 turns in phase 2. The latter can be time consuming. I do not know in how many cases actually something is done in phase 2. When I get to 17 turns in phase 1, I suspect in most cases in phase 2 it is immediately clear that it can not be solved. But I am patient. dik From dik@cwi.nl Wed Aug 25 16:00:57 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09102; Wed, 25 Aug 93 16:00:57 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA02929 (5.65b/3.10/CWI-Amsterdam); Wed, 25 Aug 1993 22:00:23 +0200 Received: by boring.cwi.nl id AA14852 (4.1/2.10/CWI-Amsterdam); Wed, 25 Aug 93 22:00:22 +0200 Date: Wed, 25 Aug 93 22:00:22 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308252000.AA14852.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu, reid@math.berkeley.edu Subject: Re: Diameter of cube group? > here's a pattern to try: > first do 6 checkerboards of order 2 (F2 B2 R2 L2 U2 D2) and then do > superfliptwist. in other words, the group product of these two elements. Was certainly one of the hardest to do. After 17 hours the best was 22 turns, but then results came in, after 18 hours 21 turns, and finally after 19 hours 20 turns: F1 R1 L2 U3 R2 L3 U3 D2 R2 F1 D1 B1 D1 F2 U3 R3 D3 F2 D2 L2 This on an SGI R4K Indigo. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From dik@cwi.nl Wed Aug 25 16:10:23 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09414; Wed, 25 Aug 93 16:10:23 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA03291 (5.65b/3.10/CWI-Amsterdam); Wed, 25 Aug 1993 22:10:07 +0200 Received: by boring.cwi.nl id AA14880 (4.1/2.10/CWI-Amsterdam); Wed, 25 Aug 93 22:10:06 +0200 Date: Wed, 25 Aug 93 22:10:06 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308252010.AA14880.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: CFF 32 table of contents Last Sunday (on Cube Day) I was handed issue #32 of Cubism For Fun. A summary the contents: 1. Short articles about the solution of the puzzle by Koos en Ton Verhoeff. (I described the puzzle before and announced the solution by Jan de Ruiter a few weeks ago.) 2. Articles about "Bob's Binary Boxes" by Hans Dockhorn and Bob Kootstra. * 3. Description by Harold Cataquet of "Alice"; a wooden packing puzzle. 4. Description by Wim Zwaan of a packing puzzle he entered in the "Hikimi Wooden Puzzle Competition". 5. Article by Jan Verbakel about "Wirrel-Warrel" puzzles (it has a different name in the US that escapes me). 6. Article by Tom Hilligers about "Kaos", a puzzle with balls in pipes. The orientation of the pipes with respect to each other can change. 7. Article by Ronald Fletterman about pretty "sculptures" with Square 1. 8. A contest announcement by Bernard Wiezorke figuring the sliding puzzle Vorsicht! (I do not know whether it is available in the US.) 9. An article by Ralph Gasser about Orbik, a puzzle introduced by Edward Hordern. 10. Results of a number of contests. * An interesting design. These are wooden boxes with in it binary switches. On top a ball can be put in, on the bottom there are a number of exits. When a ball reaches a switch it passes the switch in the given direction and puts the switch in opposite direction. The design is such that successive balls come out in successive exits (in circular numerical order). Bob Kootstra built a few of those switches, but now is asking for an optimal design of a box with 7 exits. CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch Cubists Club). It appears a bit irregular, but a few times a year. Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to approximately $ 15.-. Institutional membership is also possible. Information is available from the editor: Gerald Maurice Groen van Prinstererstraat 7-2 1051 ED Amsterdam The Netherlands Phone: +31206822943 E-mail: gm@phys.uva.nl From alan@parsley.lcs.mit.edu Wed Aug 25 21:13:44 1993 Return-Path: Received: from parsley.lcs.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AB21991; Wed, 25 Aug 93 21:13:44 EDT Received: by parsley.lcs.mit.edu id AA04367; Wed, 25 Aug 93 21:12:55 -0400 Date: Wed, 25 Aug 93 21:12:55 -0400 Message-Id: <25Aug1993.204955.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Alan@lcs.mit.edu To: Cube-Lovers@ai.mit.edu Subject: Tools lost in the mists of time... Despite being your moderator for the past 13 years, it has been quite a while since I actually played with a cube (of any order). Until last weekend, that is, when I picked up a 5x5x5 cube that someone had loaned me quite some time ago. Imagine my surprise to discover that I had in fact forgotten one of the tools I needed to solve even a 3x3x3! In particular my tool for inverting two edge cubies in place in a 3x3x3 was completely gone. It didn't take me long to develop a replacement, but I'm certain that it is nothing like what I was using years ago. I'm amazed -- there was a time when I thought I could never forget any of those tools. And by the way, it was a lot of fun to re-learn the 3x3x3 and then go on to solve the 5x5x5 -- if anyone else out there has, like me, been neglecting their cube hacking for some years, go pick up your cube again, you may be pleasantly surprised. - Alan (aka Cube-Lovers-Request) -- Alan Bawden Alan@LCS.MIT.EDU MIT Room NE43-538 (617) 253-7328 545 Technology Square Cambridge, MA 02139 06BF9EB8FC4CFC24DC75BDAE3BB25C4B From hoey@aic.nrl.navy.mil Thu Aug 26 10:31:05 1993 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA11231; Thu, 26 Aug 93 10:31:05 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA21721; Thu, 26 Aug 93 10:30:57 EDT Date: Thu, 26 Aug 93 10:30:57 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9308261430.AA21721@Sun0.AIC.NRL.Navy.Mil> To: Alan@lcs.mit.edu, cube-lovers@ai.mit.edu Subject: Tartan reborn (Re: Tools lost in the mists of time...) In-Reply-To: <25Aug1993.204955.Alan@LCS.MIT.EDU> Organization: Navy Center for Applied Research in AI Alan Bawden mentioned the joy of rediscovering his lost cube-solving techniques. This happened to me about three years ago for an unusual reason. I've become active in science fiction fandom, and fans determine where the World Science Fiction Convention (Worldcon) is held each year by running miniature political campaigns. A friend of mine was bidding for Glasgow, and she asked if I had any `plaid things'. I told her I had a plaid Rubik's cube, and a political strategy was born. The plaid cube is of course the Tartan, which Jim Saxe and I discovered and described in this group on 16 February 1981 (see archives). I blanked some old cubes, and figured out how to use spray paint to efficiently create Tartan cubes. I produced a half dozen or so, and they make good conversation pieces at conventions. Unfortunately, I seem to be the only convention-going science fiction fan who can *solve* a Tartan (with the possible exception of Phil Servita who as I recall figured out an effective method but wearied in its execution). So I would see a scrambled Tartan at a convention party, and fix it, and put it down, and five minutes later it would be scrambled again. I quickly found out how rusty I was, and through the enforced practice I've gotten about as good as I was a decade ago. But some of the Glasgow promoters took Tartan cubes over to the UK, and those cubes just never get solved. I sent them instructions for solving it, but I don't know if any of them have figured out the instructions. Well, eventually they told me they really wanted something mere mortals could deal with, and I painted some pieces of wood plaid that they could use for doorstops. I was surprised, though, to find that to make a plaid pattern going around a corner, if you only have four colors of paint, it seems the *only* thing you can do is use a coloring locally identical to the Tartan. As for the cubes in the UK, I expect to get there in 1995. For it seems the clever ploy worked, and the fans voted to have the 1995 Worldcon in Glasgow. I'm sure they owe it all to the Tartan. Sure. Dan Hoey Hoey@AIC.NRL.Navy.Mil From reid@math.berkeley.edu Fri Aug 27 22:23:04 1993 Return-Path: Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27534; Fri, 27 Aug 93 22:23:04 EDT Received: from jacobi.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math)) id AA22114; Fri, 27 Aug 93 19:22:49 PDT Date: Fri, 27 Aug 93 19:22:49 PDT From: reid@math.berkeley.edu (michael reid) Message-Id: <9308280222.AA22114@math.berkeley.edu> To: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu Subject: Re: Diameter of cube group? dik winter says > > first do 6 checkerboards of order 2 (F2 B2 R2 L2 U2 D2) and then do > > superfliptwist. in other words, the group product of these two elements. > > As they commute I did it the other way around. But I am highly > suspicious that you tried it yourself. 10 minutes and only down > to 22 turns. But continuing, possibly for weeks/months. ok, you caught me; i'd already tried this one myself. :-) but apparently i wasn't as patient as you. i just remember that it ran for a long time without doing better than 22 face turns. the point to be made here is the following: the length of time the program takes for a given position depends significantly on how far it must search in stage 1. this seems to make any claim about how long the program takes to crunch an average position meaningless. my experience is that it varies greatly depending upon the position. i think it would be more informative to stratify this information. i.e., how long it takes to search 12 moves in stage 1, and how short a solution is produced. and then the same info for 13 turns, then 14, etc. what i've been amazed by (and continue to be) is that searching only 13 or so moves in stage 1 is sufficient to produce very short solutions for many positions. something i'd thought about trying, but never got around to is trying random positions created by twist sequences such as: F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 or some random string of about 20 quarter turns of the faces F,L,B,R. a string of length 12 or 13 will be solved quickly (by the inverse of the original string). however, for length 17 or so, the program won't find the inverse of the original string until it is searching 17 moves deep in stage 1. this suggests that perhaps these positions may be harder for the program to handle. but are they harder than random positions? i don't know. mike From dik@cwi.nl Sat Aug 28 20:17:32 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00850; Sat, 28 Aug 93 20:17:32 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA17602 (5.65b/3.10/CWI-Amsterdam); Sun, 29 Aug 1993 02:17:26 +0200 Received: by boring.cwi.nl id AA01278 (4.1/2.10/CWI-Amsterdam); Sun, 29 Aug 93 02:17:23 +0200 Date: Sun, 29 Aug 93 02:17:23 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308290017.AA01278.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu, reid@math.berkeley.edu Subject: Re: Diameter of cube group? > ok, you caught me; i'd already tried this one myself. :-) > but apparently i wasn't as patient as you. i just remember that it ran > for a long time without doing better than 22 face turns. So did it here. 22 in a few minutes, 20 in a lot of hours. > the point to be made here is the following: the length of time the > program takes for a given position depends significantly on how far it > must search in stage 1. This is right, and it appears (though I have not yet thoroughly verified) that configurations that take a long time in stage 1 are a large distance from start. > this seems to make any claim about how long the > program takes to crunch an average position meaningless. Depends on how you interpret that claim. If the claim is that it turns up with a sequence that is 20 turns or shorter you are right. The claim might even be incorrect! The actual claim is that it takes a fairly short time to give a fairly short sequence (where fairly short is deliberately left unquantified). And this is true. For my set of >10000 random positions the program came up with a sequence of 27 turns or less in a short time indeed. (Actually the first solution found was 26 turns or less for all but three configurations.) Bringing that down to 20 took in a number of cases extremely long (in the order of one day). But that is still far less than when we had done a normal single phase backtracking process I think. > i think it would > be more informative to stratify this information. i.e., how long it > takes to search 12 moves in stage 1, and how short a solution is produced. > and then the same info for 13 turns, then 14, etc. Some quantification is not so very difficult I think. Without tree-pruning the time would be proportional to 18^n + 10^m for a n-turn phase1 and a m-turn phase2 solution. The tree-pruning performed is (I think) proportional to the number of turns in each phase; it will chop branches that are to large according to predefined tables. Also there are some short-cuts that make 18 not really 18 and 10 not really 10, but the reasoning remains the same. > what i've been amazed by (and continue to be) is that searching only 13 > or so moves in stage 1 is sufficient to produce very short solutions for > many positions. I do not think this is so very amazing. Each configuration can be brought in 12 turns or less to a configuration for phase 2. The proven diameter of the group of phase 2 is 25, my estimate is 19-21. So, based on my estimate a worst case would be 12 turns required in phase 1 and 21 in phase 2 giving 33 turns in an estimated time of 18^12 + 10^21, this is less than 18^17, and hence is found faster than if we had gone to 17 turns in phase 1. Actually both 12 and 21 are rare; most cases do phase 1 in 10 turns or less and phase 2 in 15 turns or less. > something i'd thought about trying, but never got around to is trying > random positions created by twist sequences such as: > F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 F1 R1 B1 L1 > or some random string of about 20 quarter turns of the faces F,L,B,R. > a string of length 12 or 13 will be solved quickly (by the inverse of > the original string). however, for length 17 or so, the program won't > find the inverse of the original string until it is searching 17 moves > deep in stage 1. this suggests that perhaps these positions may be > harder for the program to handle. but are they harder than random > positions? i don't know. I do not know, but I think not. Yes, asking the program to find the reverse of the string takes a long time. Asking the program to find an inverse of the sequence takes much less time (although the inverse found may both be shorter or longer than the original). I just tried, and after initialization it found a 10+14 turn solution in 20 seconds, going down to 11+10 after less than a minute. Getting this down to 20 might of course take considerable time (if the original sequence is minimal etc.). But I have not the time right now to check. I am busy trying to prove that 20 is minimal for superfliptwist. I have already found that there is no 19 turn solution with 16 turns in phase 1. That took about 48 hours (distributed over 6 machines). Now I am doing the same for 17 turns in phase 1; which wil obviously take much longer. (And yes, I took the precaution of allowing as the first turn only F, F2, R, R2, U, U2 in phase 1. When I go to 19 turns in phase 1, I can skip 18, I need only F, F2, R and R2, I think.) dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From reid@math.berkeley.edu Sun Aug 29 04:26:41 1993 Return-Path: Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10214; Sun, 29 Aug 93 04:26:41 EDT Received: from jacobi.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math)) id AA05585; Sun, 29 Aug 93 01:26:31 PDT Date: Sun, 29 Aug 93 01:26:31 PDT From: reid@math.berkeley.edu (michael reid) Message-Id: <9308290826.AA05585@math.berkeley.edu> To: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu Subject: Re: Diameter of cube group? > But I have not the time right now to check. I am busy trying to prove > that 20 is minimal for superfliptwist. I have already found that there > is no 19 turn solution with 16 turns in phase 1. That took about 48 > hours (distributed over 6 machines). Now I am doing the same for 17 > turns in phase 1; which wil obviously take much longer. (And yes, I > took the precaution of allowing as the first turn only F, F2, R, R2, > U, U2 in phase 1. When I go to 19 turns in phase 1, I can skip 18, > I need only F, F2, R and R2, I think.) in fact, you can eliminate the possibility of starting with F2, R2 or U2, since these each commute with superfliptwist, and may be done in stage 2. in other words, if F2 sequence = superfliptwist, then also sequence F2 = superfliptwist. also, you need not consider 19 turns in stage 1. by symmetry, you may suppose the last face turned is U, which is done in stage 2. if you use the fact that U and D commute, L and R commute and F and B commute, then the number of sequences of length n in stage 1 grows exponentially, with ratio approximately 13.35. if the runtime is proportional to the number of sequences tested in stage 1, (which may or may not be the case) that would mean testing 18 turns deep would take approximately 178.18 times as long. (eliminating the possibility of starting with F2, R2 or U2 would cut that in half.) here's something you may have already considered. if your prune tables in stage 1 consider only pairs (flip, twist), (flip, location) and (twist, location), some search paths may be pruned 8 turns early. (each of these pairs had positions 9 twists from start.) at the expense of a lot more memory, you can do some pruning 11 turns early, by storing tables for triples (flip, twist, location). you'd probably have to store these tables in very compressed form, and divide out by symmetries of the cube that preserve the U-D axis. it may turn out that the overhead of processing this compressed information does not adequately compensate for the improved foresight, but it's worth considering. it would be excellent if you could show that 20 face turns is minimal for superfliptwist! even finding a shorter solution would be great! mike From dik@cwi.nl Mon Aug 30 21:21:35 1993 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22630; Mon, 30 Aug 93 21:21:35 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA03450 (5.65b/3.10/CWI-Amsterdam); Tue, 31 Aug 1993 03:21:25 +0200 Received: by boring.cwi.nl id AA08781 (4.1/2.10/CWI-Amsterdam); Tue, 31 Aug 93 03:21:23 +0200 Date: Tue, 31 Aug 93 03:21:23 +0200 From: Dik.Winter@cwi.nl Message-Id: <9308310121.AA08781.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu, reid@math.berkeley.edu Subject: Re: Diameter of cube group? > in fact, you can eliminate the possibility of starting with F2, R2 or U2, > since these each commute with superfliptwist, and may be done in stage 2. > in other words, if F2 sequence = superfliptwist, then also > sequence F2 = superfliptwist. Right. I had not considered this in the program (it is still fairly general), but it only does mean early termination. > also, you need not consider 19 turns in stage 1. by symmetry, you may > suppose the last face turned is U, which is done in stage 2. But I have now had different thoughts. Currently phase 1 checks in 3 dimensional space. When a solution is found the program calculates the current position for phase two after which it finds a solution in a different 3 dimensional space. (I just though how I might speed up the calculations to get to the starting position for the second phase, but will not yet elaborate on that; I will first try it out.) But this does not help finding whether there are solutions of 19 turns or less. What I am now considering is to have a phase 1 program only, where phase 1 is done in an additional dimension: the permutation of the corner cubes. So to prove the non-existence of a solution of 19 turns or less, this program would seek for a phase 1 solution in 4 dimensional space of at most 19 turns and next check whether this also solves the edge cubes. This would eliminate quite a few dead alleys where the current phase 1 finds a solution and has still things to do. > if you use the fact that U and D commute, L and R commute and F and B > commute, then the number of sequences of length n in stage 1 grows > exponentially, with ratio approximately 13.35. if the runtime is > proportional to the number of sequences tested in stage 1, (which > may or may not be the case) that would mean testing 18 turns deep > would take approximately 178.18 times as long. (eliminating the > possibility of starting with F2, R2 or U2 would cut that in half.) If I use a single phase algorithm, I can eliminate much more! What I see for runtime is not entirely proportional. When looking at the number of configurations done in phase 2, this goes up by factors that start in the neighbourhood of 30 and diminish to (probably) ultimately the factor you mention. This indicates that tree pruning is much more effective with fewer turns in phase 1. > here's something you may have already considered. if your prune tables > in stage 1 consider only pairs (flip, twist), (flip, location) and > (twist, location), some search paths may be pruned 8 turns early. > (each of these pairs had positions 9 twists from start.) at the > expense of a lot more memory, you can do some pruning 11 turns early, > by storing tables for triples (flip, twist, location). you'd probably > have to store these tables in very compressed form, and divide out by > symmetries of the cube that preserve the U-D axis. it may turn out > that the overhead of processing this compressed information does not > adequately compensate for the improved foresight, but it's worth > considering. I think the overhead is much to large computation-wise and memory-wise. The size of the table would be uncompressed 2217093120 integers in the range from 0 to 12. Factoring out symmetries would reduce it by a factor of about 32 (slightly less). [4 for rotational symmetry, 4 for mirroring both U-D and F-B, 2 for inversion.] Using 3.5 bits per configuration this means > 30 MByte. The machines I am using currently are not able to handle that amount of information. But it is feasable. If we skip inversion (which is most difficult to do) we are at > 60 MByte. The problem remains to adequately number the remaining positions from 1 to max. Some configurations are inert with respect to the rotations and/or mirroring. On the other hand, we need the table in core (do not try to do this through disk access!). Some insightful thoughts are needed here. > it would be excellent if you could show that 20 face turns is minimal > for superfliptwist! even finding a shorter solution would be great! I agree to that! I have a number of machines, still going strong. From ROSEJM58@snyoneva.cc.oneonta.edu Tue Sep 21 14:33:15 1993 Return-Path: Received: from snyoneva.cc.oneonta.edu ([137.141.15.10]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22099; Tue, 21 Sep 93 14:33:15 EDT Received: from SNYONEVA.CC.ONEONTA.EDU by SNYONEVA.CC.ONEONTA.EDU (PMDF V4.2-11 #3312) id <01H37P45H2XO8WW74Q@SNYONEVA.CC.ONEONTA.EDU>; Tue, 21 Sep 1993 14:32:50 EDT Date: Tue, 21 Sep 1993 14:32:49 -0400 (EDT) From: ROSEJM58@snyoneva.cc.oneonta.edu Subject: To: cube-lovers@life.ai.mit.edu Message-Id: <01H37P45H2XQ8WW74Q@SNYONEVA.CC.ONEONTA.EDU> X-Vms-To: IN%"cube-lovers@ai.ai.mit.edu" Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII Content-Transfer-Encoding: 7BIT cube-lovers-request@ai.ai.mit.edu" From raymond@cps.msu.edu Fri Oct 1 12:37:21 1993 Return-Path: Received: from atlantic.cps.msu.