? Is it a local maximum (strong or otherwise) in? Is the length 20q in? Is it a local maximum in? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From mreid@ptc.com Tue May 30 15:39:31 1995 Return-Path:Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25072; Tue, 30 May 95 15:39:31 EDT Received: from ducie by ptc.com (5.0/SMI-SVR4-NN) id AA11138; Tue, 30 May 95 15:37:29 EDT Received: by ducie (1.38.193.4/sendmail.28-May-87) id AA15624; Tue, 30 May 1995 12:29:29 -0400 Date: Tue, 30 May 1995 12:29:29 -0400 From: mreid@ptc.com (michael reid) Message-Id: <9505301629.AA15624@ducie> To: cube-lovers@ai.mit.edu Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) Content-Length: 1649 jerry writes [ ... ] > >> >> Level Positions Strong Weak Total > >> >> Local Max Local Max Local Max > >> >> > >> >> 6 184 1 35 36 > >> > >> >beg the obvious question: what is that strong local maximum, > >> >which is unique up to symmetry? [ ... ] > Try instead, (L2R2)(U2D2)(F'B')(U2D2)(L2R2)(F'B'). > > This is a very pretty pattern which may well have a name, but I > don't know what the name is. Also, it is its own inverse. > > Is the length 12h in ? Is it a local maximum (strong or otherwise) > in? Is the length 20q in? Is it a local maximum in? no, yes (otherwise), no, and yes, respectively. we have seen this pattern several times recently. this is one of those positions with 16 symmetries. i called it "four pluses" in my message of may 11 (although i gave it in a different orientation) ) four pluses ( R2 F2 R2 U'D F2 R2 F2 UD' ) in fact, this maneuver is minimal in both the quarter turn and the face turn metrics, so its length is 16q, 10f. it is a weak local maximum in the face turn metric; one can check that no minimal maneuver ends with the face turn R. however, using the 16 symmetries which preserve the U-D axis, and inversion, we can give minimal maneuvers which end with turns of any of the six faces. this shows that it's a weak local maximum in the face turn metric. local maximality in the quarter turn metric follows in a similar manner. also, mark pointed out on april 16 that this position lies in the center of the antislice group. mike From @mail.uunet.ca:mark.longridge@canrem.com Sat Jun 3 03:39:09 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03320; Sat, 3 Jun 95 03:39:09 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <174124-7>; Sat, 3 Jun 1995 03:40:48 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA02923; Sat, 3 Jun 95 03:35:21 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E504D; Sat, 3 Jun 95 03:29:35 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Super Groups From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1148.5834.0C1E504D@canrem.com> Date: Sat, 3 Jun 1995 04:14:00 -0400 Organization: CRS Online (Toronto, Ontario) Notes on the various Super-Groups --------------------------------- I have calculated the size of the super-groups for various subgroups of the cube. I have suffixed the standard group names with the letter c to show that the centre orientations are significant. The groups are (ranked smallest to largest): Size (slice) = 768 Size (slicec) = 24,576 Size (slicec) / Size (slice) = 32 The following reference confirms this calculation and expounds further on the nature of the slice group... The Slice Group in Rubik's Cube, by David Hecker, Ranan Banerji Mathematics Magazine, Vol. 58 No. 4 Sept 1985 Size (antisl) = 6,144 Size (antislc) = 49,152 Size (antislc) / Size (antisl) = 8 Size (sq) = 663,552 Size (sqc) = 5,308,416 Size (sqc) / Size (sq) = 8 Size (ur) = 73,483,200 Size (urc) = 587,865,600 Size (urc) / Size (ur) = 8 Size (cube) = 43,252,003,274,489,856,000 Size (cubec) = 88,580,102,706,155,225,088,000 Size (cubec) / Size (cube) = 2,048 The case of the super squares group (sqc) is interesting. It is only possible to rotate opposite centres 180 degrees. There are actually 8 centres in the super square's group: (1 way) Identity (1 way) All 6 centres rotated 180 degrees (3 ways) 2 opposite centres rotated 180 degrees (3 ways) 2 pairs of opposite centres rotated 180 degrees -> Mark <- From @mail.uunet.ca:mark.longridge@canrem.com Sun Jun 4 03:28:58 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18076; Sun, 4 Jun 95 03:28:58 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <173779-8>; Sun, 4 Jun 1995 03:30:39 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA25003; Sun, 4 Jun 95 03:25:11 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E5238; Sun, 4 Jun 95 03:22:02 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Super Squares Group From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1150.5834.0C1E5238@canrem.com> Date: Sun, 4 Jun 1995 04:21:00 -0400 Organization: CRS Online (Toronto, Ontario) Way back on Thu Aug 20 20:10:13 1992 Mike wrote: > R2 F2 B2 L2 U2 L2 F2 B2 R2 ~ D2 , > so that= . I bet he didn't realize at the time he was finding a minimal sequence to rotate 2 centres in the super square's group! If we tack D2 on the end we get the sequence: R2 F2 B2 L2 U2 L2 F2 B2 R2 D2 = turn U & D centres 180 degrees in 10 face turns. -> Mark <- From bagleyd@source.asset.com Wed Jun 7 17:31:08 1995 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12024; Wed, 7 Jun 95 17:31:08 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA36858; Wed, 7 Jun 1995 17:07:22 -0400 Date: Wed, 7 Jun 1995 17:07:22 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9506072107.AA36858@source.asset.com> To: Cube-Lovers@ai.mit.edu Subject: Dinosaur Rubik's Cube Hi I just just added new modes to my xdino puzzle. It is the Rubik's Dinosaur Puzzle recently mentioned. (A cube with diagonal X cuts.) In addition to the Period 3 movement it now has a Period 2 movement with the faces cut up like: ___ |\ /| | X | |/ \| --- as opposed to just \ / X / \ In the Period 3 movement the cube turns around a corner while in Period 2 movement it turns around the center of an edge. Of course if you want to make it harder there is a "Both" mode where you can have both turning modes at once. I would like to thank Derek Bosch for suggesting the Period 2 movement -> Bosch's Cube :) I spent a good 10 minutes trying to solve Bosch's Cube and did not get anywhere. The Period 3 seems easier. Be the first in the Universe to solve it. :) All those who have X should be able to run it. There are many other puzzles in the collection as well. Any problems with the compilation or bugs ... let me know. Cheers, --__--------------------------------------------------------------- / \ \ / David A. Bagley \ | \ \ / bagleyd@source.asset.com | | \//\ Some days are better than other days. | | / \ \ -- A short lived character of Blake's 7 | \ / \_\puzzles Available at: ftp.x.org/contrib/games/puzzles / ------------------------------------------------------------------- From norgomez@itecs5.telecom-co.net Mon Jun 12 22:31:21 1995 Return-Path: Received: from ITECS5.TELECOM-CO.NET by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA11033; Mon, 12 Jun 95 22:31:21 EDT Date: Mon, 12 Jun 1995 21:28:16 -0400 Message-Id: <95061221281669@itecs5.telecom-co.net> From: norgomez@itecs5.telecom-co.net (ORLANDO GOMEZ CAMACHO, BOGOTA-COLOMBIA.) To: JCCANNEK@ukcc.uky.edu, MTY.ITESM.MX@ukcc.uky.edu, commune-list@stealth.acf.nyu.edu, COMP-CEN%UCCVMA.BITNET@vm1.nodak.edu, CONFOCAL%UBVM.BITNET@vm1.nodak.edu, CW-MAIL%TECMTYVM.BITNET@vm1.nodak.edu, CWIS-L%WUVMD.BITNET@vm1.nodak.edu, DB2-L%AUVM.BITNET@vm1.nodak.edu, DBASE-L%TECMTYVM.BITNET@vm1.nodak.edu, JO%ILNCRD.BITNET@cunyvm.cuny.edu, COMPOS01%ULKYVX.BITNET@cunyvm.cuny.edu, CRTNET%PSUVM.BITNET@cunyvm.cuny.edu, CUMREC-L%NDSUVM1.BITNET@cunyvm.cuny.edu, CVNET%YORKVM1.BITNET@cunyvm.cuny.edu, Cyber-L%Bitnic.BITNET@cunyvm.cuny.edu, DANCE-L%HEARN.BITNET@cunyvm.cuny.edu, Comp-Soc@limbo.intuitive.com, concrete-blonde@ferkel.ucsb.edu, CONS-L%MCGILL1.BITNET@cornellc.cit.cornell.edu, 441495@acadvm1.uottawa.ca, com-priv@psi.com, cjr2@cornell.edu, CORPORA-REQUEST@x400.hd.uib.no, CORRYFEE@hasara11.telecom-co.net, CPAE@catfish.valdosta.peachnet.edu, CPE-LIST@uncvm1.oit.unc.edu, CREA-CPS@nic.surfnet.nl, CREWRT-L@mizzou1.missouri.edu, CROMED-L@aearn.bitnet, CSNET-FORUM@sh.cs.net Subject: Oportunidades de Negocios X-Vms-To: @LISTA1.DIS ====================================================================== OPORTUNIDADES DE NEGOCIOS SOLO PARA PERSONAS RESIDENTES EN LOS ESTADOS UNIDOS, CON BUENOS CONTACTOS EN EMPRESAS DE ALTA TECNOLOGIA EN INFORMATICA, OFIMATICA Y TELECOMUNICACIONES ----------------------------------------------------------------------- Apreciados amigos de la lista: Nuestra empresa ha sido contratada por una firma que edita una publicaciOn especializada en las Areas de Computadoras, Telecomunicaciones, OfimAtica y Servicios relacionados que circula en Venezuela, Ecuador, Peru, Brazil y Colombia, constituyEndose en un medio Unico de informaciOn sobre estos tOpicos en las naciones descritas. Para comercializar ESPACIOS PUBLICITARIOS de esta publicaciOn, se requiere de una compan~Ia o persona que maneje presupuesto de negocios, preferible- mente con capacidad econOmica propia e idoneidad para contratar. Es nece- sario que la persona o empresa resida en los Estados Unidos de AmErica. A continuaciOn se presentan las caracterIsticas principales de la publica- ciOn: * CirculaciOn efectiva en Brazil, Colombia, Ecuador, Peru y Venezuela. * A los anunciadores se les proveen en medio magnEtico, las bases de datos con informaciOn importante de estos paises. A las Empresas y Usuarios de cada naciOn tambiEn se les harA entrega de este material que incluirA informaciOn sobre proveedores de bienes y servicios en los sectores de Telecomunicaciones, Software, Equipos de Oficina, Computadores, y Suministros. * En la publicaciOn figuran empresas Norteamericanas que fabrican, representan y proveen equipos, programas y servicios en las areas de telecomunicaciones, sector electronico, de comunicacion, computacion perifericos, componentesm accesorios, suministros, servicios y areas afines * El tiraje es de Cuarenta Mil (40,000) ejemplares para ser distribui- dos en forma gratuita. * El perfil de los anunciadores corresponde a compan~Ias Estadouniden- ses de estos sectores que se encuentren en ampliar y consolidar su operacionalidad en estos paises. La Empresa OFRECE: * Exclusividad * Participacion atractiva sobre los negocios * Excelente calidad editorial y material informativo y util para generar nuevas y mejores oportunidades de negocios en forma proactiva. ============================================================================== A las personas interesadas, se les ruega contactarnos por esta misma via, indicando los siguientes datos: Nombre : e-mail : Direccion: Ciudad : Telefono : Fax : ============================================================================== Mayores informes: COMUNICACIONES INTERACTIVAS Contacto : Orlando Gomez Camacho e-mail : norgomez@itecs5.telecom-co.net Voice Mail: (+571) 5002072 ------------------------------------------------------------------------------ From MULL4195@splava.cc.plattsburgh.edu Wed Jun 14 16:17:44 1995 Return-Path: Received: from splava.cc.plattsburgh.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29070; Wed, 14 Jun 95 16:17:44 EDT Received: from splava.cc.plattsburgh.edu by splava.cc.plattsburgh.edu (PMDF V4.2-11 #3312) id <01HRPAOJ7EWG8WYOF1@splava.cc.plattsburgh.edu>; Wed, 14 Jun 1995 16:17:58 EST Date: Wed, 14 Jun 1995 16:17:58 -0500 (EST) From: John Mullen Subject: To: Cube-Lovers@ai.mit.edu Message-Id: <01HRPAOJ87UA8WYOF1@splava.cc.plattsburgh.edu> Organization: SUNY at Plattsburgh, New York, USA X-Envelope-To: Cube-Lovers@ai.mit.edu X-Vms-To: IN%"Cube-Lovers@ai.mit.edu" X-Vms-Cc: MULL4195 Mime-Version: 1.0 Content-Transfer-Encoding: 7BIT signoff cube-lovers From comnetlu!georges.helm@eo.net Thu Jun 15 15:38:16 1995 Return-Path: Received: from eol1 (eol1.eo.lu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10184; Thu, 15 Jun 95 15:38:16 EDT Received: from comnetlu.UUCP by eol1 with UUCP (Smail3.1.28.1 #4) id m0sMKk2-000bKNC; Thu, 15 Jun 95 21:38 MET DST To: cube-lovers@ai.mit.edu Subject: Andras Mezei's book From: georges.helm@comnet.eo.lu (GEORGES HELM) Message-Id: <8AB54D5.0063000243.uuout@comnet.eo.lu> Date: Thu, 15 Jun 95 20:37:00 +1 Organization: ComNet Luxembourg BBS Reply-To: georges.helm@comnet.eo.lu (GEORGES HELM) X-Mailreader: PCBoard Version 15.21 X-Mailer: PCBoard/UUOUT Version 1.10 Mark Longridge wrote > Does anyone on Cube-Lovers have that book? Yes, I do. The illustrations cover: Rubik himself, Polytoys, Konsumex, cubes, books, cube-related items (stickers, pencils, lp's, earrings, t-shirts...) There is an article on a competition in 1982 in Budapest. And many more articles I don't understand a word of. Georges From comnetlu!georges.helm@eo.net Fri Jun 16 03:20:00 1995 Return-Path: Received: from eol1 (eol1.eo.lu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19818; Fri, 16 Jun 95 03:20:00 EDT Received: from comnetlu.UUCP by eol1 with UUCP (Smail3.1.28.1 #4) id m0sMVh1-000bKLC; Fri, 16 Jun 95 09:19 MET DST To: Cube-Lovers@ai.mit.edu Subject: Andras Mezei's book From: georges.helm@comnet.eo.lu (GEORGES HELM) Message-Id: <8AB622C.006300026F.uuout@comnet.eo.lu> Date: Fri, 16 Jun 95 09:16:00 +1 Organization: ComNet Luxembourg BBS Reply-To: georges.helm@comnet.eo.lu (GEORGES HELM) X-Mailreader: PCBoard Version 15.21 X-Mailer: PCBoard/UUOUT Version 1.10 I have the book. - Georges From BRYAN@wvnvm.wvnet.edu Fri Jun 16 14:10:52 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17367; Fri, 16 Jun 95 14:10:52 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 6542; Fri, 16 Jun 95 14:11:03 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2667; Fri, 16 Jun 1995 14:11:03 -0400 Message-Id: Date: Fri, 16 Jun 1995 14:11:02 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima There are four of them unique up to M-conjugacy (more information to come). Distance Cubes Branch Lcl {m'Xm} Branch Ratio Local from Factor Max (M-Conj. Factor of Max Start Classes) Cubes to Classes 0 1 1 0 1 12 12.000 0 1 1.000 12.000 0 2 114 9.500 0 5 5.000 22.800 0 3 1,068 9.368 0 25 5.000 42.720 0 4 10,011 9.374 0 219 8.760 45.712 0 5 93,840 9.374 0 1,978 9.032 47.442 0 6 878,880 9.366 0 18,395 9.300 47.778 0 7 8,221,632 9.355 0 171,529 9.325 47.931 0 8 76,843,595 9.347 0 1,601,725 9.338 47.976 0 9 717,789,576 9.341 0 14,956,266 9.338 47.993 0 10 6,701,836,858 9.337 42 139,629,194 9.336 47.997 4 11 62,549,615,248 9.333 1,303,138,445 9.333 47.9992 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Fri Jun 16 14:17:43 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17624; Fri, 16 Jun 95 14:17:43 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 6602; Fri, 16 Jun 95 14:17:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2819; Fri, 16 Jun 1995 14:17:52 -0400 Message-Id: Date: Fri, 16 Jun 1995 14:17:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima Search Matrix The individual cells in this chart give numbers of M-conjugacy classes. The local maxima are in column 12. Still more information to come. Number of Moves Which Go Closer to Start Level Total 0 1 2 3 4 5 6 7 8 9 1 1 1 Classes 0 1 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 5 0 2 3 0 0 0 0 0 0 0 0 0 0 3 25 0 20 4 1 0 0 0 0 0 0 0 0 0 4 219 0 182 34 2 1 0 0 0 0 0 0 0 0 5 1978 0 1677 280 20 1 0 0 0 0 0 0 0 0 6 18395 0 15642 2561 184 8 0 0 0 0 0 0 0 0 7 171529 0 145974 23773 1721 61 0 0 0 0 0 0 0 0 8 1601725 0 1362579 222235 16241 663 1 3 0 3 0 0 0 0 9 14956266 0 12719643 2077549 153026 5954 74 15 2 3 0 0 0 0 10 139629194 0 118711701 19418503 1438825 58862 925 318 11 37 0 8 0 4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Sat Jun 17 00:35:23 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23622; Sat, 17 Jun 95 00:35:23 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9876; Sat, 17 Jun 95 00:35:32 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 4531; Sat, 17 Jun 1995 00:35:32 -0400 Message-Id: Date: Sat, 17 Jun 1995 00:35:31 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima Positions 1. (UU) (F'B')(RL)(RL)(FB) 2. (UD') (F'B')(RL)(RL)(FB) 3. (UD) (F'B')(RL)(RL)(FB) 4. (UD')(FB')(LR')(FB')(FB') #4 is the one Mike Reid already found in the slice group. #3 is the one I already found in the antislice group. #1, #2, and #3 are obviously closely related. #1 and #2 appear not to be in either slice or antislice, but I have been fooled before by alternative sequences which yield the same position. #1, #2, and #3 all have the property that |{m'Xm}|=6 and |Symm(X)|=8. As has already been discussed, #4 has the property that |{m'Xm}|=24 and |Symm(X)|=2. The symmetry groups for #1, #2, and #3 are of a type Dan Hoey's taxonomy calls class P, class S, and class AX, respectively. These particular classes are hard to describe succinctly without introducing a lot of notation. But in all three cases, the symmetry groups (subgroups of M such that X=m'Xm} consist of four rotations and four reflections, and have as an axis of symmetry one of the three major axes of the cube (U-D, F-B, or R-L). There are three groups P1, P2, P3 with axis of symmetry U-D, F-B, and R-L, respectively, and similarly for S1, S2, and S3, and for AX1, AX2, and AX3. For #4, we have Symm(X)=HV in Dan's taxonomy, where HV={i,v}, and where i is the identity in M and v is the central inversion in M. If proper typography were available, the i and the v would be upper case script letters to follow Frey and Singmaster. There are relatively few positions in all of cube space for which Symm(X)=Pi or Symm(X)=Si or Symm(X)=AXi (i in 1..3). There are only 10 P positions through level 10 in the search tree (of which just one is a local maximum). There is only one S position through level 10, and only one AX position through level 10, both of which are of course local maxima. The positions are not Q-transitive, but the positions look "symmetric", and they fulfill the (incorrect) intuition that "symmetric" positions must be local maxima. We have no reason to say that other P or S or AX positions further down the search will be local maxima. I find position #4 extremely intriguing. In general, HV is not very strong symmetry, and there are relatively speaking, quite a few HV positions in cube space. We could create an HV position as follows. Put any edge cubie anywhere (say UF in RD). Put the "opposite" cubie in the "opposite" cubicle (DB in LU in this case). Continue for the remaining edge cubies, and then do the same thing for the corners, remembering only to make sure the edges and corners have the same parity. You can easily make an HV position that looks quite "random" to the casual glance, and in fact most HV positions don't look very "symmetric". But Mike's position looks very "symmetric" at a casual glance, as if its symmetry must be much stronger than HV. I certainly would not have expected to find an HV position as a local maximum close to Start. I think the "look" of Mike's position as "symmetric", and the fact that it is a local maximum close to Start are related. Without getting too long winded, I think the reasons are two-fold. First, the corners and edges have much stronger symmetry separately than they do collectively. Second, the symmetry looks much stronger if you ignore the centers (i.e., if you ignore the rotational positioning of the cubies), perhaps in the sense of Dan's CSymm function. For example, the corners are properly positioned with respect to each other, even though they are not properly positioned with respect to the fixed face centers. In the next few days, I intend to calculate Symm(X) for the corners and edges separately for Mike's position, as well as calculating CSymm(X) for the corners and edges separately and combined. I think the results will be enlightening. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From @mail.uunet.ca:mark.longridge@canrem.com Sat Jun 17 22:00:06 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03704; Sat, 17 Jun 95 22:00:06 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <177474-1>; Sat, 17 Jun 1995 21:03:06 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA05634; Sat, 17 Jun 95 20:57:30 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E7553; Sat, 17 Jun 95 19:59:47 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Crazy Corner Pattern From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1161.5834.0C1E7553@canrem.com> Date: Sat, 17 Jun 1995 05:38:00 -0400 Organization: CRS Online (Toronto, Ontario) On Mon, 17 Jan 1994 09:06:59 EST, Jerry wrote: > > Counting M-conjugacy classes of the corners of Rubik's cube > ----------------------------------------------------------- > > M-Class Number Number > Size of of > Classes Elements > > 1 1 = 1 Start > 2 1 = 2 +4 -4 Twist > 3 3 = 9 > 4 1 = 4 > 6 34 = 204 > 8 33 = 264 > 12 301 = 3612 > 16 104 = 1664 > 24 9064 = 217536 > 48 1832428 = 87956544 > > Total 1841970 88179840 I'm trying to find the 1 pattern with M-class size of 4 of the corners group. The only pattern that I can find is 4 alternate corners twisted clockwise which is in the twist orbit. It does not seem to be any pattern with just corners twisted in place. Jerry, if you could *please* identify this pattern before I go nuts...... ! -> Mark <- From BRYAN@wvnvm.wvnet.edu Sun Jun 18 00:35:58 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09680; Sun, 18 Jun 95 00:35:58 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 1914; Sat, 17 Jun 95 17:18:01 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1612; Sat, 17 Jun 1995 17:18:01 -0400 Message-Id: Date: Sat, 17 Jun 1995 17:18:00 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: A Third Way to Calculate the Real Size of Cube Space? We define the real size of cube space to be the number of M-conjugate classes {m'Ym} for m in M, set of 48 rotations and reflections of the cube, and for Y in G. Dan Hoey has calculated the real size of cube space using the Polya-Burnside theorem. Dan and I (mostly Dan) have also calculated the same result using exhaustive computer search. The computer search is much less elegant than the Polya-Burnside results, but the search does provide additional information, such as the number of positions associated with each symmetry group. The results from the computer search have not yet been posted to the list, but a draft paper is in progress. In the meantime, it occurs to me that perhaps -- but only perhaps -- there is a third way to calculate the real size of cube space. The third way would not require (much) computer searching, but would provide the same level of detail about number of positions per symmetry group as does the full blown search. The idea is based on a posting from Mike Reid. Mike calculated the number of positions in G whose symmetry preserves the U-D axis. Such positions have a symmetry group which is called X1 in Dan's taxonomy. For these positions, we say Symm(Y)=X1, where in general for Y in G we have Symm(Y) is the set (and group) of all m in M such that Y=m'Ym. X1 contains sixteen elements (eight rotations and eight reflections), and preserves the U-D axis. X2 and X3 are conjugate subgroups of X1 and similarly preserve the F-B and the R-L axes, respectively. If Y is X1-symmetric, then we have {m'Ym}={Y1,Y2,Y3}. One of the Yi is Y and is X1-symmetric, one of the Yi is X2-symmetric, and the other Yi is X3-symmetric. Mike determined (without computer search) that there are 128 X1-symmetric positions. Since four of the positions are also M-symmetric, we have 124 positions Y for which Symm(Y)=X1. Similar results hold for X2 and X3. Hence, there are 124 M-conjugacy classes containing cubes for which Symm(Y)=Xi, or perhaps we might say for which SymmClass(Y)=X. The important fact here is that we have determined that there are 124 M-conjugacy classes for symmetry class X without having to do a computer search. If we could similarly determine the number of K-symmetric positions for each of the 98 subgroups K of M without computer search, then we could calculate the real size of cube space. You really only have to determine the size of 33 subgroups. Just as the solution for X1 also gave us the solution for X2 and X3, similarly the solution for any subgroup provides the solution for all conjugate subgroups, and there are 33 classes of conjugate subgroups. I usually get myself in trouble when I delve too much into things I don't understand, but let's try a few examples. The subgroup HV={i,v} is easy to understand, where v is the central inversion. For the edges, the number of HV-symmetric positions should be 24*20*16*12*8*4. That is, put the first cubie anywhere (24 possibilities) which dictates the location of the respective "opposite" cubie. There are then 20 possibilities for the location of the third cubie which again dictates the position of the respective "opposite" cubie, and so forth. In the same manner, the number of HV-symmetric corner positions is 24*18*12*6. The number of HV-symmetric positions is then (24*20*16*12*8*4)*(24*18*12*6)/2 to take parity into account. Now we have the rub. In order to calculate the positions for which Symm(Y)=HV, we must subtract out the HV-symmetric positions which have stronger symmetry than HV, just as we subtracted out the M-symmetric positions in Mike's X1 case. But to do so, we cannot take the subgroups of M in isolation. We have to do them all, starting with M and working our way down. (And HV is pretty far down the food chain.) Some of the subgroups I can do pretty easily, and for others I have not a clue. Recall that A is the subgroup of M consisting of the 24 even rotations and reflections and that C is the subgroup of M consisting of the 24 rotations (12 even and 12 odd). As long ago as _Symmetry and Local Maxima_, Dan Hoey and Jim Saxe determined that there are only four A-symmetric positions and only four C-symmetric positions, namely the four that are also M-symmetric. Hence, there are no positions for which Symm(Y)=A nor for which Symm(Y)=C. But I haven't a clue how they knew, nor how to go about constructing an A-symmetric or a C-symmetric position from scratch. You can't get very far with my proposal unless you can figure out how to construct K-symmetric positions for any K. For one more example, consider H, the set of 12 even rotations and 12 odd reflections. I know from computer search and also from _Symmetry and Local Maxima_ that there are 24 H-symmetric positions, of which 4 are M-symmetric and 20 are H-symmetric without also being M-symmetric. The 20 H-symmetric but not M-symmetric positions form 10 M-conjugacy classes for which we would say SymmClass(Y)=H. It ought to be easy to derive this result without a computer search, but again I confess I haven't a clue as to go about constructing the 24 H-symmetric positions from scratch. Well, I could cheat and look up the Class H positions in _Symmetry and Local Maxima_, but what about the classes that haven't been figured out yet? Also, I could cheat and use the results from computer search, but that's hardly the point. One final point: just as Mike's 128 X1-symmetric positions formed a group, similarly the set of K-symmetric positions form a group for all 98 possible values of K. We have to be a little careful with our terminology. The X1-symmetric positions form a group, as do the X2-symmetric and the X3-symmetric positions. But if we want to talk about the X-symmetric positions, we no longer have a group. For example, we do not in general have closure when forming the composition of X1-symmetric positions with X2-symmetric positions. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Sun Jun 18 08:48:57 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17825; Sun, 18 Jun 95 08:48:57 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3647; Sun, 18 Jun 95 08:25:50 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 8731; Sun, 18 Jun 1995 08:25:50 -0400 Message-Id: Date: Sun, 18 Jun 1995 08:25:49 -0400 (EDT) From: "Jerry Bryan" To: Subject: Re: Crazy Corner Pattern In-Reply-To: Message of 06/17/95 at 05:38:00 from mark.longridge@canrem.com On 06/17/95 at 05:38:00 mark.longridge@canrem.com said: >On Mon, 17 Jan 1994 09:06:59 EST, Jerry wrote: >> >> Counting M-conjugacy classes of the corners of Rubik's cube >> ----------------------------------------------------------- >> >> M-Class Number Number >> Size of of >> Classes Elements >> >> 1 1 = 1 Start >> 2 1 = 2 +4 -4 Twist >> 3 3 = 9 >> 4 1 = 4 >> 6 34 = 204 >> 8 33 = 264 >> 12 301 = 3612 >> 16 104 = 1664 >> 24 9064 = 217536 >> 48 1832428 = 87956544 >> >> Total 1841970 88179840 >I'm trying to find the 1 pattern with M-class size of 4 of the >corners group. >The only pattern that I can find is 4 alternate corners twisted >clockwise which is in the twist orbit. >It does not seem to be any pattern with just corners twisted in >place. >Jerry, if you could *please* identify this pattern before I go >nuts...... ! The position is called T-symmetric in Dan's taxonomy (actually, there are four T subgroups of M, T1 through T4). The symmetry is related to opposite corners, e.g., the UFL-DRB axis, which is why there are four T subgroups. Also, the position is Q-transitive, so you can check it out in _Symmetry and Local Maxima_. I quote ".... each edge on the girdle may be swapped with the diametrically opposite edge, provided that the corners on the girdle are swapped with their opposites as well." Here, you would fix the edges and pay attention only to the swapping of the corners. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Sun Jun 18 15:55:14 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03583; Sun, 18 Jun 95 15:55:14 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4771; Sun, 18 Jun 95 15:55:23 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3480; Sun, 18 Jun 1995 15:55:23 -0400 Message-Id: Date: Sun, 18 Jun 1995 15:55:22 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/17/95 at 17:18:00 from BRYAN@wvnvm.wvnet.edu I should have said "a fourth way", I think. Martin Schoernert performed the same calculation with GAP. Hence, we have three ways in hand: 1) Dan's Polya-Burnside method, 2) Martin's GAP calculations, and 3) brute force computer search. My new proposal would then be a fourth way. Here is a question for Martin: is there any way with GAP to calculate the number of M-conjugacy classes associated with each symmetry class? It is this additional information about the "real cube space" which *is* available via computer search, and for which I am proposing an alternative which does not involve computer search. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From @mail.uunet.ca:mark.longridge@canrem.com Sun Jun 18 23:54:16 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23716; Sun, 18 Jun 95 23:54:16 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <177456-5>; Sun, 18 Jun 1995 23:55:59 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA07947; Sun, 18 Jun 95 23:50:23 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E77E1; Sun, 18 Jun 95 23:47:31 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Re: Crazy Corner Pattern From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1167.5834.0C1E77E1@canrem.com> Date: Mon, 19 Jun 1995 00:36:00 -0400 Organization: CRS Online (Toronto, Ontario) >>I'm trying to find the 1 pattern with M-class size of 4 of the >>corners group. >>The only pattern that I can find is 4 alternate corners twisted >>clockwise which is in the twist orbit. >>It does not seem to be any pattern with just corners twisted in >>place. >>Jerry, if you could *please* identify this pattern before I go >>nuts...... ! >The position is called T-symmetric in Dan's taxonomy (actually, >there are four T subgroups of M, T1 through T4). The symmetry >is related to opposite corners, e.g., the UFL-DRB axis, >which is why there are four T subgroups. >Also, the position is Q-transitive, so you can check it out >in _Symmetry and Local Maxima_. I quote ".... each edge on the >girdle may be swapped with the diametrically opposite edge, >provided that the corners on the girdle are swapped with >their opposites as well." Here, you would fix the edges and >pay attention only to the swapping of the corners. Hmmmmm, that's very interesting. Below is my interpretation of "...corners on the girdle are swapped with their opposites as well." Let's call it pattern X. D U D U U U D U D F L B R F L B R F L B R L L L F F F R R R B B B F L B R F L B R F L B R U D U D D D U D U If I understand the terminology correctly, then for this pattern X = |{m'Xm}|=3 and |Symm(X)|=16, same as the 4 spot. Also X = |{c'Xc}|=3. But perhaps this is not the same pattern.... Let's call this next cube arrangement pattern Y. F U U U U U D U L R L B R F B D R R B B D L L L F F F R R R B B B U L L F F U L R F L B F D D B D D D R D U Y = |{m'Ym}|=4 and |Symm(Y)|=12, and Y=|{c'Yc}| = 4 This pattern was created by swapping 3 pairs of diametrically opposite corners, which is in the *swap* orbit on a normal cube, but since we are dealing with corners of Rubik's cube and ignoring edges we can realize permutations with an odd number of pairs of corners swapped. -> Mark <- From BRYAN@wvnvm.wvnet.edu Mon Jun 19 09:57:45 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14163; Mon, 19 Jun 95 09:57:45 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8039; Mon, 19 Jun 95 09:57:53 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 6101; Mon, 19 Jun 1995 09:57:54 -0400 Message-Id: Date: Mon, 19 Jun 1995 09:57:53 -0400 (EDT) From: "Jerry Bryan" To: Subject: Re: Crazy Corner Pattern In-Reply-To: Message of 06/19/95 at 00:36:00 from mark.longridge@canrem.com Perhaps I should have quoted a little more from _Symmetry and Local Maxima_. Here is the T-symmetric position given by Saxe and Hoey. In their position, the UFL and RBD corners are in place, and the other three pairs are swapped. The "girdle" includes the three pairs that are swapped. Hence, there is an axis of symmetry along the UFL-RBD axis. The odd number of swaps is compensated by an odd number in the edges. The compensation is not required for corners only. R D D U U D U U B D L L F F D L L F L F F R L L F F B L R R B B F R R B R B B U R R B B U U U L U D D F D D = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Mon Jun 19 14:04:49 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29637; Mon, 19 Jun 95 14:04:49 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0056; Mon, 19 Jun 95 13:41:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3719; Mon, 19 Jun 1995 13:41:52 -0400 Message-Id: Date: Mon, 19 Jun 1995 13:41:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 3x3x3 Cubes for Sale For the first time in years, I have seen 3x3x3 cubes for sale in a store. (For American readers, the store is Hills, which is a regional chain which competes against WalMart and Kmart. The price is $8.97 -- U.S. dollars.) The box has a note signed by Erno Rubik, and gives the proper size of the cube group. Also, and I couldn't see inside the box to verify this, the Face centers seemed to be marked in such a way as to support the Supergroup. Rubik's note about the size of the problem says it is 4^4 times bigger than the regular problem. The manufacturer (or maybe distributor?) is Matchbox Toys (or is it Matchbook Toys?). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From news@nntp-server.caltech.edu Mon Jun 19 18:36:29 1995 Return-Path: Received: from piccolo.cco.caltech.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15966; Mon, 19 Jun 95 18:36:29 EDT Received: from gap.cco.caltech.edu by piccolo.cco.caltech.edu with ESMTP (8.6.7/DEI:4.41) id PAA27121; Mon, 19 Jun 1995 15:36:21 -0700 Received: by gap.cco.caltech.edu (8.6.7/DEI:4.41) id PAA16086; Mon, 19 Jun 1995 15:36:18 -0700 To: mlist-cube-lovers@nntp-server.caltech.edu Path: whuang From: whuang@cco.caltech.edu (Wei-Hwa Huang) Newsgroups: mlist.cube-lovers Subject: Re: 3x3x3 Cubes for Sale Date: 19 Jun 1995 22:36:17 GMT Organization: California Institute of Technology, Pasadena Lines: 25 Message-Id: <3s4u51$fmk@gap.cco.caltech.edu> References: Nntp-Posting-Host: accord.cco.caltech.edu X-Newsreader: NN version 6.5.0 #12 (NOV) "Jerry Bryan" writes: >For the first time in years, I have seen 3x3x3 cubes for sale in a >store. (For American readers, the store is Hills, which is a regional >chain which competes against WalMart and Kmart. The price is $8.97 -- >U.S. dollars.) The box has a note signed by Erno Rubik, and gives >the proper size of the cube group. Also, and I couldn't see inside >the box to verify this, the Face centers seemed to be marked in such >a way as to support the Supergroup. Rubik's note about the size of >the problem says it is 4^4 times bigger than the regular problem. >The manufacturer (or maybe distributor?) is Matchbox Toys (or is it >Matchbook Toys?). What you saw was "Rubik's Cube 4th Dimension." This was released some time in the early 90s. The only difference between this and the typical 3x3x3 is that four faces have pictures on their center squares, requiring the solver to orient them correctly. I belive last year there was a European Rubik Puzzle series that rereleased the cube, along with other rubik puzzles. -- -- Wei-Hwa Huang (whuang@cco.caltech.edu) Homepage (under construction): http://www.ugcs.caltech.edu/~whuang/ I have a proof that NP = P; however, it requires exponential time to write down. From pbeck@pica.army.mil Tue Jun 20 08:00:48 1995 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19315; Tue, 20 Jun 95 08:00:48 EDT Date: Tue, 20 Jun 95 8:00:16 EDT From: Peter Beck (FSAC) To: cube-lovers@ai.mit.edu Subject: cube availability Message-Id: <9506200800.aa03200@COR6.PICA.ARMY.MIL> In Feb at the NY international toy show the KOOSH people told me that they had bought the marketing rights for RUBIK items from Western Publishing (the Matchbox LABEL). They had new packaging and were changing the items to be sold (eg, they said they were going to sell Rubuik's amgic). So the items seen in Hills with Matchbox packaging is old stock that will soon no longer be around. THE FUTURE IS PUZZLING, BUT CUBING IS FOREVER !!! pete beck From mschoene@math.rwth-aachen.de Tue Jun 20 08:40:27 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21537; Tue, 20 Jun 95 08:40:27 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sO2aR-000MP0C; Tue, 20 Jun 95 14:39 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sO2aQ-00026zC; Tue, 20 Jun 95 14:39 WET DST Message-Id: Date: Tue, 20 Jun 95 14:39 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Sun, 18 Jun 1995 15:55:22 -0400 (EDT) Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? Jerry wrote in his message of 1995/06/17 We define the real size of cube space to be the number of M-conjugate classes {m'Ym} for m in M, set of 48 rotations and reflections of the cube, and for Y in G. Dan Hoey has calculated the real size of cube space using the Polya-Burnside theorem. Dan and I (mostly Dan) have also calculated the same result using exhaustive computer search. The computer search is much less elegant than the Polya-Burnside results, but the search does provide additional information, such as the number of positions associated with each symmetry group. The results from the computer search have not yet been posted to the list, but a draft paper is in progress. In the meantime, it occurs to me that perhaps -- but only perhaps -- there is a third way to calculate the real size of cube space. The third way would not require (much) computer searching, but would provide the same level of detail about number of positions per symmetry group as does the full blown search. ... detailed description of the method deleted ... And he continued in his message of 1995/06/18 I should have said "a fourth way", I think. Martin Schoernert performed the same calculation with GAP. Hence, we have three ways in hand: 1) Dan's Polya-Burnside method, 2) Martin's GAP calculations, and 3) brute force computer search. My new proposal would then be a fourth way. Well, I don't know about *four* ways. Dan used the Polya-Burnside theorem. That is, he computed the number of M-conjugacy classes as the average number of fixed points of the elements of M w.r.t. to their action on G. He computed the number of fixed points of an element m using clever arguments about the cycle structure of elements of G that m would fix. I simply observed that the number of fixed points of an element m is the size of the centralizer of in G, and then used GAP to compute those. So I don't think it is correct to call this a way of its own. The method you propose is indeed different from Dan's method that uses Polya-Burnside. I can't figure out how the brute force computer search works. So I can't tell whether it is really different from the other methods (and if indeed it is a method to compute the real size of the cube ;-). Jerry, could you say a little bit more about this computation? It appears to me that Dan and Jim Saxe must have realized all the important pieces for your new method when they wrote their seminal ``Symmetry and Local Maxima (long message)'' message of 1980/12/14. As Jerry points out, they did calculate the important values for 9 of the 33 