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15150; Fri, 1 Oct 93 12:37:21 EDT Received: from pacific (pacific.cps.msu.edu) by atlantic.cps.msu.edu (4.1/rpj-5.0); id AA07924; Fri, 1 Oct 93 12:37:10 EDT Received: by pacific (5.0/SMI-SVR4) id AA28543; Fri, 1 Oct 93 12:37:08 EDT Date: Fri, 1 Oct 93 12:37:08 EDT From: raymond@cps.msu.edu Message-Id: <9310011637.AA28543@pacific> To: cube-lovers@ai.mit.edu Subject: Seeking 5x5x5 cube Content-Length: 115 Hello cube lovers, I am looking for a 5x5x5 cube. Does anybody know where I can get one? Thanks, Carl Raymond From queiroz@eepost.uta.edu Fri Oct 1 16:25:59 1993 Return-Path: Received: from ee.uta.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA28670; Fri, 1 Oct 93 16:25:59 EDT Received: from eepost.uta.edu by ee.uta.edu (4.1/SunOS 4.1.1) id AA04334; Fri, 1 Oct 93 15:19:53 CDT Received: by eepost.uta.edu (4.1/SMI-4.1) id AA10751; Fri, 1 Oct 93 15:21:49 CDT Date: Fri, 1 Oct 1993 15:15:49 -0500 (CDT) From: Ricardoh Queiroz Subject: Re: Seeking 5x5x5 cube To: raymond@cps.msu.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <9310011637.AA28543@pacific> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Fri, 1 Oct 1993 raymond@cps.msu.edu wrote: > Hello cube lovers, > > I am looking for a 5x5x5 cube. Does anybody know where I can > get one? > > Thanks, > Carl Raymond Hi, I also have interest in a regular 3x3x3 and I can't find it. If anyone has any idea, please let us know. Thanks, Ricardo queiroz@eepost.uta.edu From raymond@cps.msu.edu Fri Oct 1 18:06:21 1993 Return-Path: Received: from atlantic.cps.msu.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04931; Fri, 1 Oct 93 18:06:21 EDT Received: from pacific (pacific.cps.msu.edu) by atlantic.cps.msu.edu (4.1/rpj-5.0); id AA18891; Fri, 1 Oct 93 18:06:14 EDT Received: by pacific (5.0/SMI-SVR4) id AA03566; Fri, 1 Oct 93 18:06:13 EDT Date: Fri, 1 Oct 93 18:06:13 EDT From: raymond@cps.msu.edu Message-Id: <9310012206.AA03566@pacific> To: queiroz@eepost.uta.edu, raymond@cps.msu.edu Subject: Re: Seeking 5x5x5 cube Cc: cube-lovers@ai.mit.edu Content-Length: 456 During the Christmas toy season last year, a local supermarket/department store chain (Meijer in Michigan) had 3x3x3 cubes. I don't recall the manufaturer, but they used the "Rubik's Cube" brand name. They also had a picture on the center cubies on each face that had to be correctly rotated for a "proper" solution. I can't recall the price, but it was reasonable. Maybe they will be easier to find as Christmas gets closer. Good luck, Carl Raymond From ncramer@bbn.com Mon Oct 4 08:51:15 1993 Return-Path: Received: from LABS-N.BBN.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19338; Mon, 4 Oct 93 08:51:15 EDT Message-Id: <9310041251.AA19338@life.ai.mit.edu> Date: Mon, 4 Oct 93 8:40:53 EDT From: Nichael Cramer To: Ricardoh Queiroz Cc: raymond@cps.msu.edu, cube-lovers@ai.mit.edu Subject: Re: Seeking 5x5x5 cube >Date: Fri, 1 Oct 1993 15:15:49 -0500 (CDT) >From: Ricardoh Queiroz >Subject: Re: Seeking 5x5x5 cube > >Hi, >I also have interest in a regular 3x3x3 and I can't find it. >If anyone has any idea, please let us know. >Thanks, >Ricardo >queiroz@eepost.uta.edu Games People Play in Harvard Square had a number of 3X3X3 that go by the name "Fourth Dimension" or something like that (they various have logos and a picture that looks like a profile of Rubik on four of the center faces). It was something on the order of $10. It also seemed more cheaply made than my other, older cubes. It felt, well, "lighter" in my hand and is rather more difficult to turn than I'm used to. It's not that they're unusable, and it's just not that they're just stiff, rather that they seem to be slightly out of alignment. Perhaps if you are someone who knows how to fine tune these things... N From ronnie@cisco.com Mon Oct 4 11:28:32 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27800; Mon, 4 Oct 93 11:28:32 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA03014 (5.67a/IDA-1.5 for ); Mon, 4 Oct 1993 08:28:29 -0700 Message-Id: <199310041528.AA03014@lager.cisco.com> To: cube-lovers@ai.mit.edu Subject: Re: Seeking 5x5x5 cube In-Reply-To: Your message of "Mon, 04 Oct 1993 08:40:53 EDT." <9310041251.AA19338@life.ai.mit.edu> Date: Mon, 04 Oct 1993 08:28:28 -0700 From: "Ronnie B. Kon" Try mailing to Peter Beck (pbeck@pica.army.mil). Ronnie From @mizzou1.missouri.edu:HOWSER@LUA6.LU.EDU Wed Oct 6 01:40:53 1993 Return-Path: <@mizzou1.missouri.edu:HOWSER@LUA6.LU.EDU> Received: from MIZZOU1.missouri.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18037; Wed, 6 Oct 93 01:40:53 EDT Received: from LUGATE.LU.EDU by MIZZOU1.missouri.edu (IBM VM SMTP V2R2) with TCP; Mon, 04 Oct 93 17:25:31 CDT Received: from LUA6.LU.EDU (p) by LUGATE.LU.EDU (4.1/6.2); Mon, 4 Oct 93 17:20:27 CDT Date: 04 OCT 93 17:32 From: To: Subject: Stiff and/or misaligned cubes Comments: Automatic Return Receipt Requested Message-Id: Back in the 'good old days' when cubing was very popular, I had a cube that was very prone to hanging up when you turned it in certain directions. I solved the problem by disassembling the cube and working on the cublets individually to remove any excess plastic and to smooth any rough spots by scraping with a razor blade and/or sanding with model car sandpaper. I raced many people with that cube and still have it after all these years. I found that the time I spent working on the bad cublets has lead to that cube wearing much more evenly than the ones I have that I never got around to working on. I also find that it gets more consistant in its movements as time goes by. As it was one of the first cubes on the market (before the BIG craze, actually) it is rather heavy but not as precisely made as the later cubes. ------------------------------------------------------------------------ Gerry Howser INTERNET: howser@lua6.lu.edu Postmaster@lua6.lul.edu Monet01@umcvmb.missouri.edu (Alternate) VOICE: (314) 681-5400 FAX: (314) 681-5566 ------------------------------------------------------------------------ From dml@hpfrcu03.france.hp.com Thu Oct 7 16:48:50 1993 Return-Path: Received: from hp.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04635; Thu, 7 Oct 93 16:48:50 EDT Received: from hpfrcu03.france.hp.com by hp.com with SMTP (16.8/15.5+IOS 3.13) id AA23157; Thu, 7 Oct 93 09:59:30 -0700 Received: by hpfrcu03.france.hp.com (1.37.109.4/15.5+IOS 3.22) id AA14854; Thu, 7 Oct 93 17:59:19 +0100 From: Patrick DEMICHEL Message-Id: <9310071659.AA14854@hpfrcu03.france.hp.com> Subject: help To: CUBE-LOVERS@life.ai.mit.edu Date: Thu, 7 Oct 93 17:59:18 MET Cc: dml@hpfrcu03.france.hp.com Mailer: Elm [revision: 72.14] help From diamond@jit081.enet.dec.com Thu Oct 7 21:54:26 1993 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21593; Thu, 7 Oct 93 21:54:26 EDT Received: by enet-gw.pa.dec.com; id AA17487; Thu, 7 Oct 93 18:54:18 -0700 Message-Id: <9310080154.AA17487@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Thu, 7 Oct 93 18:54:25 PDT Date: Thu, 7 Oct 93 18:54:25 PDT From: 08-Oct-1993 1054 To: "dml@hpfrcu03.france.hp.com"@jrdmax.enet.dec.com Cc: cube-lovers@life.ai.mit.edu Apparently-To: cube-lovers@life.ai.mit.edu Subject: RE: help dml@hpfrcu03.france.hp.com (Patrick DEMICHEL) writes: >help First you turn one side, then you turn another side. Keep it up, and soon you'll be done. -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From pbeck@pica.army.mil Fri Oct 8 08:05:19 1993 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10909; Fri, 8 Oct 93 08:05:19 EDT Date: Fri, 8 Oct 93 7:52:20 EDT From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Cc: pbeck@pica.army.mil Subject: golden solids Message-Id: <9310080752.aa13373@COR6.PICA.ARMY.MIL> i am trying to find a copy of the following: THE AESTHETICS OF THE SACRED, A HARMONIC GEOMETRY OF CONSCIOUSNESS & PHILOSOPHY OF SACRED ARCHITECTURE by ROBERT C MEURANT THE OPOUTERE PRESS BOULDER & AUCKLAND ISBN 0-908809-02-6 if you know of a bookseller who might carry this please let me know. if you have the address of opoutere press in NZ that would also be helpful. thanks From @mail.uunet.ca:mark.longridge@canrem.com Thu Oct 28 19:41:34 1993 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from ghost.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA02092; Thu, 28 Oct 93 19:41:34 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <101599(2)>; Thu, 28 Oct 1993 19:41:10 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA10332; Thu, 28 Oct 93 19:40:24 EDT Received: by canrem.com (PCB-UUCP 1.1e) id 188656; Thu, 28 Oct 93 19:26:28 -0400 To: cube-lovers@life.ai.mit.edu Subject: Cube Patterns From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.305920.104.0C188656@canrem.com> Date: Thu, 28 Oct 1993 19:20:00 -0400 Organization: CRS Online (Toronto, Ontario) Comments on Rubik's Cube Patterns --------------------------------- First some positions of theoretical interest: (F R B L)^5 = F1 L3 D2 F3 B2 R1 L3 F2 B3 R2 B1 U2 D2 R3 D2 L2 B2 L2 F2 (19 moves) So in the ht metric this is compressible. I've been thinking about new approaches to finding new patterns. To improve on the "old-fashioned" method of simply taking a cube and twisting it I wrote a module to test for legality of position and another module for arrangement entry. Thus I can doodle around with a cube pattern much more efficiently. This approach led to the discovery of the ML Checkerboard, which is to date the most involved of the pretty patterns: ML's Checkerboard = B1 U2 R1 L1 D2 B3 L2 F2 R1 F3 U3 D3 F3 B3 R2 U1 R2 D3 L2 (19 moves) Also by combining the 8 twist and the first discovered square's group antipode, a new corner's only pattern: Antwist = R1 F2 B2 D2 R1 L3 B2 R1 B2 U1 F2 U2 F2 D2 F2 R2 L2 D3 (18 moves) Also I have re-evaluated what is a complex cube position. Cube positions have different degrees (or types) of difficulty. A. A position is difficult if it is visually hard to recognize, e.g. no pattern is apparent, the cube is well mixed and random. However the pattern superfliptwist, although being 20 moves deep, IS easy to recognize. B. A position is easy with respect to computer analysis if it is cyclically decomposable. That is to say it by looking at a position a program finds it is generated by (F R B L)^5, so this position is EASY. C. A position is easy with respect to the human hand if the sequence required to solve the position can be executed rapidly. To a degree such positions are similar to positions in point (B) in that only a subset of all cube operators are required, and the sequence does not require turning all 6 sides and so the sequence is easier to memorize as well. As a result of thinking along these lines I am going to write a module to do cyclic decomposition. -> Mark <- Email: mark.longridge@canrem.com ....more patterns to follow... From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sat Dec 4 09:49:17 1993 Return-Path: <@mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06967; Sat, 4 Dec 93 09:49:17 EST Message-Id: <9312041449.AA06967@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 9070; Sat, 04 Dec 93 09:18:44 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 0830; Sat, 4 Dec 1993 09:18:44 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 0521; Sat, 4 Dec 1993 09:15:57 -0500 X-Acknowledge-To: Date: Sat, 4 Dec 1993 09:15:56 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: First Post This is my first post to Cube-Lovers, so I will introduce myself briefly. I have been cubing since about 1979 or 1980 or so when the cubes first appeared on the market. I have been cubing with a computer since about 1985, and have been active on the Internet since about 1985 (purely a coincidence of dates). I have looked for years for a cubing list, and never found one until now. I always looked for "Rubik" (or sometimes "Rubic"). For some silly reason, it never occurred to me to look for "cube". I have long since read Hofstadter's two Scientific American articles, as well as the reprints in METAMAGICAL THEMAS. The reprints, by the way, are excellent because of the additional information in the appendices. I also have a copy of Singmaster and Frey's HANDBOOK OF CUBIC MATH. I have tried unsuccessfully for years to get copies of Singmaster's earlier work -- the circulars, for example. However, I suspect that the HANDBOOK includes most if not all of the earlier material. Also, (and you won't believe this) I have just read all thirteen years of the archives of Cube-Lovers. My primary interest has been in calculating God's Algorithm. I am interested in brute force breadth first tree searches. In other words, my work is akin to the solutions of the 2x2x2 and the corners of the 3x3x3 posted by Dan Hoey and others. It is not akin to Thistlethwaite's methods. I am interested to see, however, that major recent progress appears to have been made on Thistlewaite's method. I have calculated God's Algorithm for the 2x2x2 cube and the corners of the 3x3x3. My results agree with those that have been posted here, with the exception that my search is 48 times smaller (24*2), due to the exploitation of a rotation and reflection group of the cube. I have also calculated God's Algorithm for the edges of the 3x3x3. This is a much larger problem, and took about a year running continuously on two machines. The resulting output file is about 4.2 gigabytes of data, and is stored on 31 reels of magnetic tape. This result includes the "48 times smaller" factor, else it would have been 204 gigabytes of data stored on 1464 reels of magnetic tape. I understand that this list has been very quiet of late. But assuming some modicum of interest, I will post more details of my results in subsequent messages. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sat Dec 4 21:07:09 1993 Return-Path: <@mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01155; Sat, 4 De