conjugacy classes of subgroups of M (those whose sizes are a multiple of 12). It is neither clear from their message how they found those 9 classes (in fact they apparently found all 98 subgroups of M), nor how they computed the numbers of elements of G that have a specific subgroup of M as symmetry group. Perhaps Dan can say a little bit more about this? Jerry continued in his message of 1995/06/18 Here is a question for Martin: is there any way with GAP to calculate the number of M-conjugacy classes associated with each symmetry class? It is this additional information about the "real cube space" which *is* available via computer search, and for which I am proposing an alternative which does not involve computer search. Of course there is ;-). Given a subgroup H of M, the centralizer of H in G is the set of all those elements of G that commute with all elements of H. But this is of course simply the set of those elements of G that have either H or a larger group as their symmetry group. So we can compute the numbers of elements of G with symmetry group H by computing the size of the centralizer of H in G, and then subtracting the numbers of those elements that have a symmetry group that is a proper supergroup of H. This is easy if we compute those numbers for all subgroups of M, from the larger subgroups down to the smaller. Of course it is not neccessarry to do this for all 98 subgroups of M, but only for one subgroup for each of the 33 conjugacy classes. Then if we simply divide the number of elements with symmetry group H by the index of H in M, we obtain the number of M-conjugacy classes into which those elements fall. As a GAP program this looks as follows # compute the conjugacy classes of subgroups of M classes := ConjugacyClassesSubgroups( M ); numbers := []; # for all conjugacy classes of subgroups of M for i in [Length(classes),Length(classes)-1..1] do # select a representative for this conjugacy class rep := Representative( classes[i] ); # compute how many elements have at least this symmetry group number := Size( Centralizer( G, rep ) ); # subtract the number of elements that have a larger symmetry group for k in [Length(classes),Length(classes)-1..i+1] do for sub in Elements( classes[k] ) do if IsSubgroup( sub, rep ) then number := number - numbers[k]; fi; od; od; # store the number numbers[i] := number; # print the number of the class Print( i, ":\t" ); # the size of the subgroups in the class Print( Size(rep), "\t" ); # the number of subgroups in the class Print( Size(classes[i]), "\t" ); # the number of elements whose symmetry group lies in the class Print( Size(classes[i]), " * ", number, "\t" ); # and the number of M-conjugacy classes of those elements Print( Size(classes[i]), " * ", number, " / ", Index(M,rep), "\n" ); od; *Do not try this with GAP 3.4.2 (our latest release).* GAP 3.4.2 contains several naive functions for permutation groups, that cause this computation to take a very long time. But with GAP 3.5 (our current development version), this produces in about a minute the following table. CLASS SIZE LENGHT NUMBER REAL NAME 33: 48 1 1 * 4 1 * 4 / 1 (M) 32: 24 1 1 * 0 1 * 0 / 2 (C) 31: 24 1 1 * 0 1 * 0 / 2 (AM) 30: 24 1 1 * 20 1 * 20 / 2 (H) 29: 16 3 3 * 124 3 * 124 / 3 (X1,X2,X3) 28: 12 4 4 * 12 4 * 12 / 4 (T1,T2,T3) 27: 12 1 1 * 48 1 * 48 / 4 26: 8 3 3 * 384 3 * 384 / 6 25: 8 3 3 * 1408 3 * 1408 / 6 24: 8 3 3 * 2944 3 * 2944 / 6 23: 8 3 3 * 1920 3 * 1920 / 6 22: 8 3 3 * 384 3 * 384 / 6 21: 8 3 3 * 896 3 * 896 / 6 20: 8 1 1 * 11892 1 * 11892 / 6 19: 6 4 4 * 416 4 * 416 / 8 18: 6 4 4 * 32 4 * 32 / 8 17: 6 4 4 * 7740 4 * 7740 / 8 16: 4 6 6 * 96232 6 * 96232 / 12 15: 4 6 6 * 96256 6 * 96256 / 12 14: 4 3 3 * 92928 3 * 92928 / 12 13: 4 3 3 * 437504 3 * 437504 / 12 12: 4 3 3 * 574208 3 * 574208 / 12 11: 4 3 3 * 1163520 3 * 1163520 / 12 10: 4 3 3 * 144640 3 * 144640 / 12 9: 4 3 3 * 62208 3 * 62208 / 12 8: 4 1 1 * 280272 1 * 280272 / 12 7: 3 4 4 * 3770864 4 * 3770864 / 16 6: 2 6 6 * 424415168 6 * 424415168 / 24 5: 2 6 6 * 2547748032 6 * 2547748032 / 24 4: 2 3 3 * 15285460992 3 * 15285460992 / 24 3: 2 3 3 * 18342768640 3 * 18342768640 / 24 2: 2 1 1 * 45862360944 1 * 45862360944 / 24 1: 1 1 1 * 43252003109885814336 1 * 43252003109885814336 / 48 As expected the numbers in the fourth column add to the size of G. And the numbers in the fifth column add to 901083404981813616, the real size of the cube (|M\MG/M|). For those classes that I could identify I have added their names. If somebody could describe Dan's taxonomy, I will name the other classes as well. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From mschoene@math.rwth-aachen.de Tue Jun 20 09:29:59 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23574; Tue, 20 Jun 95 09:29:59 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sO2bG-000MPBC; Tue, 20 Jun 95 14:40 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sO2bF-00026zC; Tue, 20 Jun 95 14:40 WET DST Message-Id: Date: Tue, 20 Jun 95 14:40 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Sun, 18 Jun 1995 15:55:22 -0400 (EDT) Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? Looking through the old messages about the real size of the cube group, it appeared to me that no one has shown a proof for the Polya-Burnside theorem. Since it is not difficult to prove, I decided to write one up. In the following I will use TeX notation for formulae, i.e., formulae are included in '$' signs, '{}' are used to group terms, '^' is used for superscripts, and '_' for subscripts. If $g \in G$, then I denote the set of elements that are really equivalent to $g$ by $g^M$. Jerry denotes this set by {m'gm}, but $g^M$ is the more common notation in group theory. The sum $\sum_{g \in h^M}{1/|g^M|}$ is simply 1, since it is the sum over all elements in one M-conjugacy class (h^M) of 1 over the length of that M-conjugacy class. Thus the sum $\sum_{g \in G}{1/|g^M|}$ is the number of M-conjugacy classes. Now we need a standard lemma from group theory, which tells us that the length of a class $g^M$ of an element $g$ under the action of a group $M$ is equal to the size of the group $M$ divided by the size of the subgroup of those elements of $M$ that fix $g$ (more precisely the index of that subgroup in $M$, since the lemma is true, even if $M$ is infinite). So using Jerry's notation this lemma gives $1/|g^M| = |Symm(g)|/|M|$. Applying that to the above formula we see that the number of M-conjugacy classes in $G$ is $\sum_{g \in G} {|Symm(g)|/|M|}$ Or, after a trivial change, $1/|M| \sum_{g \in G} {|Symm(g)|}$. Assume that $(g^m == g)$ is 1 if $g^m$ is equal to $g$ and 0 otherwise. Then we have $|Symm(g)| = \sum_{m \in M}{(g^m == g)}$. Thus the number of M-conjugacy classes is $1/|M| \sum_{g \in G} \sum_{m \in M} {(g^m == g)}$. Now we can simply change the order of the two summations, so we get $1/|M| \sum_{m \in M} \sum_{g \in G} {(g^m == g)}$. But of course $\sum_{g \in G} {(g^m == g)}$ is obviously the number of fixpoints of $m$. So we obtain the Polya-Burnside lemma: ``The number of M-conjugacy classes is the average number of fixpoints of the elements of $M$ w.r.t. their operation on $G$''. However, here the operation is special, so we can simplify even further. $g^m$ here means $m^{-1} g m$, so $(g^m == g)$ means $(m^{-1} g m == g)$, which is equivalent to $(m == g^{-1} m g)$ (multiply the equation first by $m$ and then by $g^{-1}$ from the left), which is $(m == m^g)$. So the number of M-conjugacy classes is $1/|M| \sum_{m \in M} \sum_{g \in G} {(m == m^g)}$. But $\sum_{g \in G} {(m == m^g)}$ is simply the size of the subgroup of those elements in $G$ that fix $m$. This is the centralizer of $m$ in $G$. So the number of M-conjugacy classes is finally $1/|M| \sum_{m \in M} |Centralizer(G,m)|$. This is the formulation that I used to compute the real size of the cube group with GAP. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From BRYAN@wvnvm.wvnet.edu Tue Jun 20 18:04:35 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25438; Tue, 20 Jun 95 18:04:35 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8764; Tue, 20 Jun 95 13:48:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 4745; Tue, 20 Jun 1995 13:48:53 -0400 Message-Id: Date: Tue, 20 Jun 1995 13:48:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/20/95 at 14:39:00 from , Martin.Schoenert@Math.RWTH-Aachen.DE On 06/20/95 at 14:39:00 Martin Schoenert said: >I can't figure out how the brute force computer search works. >So I can't tell whether it is really different from the other methods >(and if indeed it is a method to compute the real size of the cube ;-). >Jerry, could you say a little bit more about this computation? I look at corners and edges separately, and then combine the results. I can't speak for how Dan did his corner search and his edge search, but I can describe mine. Our results do match, which is always nice. Dan did most of the figuring out of "combining the results" for corners and edges. Conceptually, I look at every single corner position X and calculate Symm(X), and I look at every single edge position Y and calculate Symm(Y). In practice, there are some important shortcuts. I have a data base containing "every" corner position and a second data base containing "every" edge position with "every" defined in the following sense. The set of all positions is partitioned into equivalence classes of the form {m'Xmc} for the corners and {m'Ymc} for the edges, for m in M (48 rotations and reflections) and c in C (24 rotations). The data bases contain a representative element from each equivalence class. For all the cases where |{m'Xmc}|=1152 and |{m'Ymc}|=1152, no searching is required. For these cases, we know *a priori* that there are 24 M-conjugacy classes containing 48 elements each, and that for each of the 1152 positions we have Symm(X)=I and Symm(Y)=I. Fortunately, for the vast majority of the cases (over 97% for corners and well over 99% for edges), the so-called B-class size is 1152. Suppose the representative for {m'Xmc} is V and for {m'Ymc} is W. For the remaining cases, the idea is to use Vc and Wc for each fixed c in C as a base or representative element for an M-conjugacy class of the form {m'(Vc)m} and {m'(Wc)m}. The tricky part here is that while Vc and Vd are distinct when c and d in C are not equal, nonetheless Vc and Vd may be M-conjugate. Hence, we use each Vc and Wc which are distinct up to M-conjugacy as a representative element for an M-conjugacy class. Conceptually, we calculate Symm(T) for each T in {m'(Vc)m} for each fixed c (and for Vc distinct up to M-conjugacy) and summarize the results. In practice, it is sufficient to calculate Symm(Vc). Here comes another tricky part (at least it was until I figured it out). Initially, I assumed that Symm(T) was the same for all T in {m'(Vc)m}. However, such is not the case. Rather, if you calculate each Symm(T), you will discover that each one will be a group from a class of conjugate groups, and that each group in the class of conjugate groups will appear an equal number of times. Given that you have picked an arbitrary representative from the M-conjugacy class, you don't know which conjugate group you are going to get when you calculate Symm(m'(Vc)m). I got around this issue originally by calculating a representative group (the 98 subgroups of M then have 33 representative groups, one for each of the 33 symmetry classes). In the work I am doing now, I map directly from each of the 98 subgroups to their respective symmetry class. The exhaustive search then consists of performing the above calculations for each B-class of the form {m'Xmc} and {m'Ymc} and summarizing the results (that is, counting all the M-conjugacy classes). In addition to an overall total, you summarize by symmetry class, which gives you the additional information I have talked about, over and above Dan's Polya-Burnside results. At this point, the summary results give you the real size of the corners only problem and the real size of the edges only problem. While you are at it, you have to count the M-conjugacy classes for even positions and odd positions separately in order to put corners and edges together properly. Finally, you put the corners and edges together in all possible ways. The putting together of the corners and edges is described in the draft I have mentioned, so I will just wait until the draft is ready before posting the rest. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From hoey@aic.nrl.navy.mil Thu Jun 22 03:44:23 1995 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 AA02954; Thu, 22 Jun 95 03:44:23 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA25013; Thu, 22 Jun 95 03:43:59 EDT Date: Thu, 22 Jun 95 03:43:59 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9506220743.AA25013@Sun0.AIC.NRL.Navy.Mil> To: Cube-Lovers@ai.mit.edu, "Martin Schoenert" Subject: Ways to Calculate the Real Size of Cube Space? In-Reply-To: Martin Schoenert says: > I can't figure out how the brute force computer search works. and while Jerry Bryan gives one answer, I have another. For you see, I also ran a brute force computer search for the symmetry classes, too. And while it agrees with Jerry's answers, mine was a significantly different algorithm, which I discuss a little a the end of this message. > It appears to me that Dan and Jim Saxe must have realized all the > important pieces for your new method when they wrote their seminal > ``Symmetry and Local Maxima (long message)'' message of 1980/12/14. > As Jerry points out, they did calculate the important values for > 9 of the 33 conjugacy classes of subgroups of M (those whose sizes > are a multiple of 12). It is neither clear from their message how > they found those 9 classes (in fact they apparently found all 98 > subgroups of M), nor how they computed the numbers of elements of > G that have a specific subgroup of M as symmetry group. > Perhaps Dan can say a little bit more about this? First, to find all the subgoups of M. I represented the elements of M as a list of permutations on faces, found easily enough by finding the closure of the generators. Then I did a depth-first search for subgroups, branching by computing the closure of the current subgroup with each possible element not in that subgroup, and cutting off the search on previously-seen subgroups. It's quick, it's dirty, it works. I found the nine subgroups of order a multiple of 12, as shown in the Hasse subgroup diagram: order 48 . . . M_ /|\\_ / | \ \_ / | \ \_ order 24 . A . H . C \_ \ | / \_ \ | / \_ \|/ \ order 12 . . . E . . . . .T[1..4] (except we called E "AC" back then). The trick then was to find all the E-symmetric positions and all the T1-symmetric positions; the tasks of finding the full symmetry group of such positions and counting the positions for T2, T3, and T4 were straightforward. The best tool we had for figuring out symmetric positions was essentially the one I wrote about in ``The real size of cube space'' on 4 Nov 94. For a subgroup J of M, if a position g is J-symmetric, then g must commute with each operation m in J. Recall: The fundamental principle we use in finding whether g commutes with m can be found by examining the cycles of m. Suppose m permutes a cycle (c1,c2,...,c[k-1],ck), so that c2=m(c1), c3=m(c2), ..., ck=m(c[k-1]), and c1=m(ck). For g to commute with m, we have g(c2)=m(g(c1)), g(c3)=m(g(c2)), ..., g(ck)=m(g(c[k-1])), and g(c1)=m(g(ck)). So (g(c1),g(c2),...,g(ck)) is also a cycle of m. Thus g must map each k-cycle of m to another k-cycle of m, and in the same order. The orientation question was a lot more difficult, so we ran through a bunch of little results. The following is a cleaned-up sample of the sort of arguments, as I remember them. In it FRD is an unoriented corner cubie/cubicle and FRD.D is its down-facing color-tab/facicle. ================================================================ Lemma 1: Suppose X and Y are corners, and m is in C, m(X)=Y. Suppose g(X)=X and g(Y)=Y, and g commutes with m, Then g applies the same twist to corners X and Y. Proof: Let TX,TY be the clockwise 120-degree rotation of corners cubies X and Y, respectively. Then m(TX(.))=TY(m(.)), as can be seen by the fact that we could apply a twist to X in place (TX) before moving it to Y with m, or we can perform the same twist on Y (TY) after moving it. So if g(X)=TX^k(X), then g(Y)=g(m(X))=m(g(X))=m(TX^k(X))=TY^k(m(X))=TY^k(Y). performing the same twist on Y as on X, QED. Lemma 2: If g is E-symmetric, then each corner cubie remains in its home cubicle (not considering orientation). Proof: Supposing otherwise, take a moved cubie (without loss of generality) to be FRD, and suppose (w.l.o.g.) it moves to one of locations FDL, FLT, or BTL. Case 1. If g(FRD)=FDL, consider operation m to be the 120-degree rotation about FRD. m(FRD)=FRD, m(FDL)=FTR. So g(FRD)=g(m(FRD))=m(g(FRD))=m(FDL)=FTR, contradicting g(FRD)=FDL. Case 2. If g(FRD)=FLT, then take m as in case 1; m(FLT)=BRT, so g(FRD)=g(m(FRD))=m(g(FRD))=m(FLT)=BRT, contradicting g(FRD)=FDL. Case 3. If g(FRD)=BTL, then g(FRD.F) is BTL.B, BTL.T, or BTL.L. Case 3a. If g(FRD.F)=BTL.B, then g(FRD.R)=BTL.T, by clockwise adjacency. But m(FRD.F)=FRD.R and m(BTL.B)=BTL.L, and g(FRD.R)=g(m(FRD.F))=m(g(FRD.F))=m(BTL.B)=BTL.L contradicts G(FRD.F)=BTL.B. Cases 3b and 3c work the same way. The contradictions establish that FRD does not move, QED. Lemma 3: If g is E-symmetric, then the twists of the four corner cubies FRD, FLT, BLD, and BRT agree with each other, and and the other four also agree with each other. Proof: For any two of the four corners (e.g. FRD, FLT), there is a 120-degree rotation in E taking one to the other (e.g. the rotation about FTR). Lemma 1 applies immediately to show the twists agree, QED. Lemma 4. If g is E-symmetric, then the corner cubies are all solved, or are rotated alternately in opposite directions. Proof: From Lemma 2, all the cubies are in their home cubicles. If one of the sets from Lemma 3 is twisted, their total twist is the twist of a single cubie (since there are four of them) and so must be counteracted by having the other set twisted in the opposite direction, which is alternate corners twisted oppositely; otherwise all corner are solved, QED. Lemma 5. Let T1 refer to the group that fixes the FRD-BTL axis. Then any T1-symmetric position g must keep the FRD and BTL cubies in their solved position, and rotated by the same amount. Proof: Let m be the 120-degree rotation about FRD, which is in T1. Since FRD and BTL are the only 1-orbits of m, they are kept in place or swapped. From the proof of Lemma 2 (which uses the same m) they cannot be swapped. Otherwise, the two cubies are kept in place, and 180 degree rotation about the FL-BR axis, also in T1 fulfills the requirements of Lemma 1 to show they will both be rotated the same amount, QED. ================================================================ That's about all I feel like remembering and formalizing right now. As you can see, it's long, mechanical, and boring. That's why we never got around to writing it all down. Early last year I wrote a computer program to find *all* the symmetry groups of *all* the positions. The first part did the corners and edges separately, counting the number of positions for each symmetry group and permutation parity. For each of the 8! or 12! permutations, I checked to see if the permutation commuted with some nontrivial operation of M; if not, I just counted the appropriate number of I-symmetric positions. Otherwise I applied orientations and counted up the symmetry groups of each possible orientation. (It could probably have been made to go faster, by cutting off partial permutation or orientation generation as soon as all the non-trivial operations were ruled out.) In the above counts, I also kept track each time of whether the permutation was even or odd. Then after I had the count of even and odd permutations for corners and edges in each symmetry group, I had a program that intersected the symmetry groups, and for each pair of subgroups J,K, and each parity P, I added Corners[J,P] * Edges[K,P] to Whole[J intersect K, P], with some fancy footwork so I only needed to deal with conjugacy classes for J and K. Then for each K, the number of whole K-symmetric positions was Whole[K,Even]+Whole[K,Odd]. The program finished about the time I realized the application of the Polya-Burnside theorem. Then for most of the year, I put off writing it all up. I have Jerry to thank for reminding me to get with it, and for useful comments and discussions on the early drafts. Dan From @secyt.gov.ar,@recom.edu.ar:administ@marben.recom.edu.ar Mon Jun 26 11:47:47 1995 Return-Path: <@secyt.gov.ar,@recom.edu.ar:administ@marben.recom.edu.ar> Received: from secyt.secyt.gov.ar ([200.9.244.2]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05732; Mon, 26 Jun 95 11:47:47 EDT Received: by secyt.gov.ar with UUCP id <12188>; Mon, 26 Jun 1995 12:43:07 -0300 Received: from marben.recom.edu.ar by recom.recom.edu.ar with bsmtp (Smail3.1.28.1 #2) id m0sQFwa-0003qDC; Mon, 26 Jun 95 12:19 ARG Received: by marben.recom.edu.ar (UUPC/extended 1.11q(RAN-0.95)); Mon, 26 Jun 1995 11:57:35 ARG Received: by marben.recom.edu.ar (UUPC/extended 1.11q(REC-1.30)); Mon, 26 Jun 1995 11:57:35 ARG Date: Mon, 26 Jun 1995 08:57:35 -0300 From: "Marcelo Daniel Benveniste " Message-Id: <2feecadf.marben@marben.recom.edu.ar> To: cube-lovers@life.ai.mit.edu X-Mailer: RECMAIL [UUPC/extended 1.11q(REC-1.30)] Subject: Subscription request Please, subscribe-me to this list --- Marcelo Daniel Benveniste administ@marben.recom.edu.ar From BRYAN@wvnvm.wvnet.edu Tue Jun 27 23:22:41 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23088; Tue, 27 Jun 95 23:22:41 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 1105; Tue, 27 Jun 95 23:22:51 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9315; Tue, 27 Jun 1995 23:22:51 -0400 Message-Id: Date: Tue, 27 Jun 1995 23:22:50 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Constructing K-symmetric Cubes This is a followup on several recent messages concerning the question of K-symmetric cubes, where K is one of the 98 subgroups of M. We recall that a permutation is a special kind of function, namely a one-to-one and onto function on a set. A very common technique used with functions is to restrict the domain to a (usually proper) subset of the original domain. In the paradigm of a function as a general rule, the general rule is applied to the subset of the domain to obtain the restriction of the function. In the paradigm of a function as a set of ordered pairs, the restriction is simply a (usually proper) subset of the set of ordered pairs. A restriction of a permutation is usually not a permutation (certainly not a permutation on the original domain), but it is still a function. We will treat a permutation on the cube as a set of restrictions (functions) of the several cubicles. We take as our first example the function UFL->UFL. Unlike cycle notation, it is not assumed that the other cubicles are fixed; rather, the other function values are undefined (e.g., URF->?). Let X be any function (not necessarily a permutation) whose domain is some subset of the cubicles. We define Symm(X) in the standard fashion -- Symm(X) is the set of all m in M such that m'Xm=X. If X is the function UFL->UFL, we have Symm(X)=AT4, not Symm(X)=M as you might expect. AT4 is a subgroup in Dan's taxonomy containing six elements, and which has an axis of symmetry along the UFL-DBR axis. This definition of Symm(X) perhaps requires a minor bit of justification. In a function composition such as FG (left-to- right notation) or G(F(x)) (right-to-left "calculus" notation), it is sometimes taken as a convention that the range of F must match the domain of G. But we can also take the restriction of G to the intersection of the range of F with the domain of G, and we do so. Having done so, Symm(X) is well defined. We wish to build an M-symmetric permutation containing the function UFL->UFL, but Symm(UFL->UFL)=AT4 is not a very auspicious start. Rather, we define the conditions under which a function is K-symmetric is follows. A function X is K-symmetric if the union of the K-conjugates k'Xk is a function. Given this definition, the only M-symmetric function on UFL is in fact UFL->UFL, so we really have made a good start. Furthermore, any M-symmetric permutation that contains UFL->UFL must also contain all the M-conjugates of UFL->UFL, and the union of the M-conjugates is simply the identity permutation on the corners. The fact that Symm(UFL->UFL)=AT4 can be of some benefit in our investigations. In particular, the fact that |AT4|=6 means that there are 8 M-conjugates, so taking all the M-conjugates of UFL->UFL means that all 8 corners are specified. Our next example will be UFL->LUF (a twist of the corner). In this case, we have Symm(X)=ET4. ET4 is the subgroup of M in Dan's taxonomy which contains 3 elements including the identity plus the 1/3 and 2/3 rotations around the UFL-DBR axis. Of more import, UFL->LUF is not M-symmetric. However, it is both C-symmetric (C is the set of 24 rotations) and H-symmetric (H is the set of 12 even rotations and 12 odd reflections). As an aside, we note that it is not A-symmetric, where A is the set of 24 even rotations and reflections. But since it is both C-symmetric and H-symmetric, there is not a unique largest subgroup K for which we can say it is K-symmetric. It is easy to see that UFL->LUF is not M-symmetric. The set of M-conjugates contains both UFL->LUF and UFL->FLU, so the union of the M-conjugates is not a function. We can see the same thing from the fact that |ET4|=3. Since |ET4|=3, there are 16 M-conjugates, but there are only 8 corners to represent the 16 M-conjugates. Since UFL->LUF is C-symmetric, let's see if we can build a C-symmetric permutation. There are 8 C-conjugates (a good start!), and the 8 C-conjugates twist each of the 8 corners by 1/3 in the same direction. Hence, this is a C-symmetric but not M-symmetric permutation. Of course, it is an "illegal" position in the sense that it is not in the same orbit as Start. In the same manner, we can build a K-symmetric permutation for any K. We start with a K-symmetric function on a single cubicle. (A function which is K-symmetric is L-symmetric for any L which is a subgroup of K). We include all K-conjugates. If the cube is completely specified, we stop. Otherwise, we choose another K-symmetric function for any previously unspecified cubicle, add in the new K-conjugates, and so forth, repeating until the entire permutation is specified. Needless to say, this construction process suffers from not preserving orbit. Additional steps must be taken to assure that the constructed position is in the desired orbit (usually, the Start orbit). And some orbits do not have representatives from some subgroups, for example it is well known that there are no C-symmetric but not M-symmetric permutations in the Start orbit. To use Dan's adjectives, this process very quickly can become long, mechanical, and boring. But I now see how to build a K-symmetric permutation for any K. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Wed Jun 28 12:52:18 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20782; Wed, 28 Jun 95 12:52:18 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4454; Wed, 28 Jun 95 10:09:51 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 0089; Wed, 28 Jun 1995 10:09:52 -0400 Message-Id: Date: Wed, 28 Jun 1995 10:09:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/20/95 at 14:39:00 from , Martin.Schoenert@Math.RWTH-Aachen.DE I guess I am going to have to break down and get a copy of GAP. It is truly impressive how much GAP can do so easily. My interpretation of Martin's GAP program is that it implements the general outline of the algorithm I described, except that GAP was able to calculate the number of K-symmetric permutations in a very simple and direct way, whereas I was going to have to puzzle each one out by hand. The heart of Martin's program appears to be the following, and I have a couple of questions. > # compute how many elements have at least this symmetry group > number := Size( Centralizer( G, rep ) ); The first question is: how does the Size function work? As a simpler example than the one above, what if you simply say Size(G)? I am naively assuming that G is specified to GAP in terms of generators only, and that it makes no attempt to actually represent each element of G (too big!). And I have seen snippets of GAP libraries for G posted by Mark Longridge, and they look like generators. I have been in several group theory books lately, and I don't recall seeing a general algorithm presented for determining the size of a finite group based on its generators. The second question is like unto the first: how does the Centralizer function work? In this particular case, we don't really need the Centralizer, we only need the Size of the Centralizer, but the question remains in either case. Surely, GAP does not literally try each element of G and each element of rep to see which elements commute (too big again). So what is the general algorithm? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From rodrigo@lsi.usp.br Wed Jun 28 17:15:45 1995 Return-Path: Received: from ofelia.lsi.usp.br (lsi.poli.usp.br) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06724; Wed, 28 Jun 95 17:15:45 EDT Received: from mozart.lsi.usp.br (mozart.lsi.usp.br [143.107.3.237]) by ofelia.lsi.usp.br (8.6.12/8.6.9) with ESMTP id SAA21488; Wed, 28 Jun 1995 18:13:40 -0300 Received: (from rodrigo@localhost) by mozart.lsi.usp.br (8.6.12/8.6.9) id VAA12159; Wed, 28 Jun 1995 21:12:54 GMT Date: Wed, 28 Jun 1995 14:12:53 +48000 From: Rodrigo de Almeida Siqueira To: steinark@ifi.uio.no, bagleyd@source.asset.com, Reinaldo Augusto da Costa Bianchi Cc: cube-lovers@life.ai.mit.edu Subject: Rubik Cube... for Windows ? In-Reply-To: <199506281753.OAA19377@jaguar.lsi.usp.br> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Wed, 28 Jun 1995, steinark@ifi.uio.no wrote: > id: Email: steinark@ifi.uio.no > comments: > Just thought I would mention a Windows version Rubic Cube > program. It's located at several ftp sites and the file > to look for is called 'cubic.zip'. Has a far better > look-and-feel than XRubic I think. It has its own solving > algorithm, but this can be rather slow... > Check it out. If you don't find it I can probably send it > to you some how. > > Steinar Hello Steinar, Would you tell me where (ftp site) can I find cubic.zip ? I would like to try it and put a link in the "Robot can play with the Cube Web homepage" (http://www.lsi.usp.br/~daia/celula/cubo/) Thank you, Rodrigo Siqueira rodrigo@lsi.usp.br (http://www.lsi.usp.br/usp/rod/rod.html) From @mail.uunet.ca:mark.longridge@canrem.com Mon Jul 3 14:15:37 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06664; Mon, 3 Jul 95 14:15:37 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <181844-6>; Mon, 3 Jul 1995 14:17:04 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA20578; Mon, 3 Jul 95 14:11:10 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E9C05; Mon, 3 Jul 95 14:04:58 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Antislice Patterns From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1181.5834.0C1E9C05@canrem.com> Date: Mon, 3 Jul 1995 14:53:00 -0400 Organization: CRS Online (Toronto, Ontario) Patterns in the Anti-Slice Group -------------------------------- p4 8 flip (Op sides) (R1 L1 U1 D1 F1 B1) ^2 (12) p10a pons asinorum (L3 R1 U3 D1)^3 (12) p16a 4 cross order 2 F1 B1 U1 D1 L2 R2 U1 D1 F1 B1 U2 D2 (12) p17 4 diagonal (F1 B1 R1 L1) ^3 (12) p18a 4 diagonal,2 cross (F1 B1 R3 L3) ^3 (12) p22 2 DOT, 2 Stripe R1 L1 U2 D2 R3 L3 (6) p64a 4 Z F1 B1 L3 R3 F1 B1 L1 R1 F3 B3 L1 R1 (12) p143 Pinwheels F1 B1 L1 R1 F3 B3 U3 D3 L1 R1 U1 D1 (12) p175a 6 H order 2 U3 D3 L3 R3 F2 B2 U2 D2 L3 R3 U1 D1 (12) p198a 2 X, 4 Diag no C L1 R1 F1 B1 L3 R3 F3 B3 L1 R1 F1 B1 (12) p201 Pinwheels + Pons L1 R1 F3 B3 L1 R1 U3 D3 F1 B1 U3 D3 (12) p201 is a quite interesting position. The square's group equivalent is no shorter in q turns: p175 6 H order 2 type 2 U2 B2 L2 U2 D2 L2 F2 U2 (8) Note that p201 = |{m'Xm}|=2 and |Symm(X)|=24. From @mail.uunet.ca:mark.longridge@canrem.com Mon Jul 3 14:15:29 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06658; Mon, 3 Jul 95 14:15:29 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <181839-4>; Mon, 3 Jul 1995 14:16:59 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA20583; Mon, 3 Jul 95 14:11:12 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E9C06; Mon, 3 Jul 95 14:04:58 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Crazy Corner Pattern Revisited From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1182.5834.0C1E9C06@canrem.com> Date: Mon, 3 Jul 1995 14:59:00 -0400 Organization: CRS Online (Toronto, Ontario) Jerry writes: > In their position, the UFL and RBD corners are >in place, and the other three pairs are swapped. The "girdle" >includes the three pairs that are swapped. Hence, there >is an axis of symmetry along the UFL-RBD axis. The odd number of >swaps is compensated by an odd number in the edges. The compensation >is not required for corners only. > > R D D > U U D > U U B > > D L L F F D L L F L F F > R L L F F B L R R B B F > R R B R B B U R R B B U > > U U L > U D D > F D D Much is explained! (And this is all the way back in file "cube01" in the archives, Date: 14 December 1980 1916-EST). What does the 1916- mean in front of EST??? If we put the edges in place in Jerry's (and Dan's) position then we have a position M-conjugate to the one I posted. (re-posted below). F U U U U U D U L R L B R F B D R R B B D L L L F F F R R R B B B U L L F F U L R F L B F D D B D D D R D U Jerry Continues: >...there is an axis of symmetry along the UFL-RBD axis. The odd >number of swaps is compensated by an odd number in the edges. >The compensation is not required for corners only. Great! We both concur that this pattern is in the swap orbit. There is another position with M-class 4 in the twist orbit, but that one is *utterly* impossible unless one corner twists on it's own (perhaps due to desperate cube turning?) There are 2 other types of position with M-class 4, and that is pinwheels and pinwheels + pons asinorum. -> Mark <- (I must be cubing too much, 1916-EST has to be 7:16 pm Eastern Standard Time). From kpapado@athena.auth.gr Wed Jul 5 04:55:00 1995 Return-Path: Received: from bsa2.athena.auth.gr ([155.207.7.3]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09624; Wed, 5 Jul 95 04:55:00 EDT Received: from [155.207.48.131] by bsa2.athena.auth.gr with SMTP id AA22205; Wed, 5 Jul 1995 11:53:30 +0300 Received: by lch1.athena.auth.gr (1.38.193.4/4.7) id AA01853; Wed, 5 Jul 1995 10:52:13 +0200 Message-Id: <9507050852.AA01853@lch1.athena.auth.gr> To: cube-lovers@life.ai.mit.edu Date: Wed, 05 Jul 95 10:52:13 EET From: Kostas Papadopoulos subscribe listname Kostas Papadopoulos From hoey@aic.nrl.navy.mil Sun Jul 9 19:53:12 1995 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil ([192.26.18.51]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16871; Sun, 9 Jul 95 19:53:12 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA03807; Sun, 9 Jul 95 19:53:10 EDT Date: Sun, 9 Jul 95 19:53:10 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9507092353.AA03807@Sun0.AIC.NRL.Navy.Mil> To: "Jerry Bryan" , "Cube Lovers List" Subject: Re: 3x3x3 Cubes for Sale "Jerry Bryan" writes: > ... I couldn't see inside the box to verify this, the Face centers > seemed to be marked in such a way as to support the Supergroup. Just as well, you'd have been disappointed. As I wrote on 8 Jan 92, : While most people are content to make each face a solid color, some : cubes have markings that display whether the face centers are twisted : with respect to the rest of the cube. : [This has recently been done commercially in an spectacularly : braindamaged way, in a product known as ``Rubik's cube--the : fourth dimension'' or some such nonsense. The mfrs have marked : only four face centers, breaking symmetry while they fail to show : the surprising invariant of the Supergroup. What bagbiters!] > Rubik's note about the size of > the problem says it is 4^4 times bigger than the regular problem. And it could have been 4^(11/2). Dan Hoey@AIC.NRL.Navy.Mil From phaedrus@future.dreamscape.com Sun Jul 9 23:39:02 1995 Return-Path: Received: from zaphod.caz.ny.us (future.dreamscape.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25147; Sun, 9 Jul 95 23:39:02 EDT Received: (from phaedrus@localhost) by zaphod.caz.ny.us (8.6.11/8.6.9) id XAA01047; Sun, 9 Jul 1995 23:38:26 -0400 From: phaedrus@future.dreamscape.com Message-Id: <199507100338.XAA01047@zaphod.caz.ny.us> Subject: Re: 3x3x3 Cubes for Sale To: hoey@aic.nrl.navy.mil (Dan Hoey) Date: Sun, 9 Jul 1995 23:38:25 -0400 (EDT) Cc: BRYAN@wvnvm.wvnet.edu, Cube-Lovers@ai.mit.edu In-Reply-To: <9507092353.AA03807@Sun0.AIC.NRL.Navy.Mil> from "Dan Hoey" at Jul 9, 95 07:53:10 pm X-Mailer: ELM [version 2.4 PL24] Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 1450 > > "Jerry Bryan" writes: > > ... I couldn't see inside the box to verify this, the Face centers > > seemed to be marked in such a way as to support the Supergroup. > > Just as well, you'd have been disappointed. As I wrote on 8 Jan 92, > > : While most people are content to make each face a solid color, some > : cubes have markings that display whether the face centers are twisted > : with respect to the rest of the cube. > : [This has recently been done commercially in an spectacularly > : braindamaged way, in a product known as ``Rubik's cube--the > : fourth dimension'' or some such nonsense. The mfrs have marked > : only four face centers, breaking symmetry while they fail to show > : the surprising invariant of the Supergroup. What bagbiters!] > > > Rubik's note about the size of > > the problem says it is 4^4 times bigger than the regular problem. > > And it could have been 4^(11/2). And worse, it's not even 4^4, since there are two ways to align the face centers so that the puzzle looks solved. The two unmarked faces are opposite each other, and the four marked faces are marked with symbols (Rubik's signature, his silouette, "C*4^4", and the "Rubik's Cube" trademark) that have a definite "up", but don't tie in to the rest of the cube at all. If all four are upside down, to "solve" the cube simply requires turning the cube over. > > Dan > Hoey@AIC.NRL.Navy.Mil > From BRYAN@wvnvm.wvnet.edu Mon Jul 10 10:46:59 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14144; Mon, 10 Jul 95 10:46:59 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4200; Mon, 10 Jul 95 10:46:57 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9475; Mon, 10 Jul 1995 10:46:57 -0400 Message-Id: Date: Mon, 10 Jul 1995 10:46:56 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Run Times, Storage Requirements, etc. In-Reply-To: Message of 04/21/95 at 11:37:18 from mreid@ptc.com On 04/21/95 at 11:37:18 mreid@ptc.com said: >jerry writes >> What would be really nice (and which may not be possible) is some >> representation for the cube such that a cube Z and its neighbors >> Zq or Zh are stored very close together. >remember that the diameter of the group is small. (my guess is >21 face turns, 24 quarter turns.) so this isn't possible without >resorting to a liberal definition of "very close". This is something I have been thinking about for a long time. The idea is that if large searches were run on either massively parallel machines, or even on farms of workstations, it would be really nice if most neighbors stayed on the same machine. Some of my searches get so large that I have to decompose them in a manner somewhat similar to the manner in which I envision decomposing searches for parallel processing. Let me use an example a project I am working on as we speak. I am trying to do a complete search for edges only (with centers). The data base for level 10 of the search is on four tapes (not so big, really). The data base is sorted. The neighbors will be sorted according to the same collating sequence. In order to create level 11, all neighbors have to be generated and sorted (deleting duplicates), and then matched against the level 9 data base (again deleting duplicates). I desire to partition the sorted neighbors using as boundary points for the partition the first record on each of the four tapes for level 10. My experience is that boundary points that partition one level of the data base equally also partitions deeper levels of the data base equally. Having partitioned level 11, the sizes of the output files are rather striking: Neighbors in Neighbors in Neighbors in Neighbors in Tape 1 Tape 2 Tape 3 Tape 4 Partition Partition Partition Partition Lvl 10 Tape 1 5.2 tapes 2.0 tapes 1.2 tapes 0.2 tapes Lvl 10 Tape 2 1.3 tapes 5.0 tapes 1.9 tapes 1.2 tapes Lvl 10 Tape 3 1.2 tapes 1.4 tapes 5.1 tapes 2.2 tapes Lvl 10 Tape 4 0.2 tapes 1.1 tapes 1.6 tapes 6.0 tapes In real round numbers, the level 11 data base is going to have about 32 tapes, with about 8 tapes in each partition (the branching factor is about 8 at this level of the search). But as the chart above shows, there is a strong tendency for neighbors to stay in the same partition. (The chart does not reflect it, but to complete the processing for level 11, the "Tape 1 partition" will all have to be merged together, as will the "Tape 2 partition", etc.) I would emphasize that these "partitions" I am talking about are totally arbitrary subdivisions of the data into smaller chunks to make the problem manageable. But imagine if you will that instead of four tapes, I had four machines, each with a sizable hard disk. Each machine would be assigned one of the four partitions. As it generated neighbors, each machine would either store the neighbor locally or would send the neighbor on to one of the other three machines as required. Obviously, the more of the neighbors you can keep locally the better. With various problems I have worked on, I have had various numbers of partitions of the data -- 10, 16, 32, 128, etc. The effect I am describing is always there, and I am not totally sure why. I have some theories, but nothing definitive. I think the effect is stronger because I am using representative elements of conjugacy classes or other equivalence classes than if I were storing all individual cubes. Also, the effect I am describing can be characterized by the almost silly statement that "one twist of the cube doesn't change the cube very much". But it's true. For example, the dominant part of the sort is the Front face, which isn't changed by twists of the Back. Also, twists of the Front don't change the Front very much when you consider that representative elements are being stored. It is only twists of the Up, Down, Right, and Left faces which change the Front very much. Finally, the sort order for the Front face is the Upper part, the Right part, the Down part, and the Left part, so that even a twist of the Left face doesn't change the Front face very much with respect to its sort order. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Mon Jul 10 19:25:51 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16842; Mon, 10 Jul 95 19:25:51 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 7039; Mon, 10 Jul 95 16:20:30 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9562; Mon, 10 Jul 1995 16:20:30 -0400 Message-Id: Date: Mon, 10 Jul 1995 16:20:29 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Partial Results, Edges only (with Face Centers), Qturns Level M-Conjugacy Branching Lcl Positions Branching Lcl Classes Factor Max Factor Max 0 1 0 1 0 1 1 1.000 0 12 12.000 0 2 5 5.000 0 114 9.500 0 3 25 5.000 0 1068 9.368 0 4 215 8.600 0 9819 9.194 0 5 1886 8.772 0 89392 9.104 0 6 16902 8.962 0 807000 9.028 0 7 150442 8.900 0 7209384 8.934 0 8 1326326 8.816 1 63624107 8.825 2 9 11505339 8.675 552158812 8.678 10 96755918 8.410 4643963023 8.411 The local maximum unique up to M-conjugacy is the 6-H position (see Symmetry and Local Maxima). I do not yet have a process for the 6-H, but I should be able to have one soon. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From mschoene@math.rwth-aachen.de Wed Jul 12 06:50:38 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16694; Wed, 12 Jul 95 06:50:38 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sVzLJ-000MPGC; Wed, 12 Jul 95 12:48 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sVzLJ-00025fC; Wed, 12 Jul 95 12:48 WET DST Message-Id: Date: Wed, 12 Jul 95 12:48 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Wed, 28 Jun 1995 10:09:51 -0400 (EDT) Subject: How to compute the size of a permutation group Jerry Bryan wrote in his message of 1995/06/28 I guess I am going to have to break down and get a copy of GAP. It is truly impressive how much GAP can do so easily. Hah, great, another user for GAP. Now maybe I can get a raise from my boss ;-) Jerry continued My interpretation of Martin's GAP program is that it implements the general outline of the algorithm I described, except that GAP was able to calculate the number of K-symmetric permutations in a very simple and direct way, whereas I was going to have to puzzle each one out by hand. Let me rephrase that a little bit. My GAP prgram implements the general outline of the algorithm you described, except that I am able to tell GAP to calculate the number of K-symmetric permutations in a very simple and direct way (but GAP is going to puzzle each one out using an non-simple and non-direct algorithm I shall outlined in another message). Jerry continued The heart of Martin's program appears to be the following, and I have a couple of questions. > # compute how many elements have at least this symmetry group > number := Size( Centralizer( G, rep ) ); The first question is: how does the Size function work? As a simpler example than the one above, what if you simply say Size(G)? I am naively assuming that G is specified to GAP in terms of generators only, and that it makes no attempt to actually represent each element of G (too big!). And I have seen snippets of GAP libraries for G posted by Mark Longridge, and they look like generators. I have been in several group theory books lately, and I don't recall seeing a general algorithm presented for determining the size of a finite group based on its generators. For this particular case (asking for the size of a centralizer in a permutation group), the answer is trivial. The algorithm that computes (generators for) the centralizer computes the size of the centralizer as a byproduct. I will try to outline this algorithm in another message. But when you say 'Size(G)', then GAP will indeed use an algorithm that computes the size of a permutation group given by a set of generators. The algorithm is called the ``Schreier--Sims'' algorithm. It was developed by C.Sims for his investigations of sporadic simple groups, and is mostly based on a lemma by N.Schreier. Unfortunately it is *not* described in any of the generally available textbooks on group theory. The funny thing is, that this algorithm works very much like the algorithm puzzlers use to find a method to solve the cube. Of course it is lacking the cleverness many puzzlers use to prove that their method is complete (i.e., solves all possible states), but then the algorithm (resp. the computer running the algorithm) is a lot more patient. The first step (for puzzlers and the algorithm) is to find a method to bring the first facelet to its home place. The algorithm does this by finding all (24) places to which this facelet can be moved. This is done by applying the (6) generators to all the places found so far, and repeating until no new places are found. For each place it also remembers a process that moves the choosen facelet from its home place to this place, called a representative for the place. The second step (for puzzlers and the algorithm) is to find enough processes that leave the first facelet in its home place. If one has enough such processes, one can simply start another round of the algorithm. I.e., one uses those processes to bring the second facelet to its home place, finds enough processes that leave the first and the second facelet in their respective home places, and so on. In group theoretic terms the set of elements that leave the first facelet is in its home place is a subgroup (called the stabilizer of that point), and the problem is to find a set of generators for this subgroup. So how does the algorithm find such a set of generators? This is where Schreier's lemma kicks in. Let us first take a representative of a place (so moves the first facelet to the place ). Then we multiply this by a generator of the original group. Say moves the first facelet from the place to the place . Finally we multiply the product * by the inverse of the representative of the place . This obviously moves the first facelet back to its home place, so the product * * ^-1 is a process that leaves the first facelet in its home place, i.e., it is an element of the subgroup. Now Schreier's lemma says that if we take all 24 times 6 such elements, then the resulting set is a set of generators for the subgroup. In fact Schreier's lemma says that if you select the representatives carefully, then 24 * (6 - 1) + 1 generators will suffice, and for certain subgroups (subgroups of free groups) no smaller set of generators will do. So we now start another round of the algorithm with this (smaller) subgroup. How much smaller is this subgroup? The size of the whole group is the size of the subgroup times 24 (we get each element of the whole group exactely once by multiplying each subgroup element by each of the 24 representatives). So the subgroup is smaller by a factor of 24. After at most 48 rounds (for the 48 facelets) we are done. We can now easily compute the size of the entire group. It is the size of the subgroup of those elements that leave the first facelet in its home place times 24 (the number of possible places for the first facelet). The size of this subgroup is the size of the subgroup that leaves the first and the second facelet in their respective home places times the number of possible places for the second facelet. And so on. We also have a method to solve any state as follows. We first find the place of the first facelet and multiply by the inverse of the representative of , to move that facelet back to its home place. Then we find the place of the second facelet and multiply by the inverse of the representative found in the second round of the algorithm, to move that facelet back to its home place. Since that representative was found in the second round it doesn't move the first facelet, so now the first and the second facelet are in their respective home places. After at most 48 rounds, we have moved all facelets to their home places. As has been pointed out before, such a method to solve any state gives us a membership test for . Suppose we are given an arbitrary permutation of the facelets. Then we simply try to solve . If we can solve it, then is an element of . If we cannot solve it, then is not an element of . Now the algorithm as described above has a fatal flaw. The number of generators increases with each round. In the first round we have 6 generators, in the second round we have about 24 * 6 generators, in the third round we have about 24^2 * 6 generators, and so on. Most of these generators will be redundant, i.e., they will lie in the subgroup generated by the other generators. The solution is as follows. Instead of taking all 24 * 6 generators for the second round, we randomly select some (say 6) of them. Those will generate a subgroup ~~of the whole stabilizer (and our hope is that this subgroup will be equal to the whole stabilizer). Then we apply the algorithm to~~~~, and compute a method to solve any element of~~~~. As described above, this gives us a way to test membership in~~~~. We now compute all 24 * 6 generators for the stabilizer, and for each one test whether it is an element of~~~~. If they are all elements of~~~~, then~~~~is indeed the whole stabilizer, and we are done. If not, then we have found a new non-redundant generator for the stabilizer, and we start anew, this time with 7 generators instead of the originally choosen 6. The first algorithm is sometimes called the iterative Schreier--Sims, because it iterates over the stabilizers. The second algorithm is called the recursive Schreier--Sims, because it assumes that we have solved the problem for~~~~recursively. This is the algorithm currently implemented in GAP. There is a third variant called Schreier--Todd--Coxeter--Sims, that uses a Todd--Coxeter algorithm to prove that~~~~is the whole stabilizer without computing all Schreier generators, which is important when there are many (say 1000000) possible places for the first facelet. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From BRYAN@wvnvm.wvnet.edu Wed Jul 12 10:31:12 1995 Return-Path:~~Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26378; Wed, 12 Jul 95 10:31:12 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0082; Wed, 12 Jul 95 10:31:11 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 5371; Wed, 12 Jul 1995 10:31:11 -0400 Message-Id: Date: Wed, 12 Jul 1995 10:31:10 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Partial Results, Edges only (with Face Centers), Qturns In-Reply-To: Message of 07/10/95 at 16:20:29 from BRYAN@wvnvm.wvnet.edu On 07/10/95 at 16:20:29 Jerry Bryan said: >The local maximum unique up to M-conjugacy is the 6-H position >(see Symmetry and Local Maxima). I do not yet have a process >for the 6-H, but I should be able to have one soon. I can now give one way to do it as (UD)(RR)(LL)(UD). It seems too simple once you chase it down. Note that this is *not* the 6-H position when you apply the process to the whole cube; you have to omit the corners for this process to yield the 6-H. Nonetheless, the pattern is a nice one when the process is applied to the whole cube, one that looks familiar, although I cannot place it. It is *almost* what I described as the "interesting" part of three of the 10q local maxima on the whole cube, but the "interesting" part of the 10q local maxima is (U'D')(RR)(LL)(UD) instead. It might be noted that the length of this position is also 8q on an edges-only-without-centers cube (see my note of 8 Dec 1993 22:41:38). I did not actually provide a process for the without-centers case, but the same process works for the 6-H edges-only with or without centers for this position. Such is not always true. I have talked about it before, but many minimal processes for without-center cubes induce an invisible rotation which becomes visible when the Face centers are included. This is probably as good a time as any to correct an old error, pointed out to me by Dan Hoey. The length of a position without centers is the minimum taken over C of the length of the same position with centers -- that is, the minimum of the respective lengths of the same position rotated 24 different ways. For searches without centers I store representatives of the form Y=Repr{m'Xmc}. At one point, I said |Y|=min{|Yc|}. This is certainly not true. Y is just one of the {Yc}, and it is totally arbitrary which one it is. The difficulty is really a notational one. It is the length of Y without centers which is min{|Yc|}, not the length of Y itself (with centers). But I don't have a good way to say "Y without centers" or especially to say "length of Y without centers". But in any case, the most interesting cases to me are the ones where the length without centers matches the length with centers, so that the minimal process for the without centers case does not induce an invisible rotation. The position at hand is such a case. Finally, the position is in the anti-slice group (i.e., (UD)(RL)(RL)(UD)), so the position is a local maximum in the anti-slice edges only group with a length 4a. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From BRYAN@wvnvm.wvnet.edu Wed Jul 12 18:04:52 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23708; Wed, 12 Jul 95 18:04:52 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4002; Wed, 12 Jul 95 18:04:54 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 0388; Wed, 12 Jul 1995 18:04:54 -0400 Message-Id: Date: Wed, 12 Jul 1995 18:04:53 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Partial Results, Edges only (with Face Centers), Qturns In-Reply-To: Message of 07/12/95 at 10:31:10 from BRYAN@wvnvm.wvnet.edu On 07/12/95 at 10:31:10 Jerry Bryan said: >It might be noted that the length of this position >is also 8q on an edges-only-without-centers cube (see my note >of 8 Dec 1993 22:41:38). Uh, the 8 Dec 1993 22:41:38 note concerns the Superflip composed with the 6-H. The plain old 6-H was 8 Dec 1993 23:16:50. The length of the Superflip composed with the 6-H in edges-without- centers was 13. The position is even, so the length must be at least 14 in edges-with-centers. My search of edges-with-centers has not gotten that far yet. Also, the length 13 Superflip 6-H process for edges-without-centers induces an invisible rotation, so it is not as pretty as the length 8 process for 6-H which does not induce an invisible rotation. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From hoey@aic.nrl.navy.mil Wed Jul 12 18:02:55 1995 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 AA23623; Wed, 12 Jul 95 18:02:55 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA00318; Wed, 12 Jul 95 18:02:51 EDT Date: Wed, 12 Jul 95 18:02:51 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9507122202.AA00318@Sun0.AIC.NRL.Navy.Mil> To: Cube-Lovers@ai.mit.edu Subject: Happy birthday References: <12Jul81_134343_DH51@CMU-10A> (Sorry to have neglected the last thirteen). \ \ \ \ /\ /\ /\ /\ / / \ / / \ / / \ / / \ \/ /\ \/ /\ \/ /\ \/ /\ \ / / \ \ / / \ \ / / \ \ / / \ {} \/ {}/\ \/ {}/\ \/ {}/\ \/ /\ {} \ {/ / \ \ {/ / \ \ {/ / \ \ / / {} {} {\/ {}/\{\/ {}/\{\/ {}/\ \/ {} {} {}\ {/ /{}\ {/ /{}\ {/ / \ +_------{}+_--{}--{}{}--{\/-{}{}--{\/-{}{} {\/ {} | `-_ {} `-{} {}{}`-{}\ {}{}`-{}\ {}{}`-{}\ {} |S `+_------{}+_--{}--{}{}--{}--{}{}--{}--{}{} {} | | `-_ {} `-{} {}{}`-{} {}{}`-{} {}{}`-{} | S |A `+_------{}+_--{}--{}+_--{}--{}+_--{}--{}+_ | | A|M`-_ {} `-{} {} `-{} {} `-{} {} `-_ +_ S| | M `+_------{}+_------{}+_------{}+_-------`+_ |L`-_ |A |M M|*`-_ {} `-_ {} `-_ {} `-_ `-_ | L `+_ A| M |* * `+--------`+--------`+--------`+--------`+ |L L|O`-_ |M M|* * *| | | | | | |O O `+_ M |* *| | | | | |L L|O O O| `-_M|* * *| Happy | Birth | day | to | +_ L |O O O| `+_ * *| | | | | | `-_L|O O O| Y | `-_*| | | | | |H `+_ O O| |D `+---------+---------+---------+---------+ | H | `-_O| Y | D| | | | | |H H| `+_ | D | | | | | | H | |R`-_ |D | Cube | Lovers | @ | MIT | +_ H| E |R `+_ D| | | | | |W`-_ | |R R R|E`-_ | | | | | |W `+_ |R R|E `+---------+---------+---------+---------+ |W W W|A`-_ |R R R| E E| | | | | |W W W|A `+_ R|E E E| | | | | |W W W| A A| `-_R|E E | 15 | Years | 1659 | Messages| +_ W|A A|S `+_ E| | | | | `-_W|A A | S | `-_E| | | | | `+_ A|S S S| `+---------+---------+---------+---------+ `-_A| S | | | | | | `+_ S| | | | | | `-_ | | Many | Happy |Restores | | `+_ | | | | | `-_ | | | | | `+---------+---------+---------+---------+ --Dan From vorms@iprolink.ch Sun Jul 16 07:00:55 1995 Return-Path: Received: from badboy.iprolink.ch by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05788; Sun, 16 Jul 95 07:00:55 EDT Received: from port26.iprolink.ch (port26.iprolink.ch [194.41.63.26]) by badboy.iprolink.ch (8.6.12/8.6.12) with SMTP id MAA27838 for ; Sun, 16 Jul 1995 12:56:34 +0200 Date: Sun, 16 Jul 1995 12:56:34 +0200 Message-Id: <199507161056.MAA27838@badboy.iprolink.ch> X-Sender: vorms@mail.iprolink.ch (Unverified) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Cube-Lovers@life.ai.mit.edu From: vorms@iprolink.ch (Beffa Raphael) X-Mailer: Hello my name is Raphael Beffa I am a mew user on the web. I am really happy to realise that peaple interested on the cube have a space to talk about on the net. I spent manny hours to develop different prototypes: An 5^3 cube in 1981 a symetrical one 2^3 cube in 1981 and one more sophisticated 5^3 cube 1983. I realised a simple simulator on my PC for a 3^3 cube . One representation is a classical isometrical perspective , I realised an other representation I called a "Planicube" view (6 faces on the same view). Is somebody intersted on cube rotating on the summit (4 axes) insted of rotating on the face (3 axes the original one). Or even rotating on the face and on the summit (7 axes). Maybe those prototypes allready exist. Please let me now. Is anybody able to give me the solution of the Masterball. I am happy to meet new frends, Reguards Raphael From vorms@iprolink.ch Sun Jul 16 16:16:22 1995 Return-Path: Received: from badboy.iprolink.ch by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25739; Sun, 16 Jul 95 16:16:22 EDT Received: from port33.iprolink.ch (port33.iprolink.ch [194.41.63.33]) by badboy.iprolink.ch (8.6.12/8.6.12) with SMTP id WAA68668 for ; Sun, 16 Jul 1995 22:12:02 +0200 Date: Sun, 16 Jul 1995 22:12:02 +0200 Message-Id: <199507162012.WAA68668@badboy.iprolink.ch> X-Sender: vorms@mail.iprolink.ch (Unverified) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Cube-Lovers@life.ai.mit.edu From: vorms@iprolink.ch (Beffa Raphael) X-Mailer: Hello my name is Raphael Beffa I am a mew user on the web. I am really happy to realise that peaple interested on the cube have a space to talk about on the net. I spent manny hours to develop different prototypes: An 5^3 cube in 1981 a symetrical one 2^3 cube in 1981 and one more sophisticated 5^3 cube 1983. I realised a simple simulator on my PC for a 3^3 cube . One representation is a classical isometrical perspective , I realised an other representation I called a "Planicube" view (6 faces on the same view). Is somebody intersted on cube rotating on the summit (4 axes) insted of rotating on the face (3 axes the original one). Or even rotating on the face and on the summit (7 axes). Maybe those prototypes allready exist. Please let me now. Is anybody able to give me the solution of the Masterball. I am happy to meet new frends, Reguards Raphael From vorms@iprolink.ch Sun Jul 16 16:16:19 1995 Return-Path: Received: from badboy.iprolink.ch by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25737; Sun, 16 Jul 95 16:16:19 EDT Received: from port33.iprolink.ch (port33.iprolink.ch [194.41.63.33]) by badboy.iprolink.ch (8.6.12/8.6.12) with SMTP id WAA67592 for ; Sun, 16 Jul 1995 22:11:54 +0200 Date: Sun, 16 Jul 1995 22:11:54 +0200 Message-Id: <199507162011.WAA67592@badboy.iprolink.ch> X-Sender: vorms@mail.iprolink.ch (Unverified) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Cube-Lovers@ai.mit.edu From: vorms@iprolink.ch (Beffa Raphael) Subject: Re: X-Mailer: >Hello my name is Raphael Beffa > > I am a mew user on the web. I am really happy to realise that peaple >interested on the cube have a space to talk about on the net. > > I spent manny hours to develop different prototypes: >An 5^3 cube in 1981 a symetrical one 2^3 cube in 1981 and one more >sophisticated 5^3 cube 1983. > I realised a simple simulator on my PC for a 3^3 cube . One >representation is a classical perspective , I realised an other representation I called a "Planicube" view (6 faces on the same view). > > Is somebody intersted on cube rotating on the summit (4 axes) insted of >rotating on the face (3 axes the original one). > Or even rotating on the face and on the summit (7 axes). > Maybe those prototypes allready exist. Please let me now. > > Is anybody able to give me the solution of the Masterball. > > I am happy to meet new frends, Reguards Raphael From BRYAN@wvnvm.wvnet.edu Wed Jul 19 21:47:46 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04627; Wed, 19 Jul 95 21:47:46 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8401; Wed, 19 Jul 95 21:47:48 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 8646; Wed, 19 Jul 1995 21:47:48 -0400 Message-Id: Date: Wed, 19 Jul 1995 21:47:47 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Level 11, Edges-with-Face-Centers, Qturns Level M-Conjugacy Branching Lcl Positions Branching Lcl Classes Factor Max Factor Max 0 1 0 1 0 1 1 1.000 0 12 12.000 0 2 5 5.000 0 114 9.500 0 3 25 5.000 0 1068 9.368 0 4 215 8.600 0 9819 9.194 0 5 1886 8.772 0 89392 9.104 0 6 16902 8.962 0 807000 9.028 0 7 150442 8.900 0 7209384 8.934 0 8 1326326 8.816 1 63624107 8.825 2 9 11505339 8.675 0 552158812 8.678 0 10 96755918 8.410 4643963023 8.411 11 750089528 7.752 36003343336 7.753 Note that there are no local maxima at level 9, although one showed up at level 8. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 From FOMEQUE@aol.com Sat Jul 22 18:25:28 1995 Return-Path: Received: from emout04.mail.aol.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19550; Sat, 22 Jul 95 18:25:28 EDT Received: by emout04.mail.aol.com (1.37.109.11/16.2) id AA060581718; Sat, 22 Jul 1995 18:21:58 -0400 Date: Sat, 22 Jul 1995 18:21:58 -0400 From: FOMEQUE@aol.com Message-Id: <950722182158_120701269@aol.com> To: cube-lovers@life.ai.mit.edu Subject: Requesting Information Dear Cube lovers, How do I access your service ? Sincerely, Fomeque From nichael@sover.net Sun Jul 23 14:08:25 1995 Return-Path: Received: from maple.sover.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21318; Sun, 23 Jul 95 14:08:25 EDT Received: from [204.71.18.82] (st2.sover.net [204.71.18.82]) by maple.sover.net (8.6.12/8.6.12) with SMTP id OAA26687 for ; Sun, 23 Jul 1995 14:03:19 -0400 Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Sun, 23 Jul 1995 14:12:48 -0400 To: Cube-Lovers@ai.mit.edu From: nichael@sover.net (Nichael Lynn Cramer) Subject: Rubik's Race [I don't know if this has been discussed before. Maybe some of the old timers know about this --the copyright on the box is 1982-- but I don't recall having seen it mention since I've been on the list...] Anyway, Saturday we hit a rumage sale at the next town over (Hinsdale VT) and I picked up an interesting game in the "toys" section. It's called "Rubik's Race". (It was put out by Ideal and seems to be reasonably well made, so I assume it's "authorized".) The main "board" is about 18in long by about 7in wide. The players set at either end and their section of the board has a recessed area that holds 24 plastic pieces (about 1in square) aranged in a 5X5 grid with one empty space. (The square pieces are divided into six groups of four, each four-group being one of the standard colors on the side of the Cube.) At the start of a game, one player picks up a small box that contains nine small "cubies" (about 3/8in on a side), shakes the box and sets it down. The cubies settle onto the bottom of the box in 3X3 grid (thereby resembling the face of cube). Each player then proceeds to slide his pieces around (using the blank as maneourvering space) until the center 3X3 grid (out of the original 5X5 grid) matches the 3X3 "face" in the small box. The board is also divided between the two players' area by a vertical, hinged piece which can fall towards either end, and which has a "window" in the center. When the player finishes his central 3X3 grid, he pulls the hinged pieces towards himself --the window of which exposes his central grid-- and declares himself the winner. An obvious variation is for one player to scramble a cube, set it on the table and have each player try to match the topmost face. So, as such the games provides an interesing combination of the Cube, the 15-puzzle and Battleship. ;-) (But probably the most amazing part is that, given that I picked this up at rummage sale, *all* of the nearly sixty small plastic pieces were still there! All in all not a bad investment for 10cents.) Nichael - "...did I forget, forget to mention Memphis? nichael@sover.net Home of Elvis, and the ancient Greeks." From munafo@vgi.com Mon Jul 24 14:59:46 1995 Return-Path: Received: from vgi.com (hoss.vgi.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16862; Mon, 24 Jul 95 14:59:46 EDT Received: from frank.vgi.com ([1.0.2.139]) by vgi.com (4.1/SMI-4.1) id AA11341; Mon, 24 Jul 95 14:59:42 EDT Received: by frank.vgi.com (1.38.193.4/SMI-4.1) id AA06570; Mon, 24 Jul 1995 15:00:02 -0400 Date: Mon, 24 Jul 1995 15:00:01 -0400 (EDT) From: Robert Munafo X-Sender: munafo@frank To: CUBE-LOVERS List Subject: Little keychain cubes Message-Id: