From dik@cwi.nl Thu May 28 12:33:49 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02289; Thu, 28 May 92 12:33:49 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA03118 (5.65b/2.10/CWI-Amsterdam); Thu, 28 May 1992 17:18:31 +0200 Received: by boring.cwi.nl id AA00307 (5.65b/2.10/CWI-Amsterdam); Thu, 28 May 1992 15:00:49 +0200 Date: Thu, 28 May 1992 15:00:49 +0200 From: Dik.Winter@cwi.nl Message-Id: <9205281300.AA00307.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Corrected calculations are now done. After an initial false start I have now calculated the path-lengths in phase 1 of Kociemba's algorithm. The figures are as follows: path configurations 0: 1 1: 4 2: 50 3: 592 4: 7156 5: 87236 6: 1043817 7: 12070278 8: 124946368 9: 821605960 10: 1199128738 11: 58202444 12: 476 The figure 50 for path length 2 is easily verified by hand. I have a list with information about the configurations requiring a path-length of 12 (actually the paths leading to such a configurations). As should be true for each minimal path in phase 1, all paths start and terminate with a quarter turn of F, R, B or L. Some details. Phase 1 of the algorithm brings the cube in the subgroup generated by [F^2, R^2, B^2, L^2, U, D]. There are in this case 2,217,093,120 (2048 * 2187 * 495) cosets. This can be (and has been) reduced largely by observing symmetries. In this case rotating the complete cube along the UD axis by a quarter turn, rotating the cube along the RL axis by a half turn and mirroring through the FRBL plane reveal equivalent cosets. Although it is possible to remove *all* cosets that are equivalent to some canonical coset this was not done. The removal has only been done for the twists of corner cubes, reducing the factor 2187 to 168, and reducing the number of configurations to be handled to 170,311,680. For each configuration a minimal path was calculated. This was done starting with an absolute minimum found through the coordinate axis and through the 2-dimensional coordinate spaces. When a path of that length was not found the path length was increased and a new attempt was made. This was done until a path was found. All searches were exhaustive. On average paths were searched for 3 different lengths (519,177,716 attempts for 170,311,679 configurations). The computations were done on a farm of workstations where each workstation got a portion of the flip dimension (2048 cases of 83,160 configurations). Computation time for one portion was from 1 to 2 hours (1.5 on average), so the total computation was about 3000 hours. On a system with enough memory (50 MByte) it would have taken only 1 hour (this based on experiments with the corner cubes-only part). It could also have been with a single processor and a 50 MByte file, in that case CP time would also be about 1 hour, but the I/O time would exceed the 3000 hours very much. Using this result and the result by Hans Kloosterman the diameter of the cube group is at most 37. I conjecture the maximal path length in phase 2 of Kociemba's algorithm is 16, although the requirements on computer time cq. memory do inhibit calculations at this moment (only in memory would be feasible, but that requires 500 to 1000 MByte and computation time would be about one day). This figure of 16 would reduce the upperbound of the groups diameter to 28. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland dik@cwi.nl From @mitvma.mit.edu:hans@freyr.research.ptt.nl Fri May 29 13:21:10 1992 Return-Path: <@mitvma.mit.edu:hans@freyr.research.ptt.nl> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) id AA09766; Fri, 29 May 92 13:21:10 EDT Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 3176; Fri, 29 May 92 13:22:18 EDT Received: from research.ptt.nl by MITVMA.MIT.EDU (Mailer R2.08 R208004) with BSMTP id 8330; Fri, 29 May 92 13:22:17 EDT Received: from dnlunx.research.ptt.nl (DNLUNX) by research.ptt.nl (PMDF #12085) id <01GKLFC81PO0DQGQ2Q@research.ptt.nl>; Fri, 29 May 1992 19:21 +0100 Received: by gefjon.dnl (4.1/SMI-4.1) id AA03306; Fri, 29 May 92 19:20:23 +0200 Date: Fri, 29 May 92 19:20:22 MET DST From: J.M.Kloosterman@research.ptt.nl (Hans Kloosterman) Subject: Lower-bound Kociemba's algorithm To: cube-lovers@life.ai.mit.edu Message-Id: <9205291720.AA03306@gefjon.dnl> X-Envelope-To: cube-lovers@life.ai.mit.edu X-Mailer: ELM [version 2.3 PL11] Dik Winter writes: > Using this result and the result by Hans Kloosterman the diameter of the > cube group is at most 37. I conjecture the maximal path length in phase 2 > of Kociemba's algorithm is 16, although the requirements on computer time > cq. memory do inhibit calculations at this moment (only in memory would be > feasible, but that requires 500 to 1000 MByte and computation time would be > about one day). This figure of 16 would reduce the upperbound of the groups > diameter to 28. Unfortunately Dik's conjecture for phase 2 is too optimistic. Recall the maximum distances of the 4 stages of my algorithm: 1. 7 moves within the group 2. 10 moves within the group 3. 8 moves within the group 4. 18 moves within the group (Stage 3 and 4 together requires at most 25 moves.) These number of moves are minimal and cannot be improved within their group of moves. (Stage 2 can also not be improved using all moves.) From this we may conclude that the maximum path length in stage 2 of the algorithm of Kociemba is at least 18 moves. Taking the results of Dik Winter for stage 1 into account, the lower-bound for the mximum of Kociemba's algorithm becomes 30 moves. Hans Kloosterman From dik@cwi.nl Fri May 29 20:32:28 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA23171; Fri, 29 May 92 20:32:28 EDT Received: from steenbok.cwi.nl by charon.cwi.nl with SMTP id AA17356 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:32:23 +0200 Received: by steenbok.cwi.nl id AA01086 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:32:22 +0200 Date: Sat, 30 May 1992 02:32:22 +0200 From: Dik.Winter@cwi.nl Message-Id: <9205300032.AA01086.dik@steenbok.cwi.nl> To: J.M.Kloosterman@research.ptt.nl, cube-lovers@life.ai.mit.edu Subject: Re: Lower-bound Kociemba's algorithm (About my conjecture of 16 moves for phase 2:) > Unfortunately Dik's conjecture for phase 2 is too optimistic. > Recall the maximum distances of the 4 stages of my algorithm: > 1. 7 moves within the group > 2. 10 moves within the group > 3. 8 moves within the group > 4. 18 moves within the group > (Stage 3 and 4 together requires at most 25 moves.) > These number of moves are minimal and cannot be improved within their > group of moves. Did you (since your article) do an exhaustive search? In your article you mentioned that you had 6 positions that still do require 18 moves. And you mention that you doubted that there would be 17 move solvers. Have you proven since then that it can not be done in less than 18? If not, send me your positions and I will try. I have currently a program running that tries all phase 4 positions. It is possible to reduce the number of searches from 3,981,312 (the article contains a typo here) to 428,544 by observing equivalent positions (as I did mention in a previous article (*)). Assuming my conjecture of 16 the complete calculations would take about 1000 to 1500 hours (%). Not unprecedented ;-). (There must be a reason that I am a member of the CWI research group on large scale computing.) There are now only two machines munching at the problem, but there would be no problem to start up a few more again. I just did it to see what happens. dik -- * The equivalent positions are found by rotation of the complete cube along the UD axis for a quarter turn, along the RL axis through a half turn and mirroring along the FRBL plane. When looking at one dimension only this reduces the number from 40320 to 2768. Restricting to Hans's initial positions in phase 4, this reduces the number from 576 to 62. So the count becomes: 62 * 576 * 24 / 2 in stead of 576 * 576 * 24 / 2 (= ((4!)^5) / 2). -- % I found that an exhaustive search upto 16 moves takes about 10 seconds. Increasing to 17 would up the time to 110 seconds. So if you mail me the situations for which you do not yet have less than 18 moves I will have an attempt at them. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland dik@cwi.nl From dik@cwi.nl Fri May 29 20:44:04 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA23423; Fri, 29 May 92 20:44:04 EDT Received: from steenbok.cwi.nl by charon.cwi.nl with SMTP id AA17413 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:44:02 +0200 Received: by steenbok.cwi.nl id AA01102 (5.65b/2.10/CWI-Amsterdam); Sat, 30 May 1992 02:44:01 +0200 Date: Sat, 30 May 1992 02:44:01 +0200 From: Dik.Winter@cwi.nl Message-Id: <9205300044.AA01102.dik@steenbok.cwi.nl> To: J.M.Kloosterman@research.ptt.nl, cube-lovers@life.ai.mit.edu Subject: Re: Lower-bound Kociemba's algorithm As an afterthough, it would be interesting if it is possible to reduce the number of moves in your fourth phase. The main difference between your algorithm and Kociemba's is that yours is deterministic. Kociemba's algorithm performs quite a number of searches before finding the optimal solution. And even than it is not known whether the solution is indeed optimal, longer searches might reveal better solutions. Your algorithm gives an upper bound to the number of moves, and the solution is reached in limited time. Kociemba's algorithm is in theory unlimited in time. My experience is that it is best to limit the first phase in Kociemba's algorithm to 13 moves. But that is only because of time constraints. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland dik@cwi.nl From @mitvma.mit.edu:hans@freyr.research.ptt.nl Sat May 30 14:26:53 1992 Return-Path: <@mitvma.mit.edu:hans@freyr.research.ptt.nl> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) id AA07529; Sat, 30 May 92 14:26:53 EDT Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 6142; Sat, 30 May 92 14:27:59 EDT Received: from research.ptt.nl by MITVMA.MIT.EDU (Mailer R2.08 R208004) with BSMTP id 1345; Sat, 30 May 92 14:27:58 EDT Received: from dnlunx.research.ptt.nl (DNLUNX) by research.ptt.nl (PMDF #12085) id <01GKMUCOZXDCDQGRT4@research.ptt.nl>; Sat, 30 May 1992 19:41 +0100 Received: by gefjon.dnl (4.1/SMI-4.1) id AA03556; Sat, 30 May 92 19:40:37 +0200 Date: Sat, 30 May 92 19:40:36 MET DST From: J.M.Kloosterman@research.ptt.nl (Hans Kloosterman) Subject: Re: Lower-bound Kociemba's algorithm In-Reply-To: <9205300044.AA01102.dik@steenbok.cwi.nl>; from "Dik.Winter@CWI.NL" at May 30, 92 2:44 am To: Dik.Winter@cwi.nl Cc: cube-lovers@life.ai.mit.edu Message-Id: <9205301740.AA03556@gefjon.dnl> X-Envelope-To: cube-lovers@life.ai.mit.edu X-Mailer: ELM [version 2.3 PL11] > Did you (since your article) do an exhaustive search? In your article you > mentioned that you had 6 positions that still do require 18 moves. And you > mention that you doubted that there would be 17 move solvers. Have you > proven since then that it can not be done in less than 18? If not, send me > your positions and I will try. I have done an exhaustive search and none of the 6 situations of 18 moves could be reduced to 17 moves (within the group of ). For the case you want to verify, one of them is: L2 U R2 B2 U2 B2 L2 D2 L2 F2 D' L2 B2 F2 L2 F2 U' D Hans Kloosterman From dik@cwi.nl Sat May 30 18:12:35 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA10599; Sat, 30 May 92 18:12:35 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA26864 (5.65b/2.10/CWI-Amsterdam); Sun, 31 May 1992 00:12:31 +0200 Received: by boring.cwi.nl id AA02915 (5.65b/2.10/CWI-Amsterdam); Sun, 31 May 1992 00:12:30 +0200 Date: Sun, 31 May 1992 00:12:30 +0200 From: Dik.Winter@cwi.nl Message-Id: <9205302212.AA02915.dik@boring.cwi.nl> To: J.M.Kloosterman@research.ptt.nl Subject: Re: Lower-bound Kociemba's algorithm Cc: cube-lovers@life.ai.mit.edu > I have done an exhaustive search and none of the 6 situations of 18 moves > could be reduced to 17 moves (within the group of ). > For the case you want to verify, one of them is: > > L2 U R2 B2 U2 B2 L2 D2 L2 F2 D' L2 B2 F2 L2 F2 U' D > Of course I verified it ;-). This one does indeed kill Kociemba's algorithm. On a fast processor (65 MHz SPARC) with a larger limit database than Kociemba is using himself (the database is about 5 MByte for the second phase), it took 3 hours 15 minutes to find a minimal solution. Of 18 moves. From dik@cwi.nl Mon Jun 8 16:40:43 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA20905; Mon, 8 Jun 92 16:40:43 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA09954 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:40:42 +0200 Received: by boring.cwi.nl id AA12825 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:40:41 +0200 Date: Mon, 8 Jun 1992 22:40:41 +0200 From: Dik.Winter@cwi.nl Message-Id: <9206082040.AA12825.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: How big is the Magic Domino? (part2, data) 2x2x2 cube q+h loc max q only loc max 0: 1 - 1 - 1: 9 - 6 - 2: 54 - 27 - 3: 321 - 120 - 4: 1847 11 534 - 5: 9992 8 2256 - 6: 50136 96 8969 - 7: 227536 904 33058 16 8: 870072 13212 114149 53 9: 1887748 413392 360508 260 10: 623800 604516 930588 1460 11: 2644 2644 1350852 34088 12: 782536 402260 13: 90280 88636 14: 276 276 3x3x3 corners only q+h loc max q only loc max 0: 1 - 1 - 1: 18 - 12 - 2: 243 - 114 - 3: 2874 6 924 - 4: 28000 88 6539 - 5: 205416 792 39528 - 6: 1168516 15304 199926 114 7: 5402628 46068 806136 600 8: 20776176 325680 2761740 17916 9: 45391616 9757376 8656152 10200 10: 15139616 14665856 22334112 35040 11: 64736 64736 32420448 818112 12: 18780864 9654240 13: 2166720 2127264 14: 6624 6624 magic domino, 1 solution q+h loc max q only loc max 0: 1 - 1 - 1: 10 - 8 - 2: 67 - 48 - 3: 420 - 260 - 4: 2335 - 1330 - 5: 12260 - 6556 - 6: 61038 3 31301 - 7: 291004 12 144392 - 8: 1327429 793 638407 2 9: 5821374 6170 2709620 64 10: 24141784 87202 10873023 1261 11: 89480354 990826 39768668 15728 12: 262907144 13212972 124815946 214530 13: 485409604 91824956 296531984 2741192 14: 508704668 161596512 460831364 23949864 15: 232904952 175407548 435219080 72423024 16: 14508468 13668852 215035460 91647012 17: 129376 128592 38469576 35228568 18: 112 112 624320 618368 19: 1056 1056 magic domino, 4 solutions q+h loc max q only loc max 0: 4 - 4 - 1: 28 - 24 - 2: 136 - 108 - 3: 672 - 480 - 4: 3228 - 2116 - 5: 15072 - 9120 - 6: 69000 - 39188 - 7: 310784 92 166408 - 8: 1369220 1052 691508 56 9: 5888676 8656 2812496 192 10: 24209988 92284 11015008 1860 11: 89458152 976008 39837904 16104 12: 262772436 13124304 124673780 202940 13: 485358148 91776620 296336800 2667824 14: 508703948 161595792 460769708 23896632 15: 232904952 175407548 435217336 72421280 16: 14508468 13668852 215035460 91647012 17: 129376 128592 38469576 35228568 18: 112 112 624320 618368 19: 1056 1056 magic domino, 8 solutions q+h loc max q only loc max 0: 8 - 8 - 1: 56 - 48 - 2: 272 - 216 - 3: 1344 - 960 - 4: 6456 - 4232 - 5: 30144 - 18240 - 6: 138000 - 78376 - 7: 621568 184 332816 - 8: 2732664 3096 1383016 112 9: 11649816 28672 5612576 384 10: 46553800 331960 21772432 4584 11: 158726064 3909520 75752384 47792 12: 377277280 46692640 208971608 783864 13: 507933248 129847936 388348544 11790688 14: 414571632 181149888 466373488 54544928 15: 102181280 86967456 334811104 78445984 16: 3271456 3221680 114248208 79836432 17: 7312 7312 7974528 7869280 18: 19616 19616 From dik@cwi.nl Mon Jun 8 16:38:55 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA20835; Mon, 8 Jun 92 16:38:55 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA09864 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:38:52 +0200 Received: by boring.cwi.nl id AA12813 (5.65b/2.10/CWI-Amsterdam); Mon, 8 Jun 1992 22:38:51 +0200 Date: Mon, 8 Jun 1992 22:38:51 +0200 From: Dik.Winter@cwi.nl Message-Id: <9206082038.AA12813.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: How big is the Magic Domino? Having done a number of calculations on maximal distances I thought about getting at newer pastures. The Magic Domino. The results follow in the next mailing, this post discusses a bit about the information found there. I added to the next mail also the previous results for the 2x2x2 and the corners of the 3x3x3 together with some additional results not presented previously. There are for each puzzle five columns. The first one enumerates the number of moves, the next two give the results if both quarter and half turns are accepted as moves, the last two give the information if only quarter turns are accepted (of course, on the Domino this distinction is there only for the U and D faces, the others know half turns only). For each case there are two columns, the first giving the number of positions requiring the stated number of moves, the second column gives the number of local maxima (i.e. each move brings you closer to a solution). There are three tables for the Domino. The one you want to pick depends on how you view the puzzle. The first view is that there is only one solution with on top 1 to 3 running from left back to right back. The second view is that rotation of the puzzle makes different configurations indistinguishable, so the total number of configuration is (8!)^2 / 4. An alternative way to look at it is that there are 4 solutions. One the standard solution, the others obtained by rotating the domino along the UD axis. The distinction between the two views is only a factor of four in the number of configurations for the different path-lengths. Finally, we can view as a solution the configuration with on top 1 to 3 running from right back to left back in stead of the other way around. Actually this solution is not worse than the other, because, if we turn over a solved Domino we go from one to the other. This view can also be expressed by saying there are 8 solutions. I give results for all three cases. The numbers upto (and including) path-length 2 have been checked by hand. Some remarkable observations. When we compare the tables for 1 solution and those for 4 solutions we see that for short path-lengths the number of configurations is multiplied by 4. On the other hand, for long path-lengths the number of configurations is equal! We can say that rotation of the Domino has only a short range effect. On the other hand, if we compare both with the 8-solutions tables we see that the latter allows shorter solutions in general, so mirroring has a long range effect. Each of the 6 calculations on the Magic Domino took 2 to 2.5 hours on one processor of an SGI 4D-420S. The program is completely memory bound (and the cache does not help). It needs at least 31 MByte of core (and must be resident) otherwise you will get no results at all in reasonable time. I tried it on the 32 MByte FPS; while it will happily give results initially at some stage it will not longer run. Not only that it will not walk either, and also not crawl. It is just sitting there paging in and paging out (a phenomenon known as page thrashing). I found that the program would get less than 0.005 % of the CPU on an otherwise unloaded machine. The program would enable me to write a 27901440 byte file that would assist in an optimal solver for the Domino. dik From dik@cwi.nl Mon Jun 8 20:48:48 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA28679; Mon, 8 Jun 92 20:48:48 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA26132 (5.65b/2.10/CWI-Amsterdam); Tue, 9 Jun 1992 02:48:46 +0200 Received: by boring.cwi.nl id AA13958 (5.65b/2.10/CWI-Amsterdam); Tue, 9 Jun 1992 02:48:46 +0200 Date: Tue, 9 Jun 1992 02:48:46 +0200 From: Dik.Winter@cwi.nl Message-Id: <9206090048.AA13958.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Magic Domino part3 Considering my previous fiasco, I have now incorporated the changes needed to make the magic domino calculating program working into the program that calculated the corners on a 3x3x3 cube. The results still match, which gives me confidence that the algorithms are correct indeed. Moreover, it reduced the time to do the 3x3x3 corner calculations to 8 minutes. dik From wft@math.canterbury.ac.nz Wed Jun 10 00:37:06 1992 Return-Path: Received: from CANTVA.CANTERBURY.AC.NZ by life.ai.mit.edu (4.1/AI-4.10) id AA08570; Wed, 10 Jun 92 00:37:06 EDT Received: from math.canterbury.ac.nz by csc.canterbury.ac.nz (PMDF #12052) id <01GL2137V0CW90NKNZ@csc.canterbury.ac.nz>; Wed, 10 Jun 1992 16:36 +1200 Received: by math.canterbury.ac.nz (4.1/SMI-4.1) id AA27010; Wed, 10 Jun 92 16:36:33 NZS Date: Wed, 10 Jun 92 16:36:33 NZS From: wft@math.canterbury.ac.nz (Bill Taylor) Subject: Name query. To: Cube-Lovers@life.ai.mit.edu Message-Id: <9206100436.AA27010@math.canterbury.ac.nz> X-Envelope-To: Cube-Lovers@life.AI.MIT.EDU Can anyone tell me:- Why is the "Pons Asinorum" pattern so called ? --------------------------------------------------------------------- Bill Taylor wft@math.canterbury.ac.nz Artificial intelligence beats real stupidity. --------------------------------------------------------------------- From ronnie@cisco.com Wed Jun 10 15:12:01 1992 Return-Path: Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA01827; Wed, 10 Jun 92 15:12:01 EDT Received: from lager.cisco.com by wolf.cisco.com with TCP; Wed, 10 Jun 92 12:11:36 -0700 Message-Id: <9206101911.AA01765@wolf.cisco.com> To: wft@math.canterbury.ac.nz (Bill Taylor) Cc: Cube-Lovers@life.ai.mit.edu Subject: Re: Name query. From: ronnie@cisco.com Date: Wed, 10 Jun 92 12:11:35 PDT Sender: ronnie@cisco.com > Can anyone tell me:- > > Why is the "Pons Asinorum" pattern so called ? Pons Asinorum is Latin for "Asses' Bridge," and is the name of the proposition that the base angles of an isoceles triangle are equal. It is more generally any test of ability imposed upon the inexperienced or ignorant. Ronnie (who has xwebster) From GOET@rcl.wau.nl Thu Jun 11 02:46:21 1992 Return-Path: Received: from NET.WAU.NL by life.ai.mit.edu (4.1/AI-4.10) id AA18013; Thu, 11 Jun 92 02:46:21 EDT Received: from RVD.WAU.NL by NET.WAU.NL (PMDF #12413) id <01GL2YYMEHS0001ZSZ@NET.WAU.NL>; Thu, 11 Jun 1992 08:46 GMT +01:00 Received: from RCL.WAU.NL by RCL.WAU.NL (PMDF #12413) id <01GL2YSCT30W9ED95T@RCL.WAU.NL>; Thu, 11 Jun 1992 08:41 GMT +01:00 Date: Thu, 11 Jun 1992 08:41 GMT +01:00 From: "Kees Goet - Landbouwuniversiteit, Afd. I&D" Subject: Unsubscribe To: cube-lovers@life.ai.mit.edu Message-Id: <01GL2YSCT30W9ED95T@RCL.WAU.NL> X-Envelope-To: cube-lovers@life.ai.mit.edu X-Vms-To: IN%"cube-lovers@life.ai.mit.edu" X-Vms-Cc: GOET Could somebody please remove me from this list. Thanks in advance. Kees Goet. From STEFANO@agrclu.st.it Thu Jun 11 10:14:42 1992 Return-Path: Received: from pol88a (pol88a.polito.it) by life.ai.mit.edu (4.1/AI-4.10) id AA25505; Thu, 11 Jun 92 10:14:42 EDT Received: from AG-IN by POLITO.IT (PMDF #12666) id <01GL3B97N2G095MMGN@POLITO.IT>; Thu, 11 Jun 1992 14:38 GMT+1 Date: Thu, 11 Jun 92 14:36 CET From: STEFANO BONACINA Subject: Signoff To: cube-lovers@life.ai.mit.edu Message-Id: <40E32DA5DC3F00DAC9@agr04.ST.IT> X-Organization: SGS-THOMSON Microelectronics X-Envelope-To: cube-lovers@life.ai.mit.edu X-Vms-To: IN%"cube-lovers@life.ai.mit.edu" Could anyone tell me how can I unsubscribe from this mailing list? Thanks in advance. Stefano From alan@ai.mit.edu Thu Jun 11 15:20:31 1992 Return-Path: Received: from august (august.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA09200; Thu, 11 Jun 92 15:20:31 EDT Received: by august (4.1/AI-4.10) id AA04098; Thu, 11 Jun 92 15:21:19 EDT Date: Thu, 11 Jun 92 15:21:19 EDT Message-Id: <9206111921.AA04098@august> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: Kees.Goet@rcl.wau.nl, STEFANO@agrclu.st.it Cc: Cube-Lovers@ai.mit.edu In-Reply-To: "Kees Goet - Landbouwuniversiteit, Afd. I&D"'s message of Thu, 11 Jun 1992 08:41 GMT +01:00 <01GL2YSCT30W9ED95T@RCL.WAU.NL> Subject: Please do not bother everybody with administrative requests Date: Thu, 11 Jun 92 14:36 CET From: STEFANO BONACINA Could anyone tell me how can I unsubscribe from this mailing list? Thanks in advance. Stefano Date: Thu, 11 Jun 1992 08:41 GMT +01:00 From: "Kees Goet - Landbouwuniversiteit, Afd. I&D" Could somebody please remove me from this list. Thanks in advance. Kees Goet. As everyone is informed when they subscribe, administrative requests should be directed to Cube-Lovers-Request@AI.MIT.EDU (me). Even if you lost my original greeting message, the "-Request" suffix is a sufficiently widespread convention for mailing lists that you should have tried it first, before bothering the entire mailing list. STEFANO@agrclu.st.it, I have removed you. Kees.Goet@rcl.wau.nl, I will be sending you separate mail about your subscription. - Alan From gls@think.com Thu Jun 11 17:11:55 1992 Received: from mail.think.com (Mail1.Think.COM) by life.ai.mit.edu (4.1/AI-4.10) id AA12924; Thu, 11 Jun 92 17:11:55 EDT Return-Path: Received: from Strident.Think.COM by mail.think.com; Thu, 11 Jun 92 16:52:32 -0400 From: Guy Steele Received: by strident.think.com (4.1/Think-1.2) id AA18593; Thu, 11 Jun 92 16:52:31 EDT Date: Thu, 11 Jun 92 16:52:31 EDT Message-Id: <9206112052.AA18593@strident.think.com> To: ronnie@cisco.com Cc: wft@math.canterbury.ac.nz, Cube-Lovers@life.ai.mit.edu In-Reply-To: ronnie@cisco.com's message of Wed, 10 Jun 92 12:11:35 PDT <9206101911.AA01765@wolf.cisco.com> Subject: Name query. From: ronnie@cisco.com Date: Wed, 10 Jun 92 12:11:35 PDT > Can anyone tell me:- > > Why is the "Pons Asinorum" pattern so called ? Pons Asinorum is Latin for "Asses' Bridge," and is the name of the proposition that the base angles of an isoceles triangle are equal. It is more generally any test of ability imposed upon the inexperienced or ignorant. The term also carries the connotation that the test is in fact of the simplest and most elementary kind. If you can't prove the Pons Asinorum of geometry, then you don't know even the most elementary concept of geometry--i.e., as a geometer, you know as much as a donkey. And if you cannot form the Pons Asinorum pattern, you sure don't know much about cubing. --Guy Steele From ACW@riverside.scrc.symbolics.com Thu Jun 11 17:44:29 1992 Return-Path: Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA13947; Thu, 11 Jun 92 17:44:29 EDT Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 834944; 11 Jun 1992 17:39:46-0400 Date: Thu, 11 Jun 1992 17:39-0400 From: Allan C. Wechsler Subject: Name query. To: gls@think.com, ronnie@cisco.com Cc: wft@math.canterbury.ac.nz, Cube-Lovers@life.ai.mit.edu In-Reply-To: <9206112052.AA18593@strident.think.com> Message-Id: <19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM> Date: Thu, 11 Jun 1992 16:52 EDT From: Guy Steele From: ronnie@cisco.com Date: Wed, 10 Jun 92 12:11:35 PDT > Can anyone tell me:- > > Why is the "Pons Asinorum" pattern so called ? Pons Asinorum is Latin for "Asses' Bridge," and is the name of the proposition that the base angles of an isoceles triangle are equal. It is more generally any test of ability imposed upon the inexperienced or ignorant. The term also carries the connotation that the test is in fact of the simplest and most elementary kind. If you can't prove the Pons Asinorum of geometry, then you don't know even the most elementary concept of geometry--i.e., as a geometer, you know as much as a donkey. And if you cannot form the Pons Asinorum pattern, you sure don't know much about cubing. --Guy Steele I think the metaphorical leap from geometry to cubing was probably made by Bernie Greenberg, in whatever year it was that Hofstadter did his Sci Am column. Hofstadter came to MIT to talk to a bunch of cubers, gathering material for his article. I was in the group and my name is mentioned in the article -- the only time I have ever gotten my name into Sci Am. "Pons Asinorum" has a lot of Bernie's style about it -- casual use of Latin, whimsical metaphor, fondness for naming things. He had a bunch of cube operators with Latin names, and also some wacky English ones. I remember the Spratt Wrench (F R'L D R'L B R'L U R'L) which flips four edges and was what everyone used before monoflips were discovered. Bernie also had things with names like the Lesser Hammer of the Right and the Greater Hammer of the Right; his "patter" was fabulous. I regret not having a videotape of Bernie solving the cube in, say, 1978. (I hope I've got the year right.) While I'm reminiscing, I should confess that my standard corner operator is still the same as it was then: (FUR)^5, which exchanges two corners, leaves the rest of the corners alone, and fucks the edges completely. (Prudes, do not hassle me. This has been a technical term in cubing around MIT since The Beginning.) Because of this property of "furry five", I have to home and orient all the corners first, before I touch the edges. It's the kind of quirky algorithm you don't see among younger cubers, because everybody these days learns how to solve the thing from a book. In the Beginning, there were no books, and I proudly state that I solved the cube from scratch, by brainpower. Later I discovered that there were easier ways to do things than (FR)^105! I had pages and pages covered with little cube diagrams with arrows showing how the stickers were permuted by a particular sequence. I'm interested in hearing other reminiscences from people who actually solved the cube -- you're disqualified if you learned how to solve it from somebody else, or from a book. From dik@cwi.nl Thu Jun 11 18:46:53 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA16508; Thu, 11 Jun 92 18:46:53 EDT Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA05569 (5.65b/2.10/CWI-Amsterdam); Fri, 12 Jun 1992 00:46:44 +0200 Received: by boring.cwi.nl id AA22860 (5.65b/2.10/CWI-Amsterdam); Fri, 12 Jun 1992 00:46:42 +0200 Date: Fri, 12 Jun 1992 00:46:42 +0200 From: Dik.Winter@cwi.nl Message-Id: <9206112246.AA22860.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Re: Name query. Actually reminiscences. > I > regret not having a videotape of Bernie solving the cube in, say, 1978. > (I hope I've got the year right.) I think it must be later. The cube was put on the market in Hungary in 1977 and first exported in 1980. Although earlier examples were privately exported I presume. > While I'm reminiscing, I should confess that my standard corner operator > is still the same as it was then: (FUR)^5, which exchanges two corners, > leaves the rest of the corners alone, and fucks the edges completely. Happens to me also. I still use operators I found myself in favour of (shorter) processes found later in books. I remember them better! > I'm interested in hearing other reminiscences from people who actually > solved the cube -- you're disqualified if you learned how to solve it > from somebody else, or from a book. I got one for my birthday in 1981 (yes, I was late). By the end of the party it was completely scrambled. One long night and a long day afterwards had me solve the cube. Although at that moment I had not completely lined up procedures to do it. Later I more or less procedurized it. Much stranger was my first encounter with Square 1. As all puzzles it was scrambled within minutes after I brought it home. I tried to solve it, but for some reason I did not yet see how to bring it back in the shape of a cube. The next day when I came home from work it was in the shape of a cube. It appears that my 8 year old daughter had done that! Solving the remainder was fairly simple. dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland dik@cwi.nl From sjfc!ggww@cci632.cci.com Sun Jun 14 22:54:44 1992 Return-Path: Received: from uu.psi.com by life.ai.mit.edu (4.1/AI-4.10) id AA27747; Sun, 14 Jun 92 22:54:44 EDT Received: from sjfc.UUCP by uu.psi.com (5.65b/4.1.031792-PSI/PSINet) id AA14756; Sun, 14 Jun 92 22:36:33 -0400 Received: by cci632.cci.com (5.54/5.17) id AA07588; Sun, 14 Jun 92 22:03:16 EDT Received: by sjfc.UUCP (5.51/4.7) id AA00173; Sun, 14 Jun 92 21:37:14 EDT Date: Sun, 14 Jun 92 21:37:14 EDT From: sjfc!ggww@cci632.cci.com (Gerry Wildenberg) Message-Id: <9206150137.AA00173@sjfc.UUCP> To: cube-lovers@life.ai.mit.edu Subject: Remove me from this list. Please remove me from the mailing list. Gerry Wildenberg ggww@sjfc.uucp St. John Fisher College sjfc!ggww@cci.com Rochester, NY 14618 ggww@sjfc.edu (New, may not yet work.) From ACW@riverside.scrc.symbolics.com Tue Jun 16 16:12:32 1992 Return-Path: Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA28765; Tue, 16 Jun 92 16:12:32 EDT Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 838315; 16 Jun 1992 16:14:52-0400 Date: Tue, 16 Jun 1992 16:13-0400 From: Allan C. Wechsler Subject: Re: Name query. Actually reminiscences. To: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu In-Reply-To: <9206112246.AA22860.dik@boring.cwi.nl> Message-Id: <19920616201302.7.ACW@PALLANDO.SCRC.Symbolics.COM> Date: Thu, 11 Jun 1992 18:46 EDT From: Dik.Winter@cwi.nl Much stranger was my first encounter with Square 1. As all puzzles it was scrambled within minutes after I brought it home. I tried to solve it, but for some reason I did not yet see how to bring it back in the shape of a cube. The next day when I came home from work it was in the shape of a cube. It appears that my 8 year old daughter had done that! Solving the remainder was fairly simple. Our four-year-old managed to assemble our Snafooz into a cube once. No one else has been able to do it, and he can't duplicate his success. (He can't even read.) From wft@math.canterbury.ac.nz Fri Jun 19 03:07:51 1992 Return-Path: Received: from CANTVA.CANTERBURY.AC.NZ by life.ai.mit.edu (4.1/AI-4.10) id AA19564; Fri, 19 Jun 92 03:07:51 EDT Received: from math.canterbury.ac.nz by csc.canterbury.ac.nz (PMDF #12052) id <01GLEQZFO09S9X3ZMD@csc.canterbury.ac.nz>; Fri, 19 Jun 1992 19:07 +1200 Received: by math.canterbury.ac.nz (4.1/SMI-4.1) id AA12708; Fri, 19 Jun 92 19:07:30 NZS Date: Fri, 19 Jun 92 19:07:30 NZS From: wft@math.canterbury.ac.nz (Bill Taylor) Subject: reminiscences To: Cube-Lovers@life.ai.mit.edu Cc: wft@math.canterbury.ac.nz Message-Id: <9206190707.AA12708@math.canterbury.ac.nz> X-Envelope-To: Cube-Lovers@AI.AI.MIT.EDU Allan C. Wechsler asks for general reminiscences from people who solved the cube. It's just as well hardly anyone's replied, or the list would be swamped with boring anecdotes ! So maybe I'll add an anecdote or two of my own. Dik.Winter@cwi.nl writes > > While I'm reminiscing, I should confess that my standard corner operator > > is still the same as it was then: (FUR)^5, which exchanges two corners, > > leaves the rest of the corners alone, and fucks the edges completely. > >Happens to me also. I still use operators I found myself in favour of >(shorter) processes found later in books. I remember them better! Very true. This reminds me of what I read in (I think) the math games column of Scientific American, about mid-to-late 80's. The cubing craze had largely passed, and someone who had been an addict, but hadn't touched it for some years, had occasion to try it again. He realized with horror, that he couldn't remember a single thing! However, as he began to fiddle with the cube rather disconsolately, he found himself automatically doing the right things. "I couldn't remember how to do it, but my fingers could !!", he said. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ This was my experience too, a few years ago. It's quite uncanny, like starting to ride a bike again after decades of not doing; only more so. That's what comes of sticking loyally to your first halfway-decent discoveries on cube-solving. From madly over-addictive use, they become burned into your brain tissue. My own pet method has always been to put all the edges right first, using only common sense (except maybe at the very end some slight method needed); then put the corners right using the basic "8-fold way"..... R~ U L U~ R U L~ U~ . We found this eightfold way quite easy to remember, the face order is very natural, and the sequence of clockwise vs anti-clockwise turns, i.e. ACCA CCAA, seems somehow like a sonnet rhyming scheme (now burnt irrevocably into my finger-moving cortex). This eightfold way is just a commutator of a face move and (a commutator of two face moves)); so it turns out to be group-theoretically natural, as commutators do "as little as possible". The eight-fold way can also be viewed in a natural geometric light, as just a standard 3-permutation of corners, similaritied away from one another to avoid interference. (Don't know what the standard technical terms for all this are, sorry; it's probably old hat to most readers here.) Viewing it this way, one can quickly re-create several (8-fold) variants, and some 10-fold ones, all of the same type, and all variously useful. By similarities, one can usually put a corner into a more useful spot, so as to get two corners done at a time, with one 8-fold. ENOUGH; of teaching grandmothers to suck eggs. I was going to reminisce. Not many people seem to do the cube this way, that is, edges first. It was shown to me by my late colleague Brent Wilson (the other of the "we" refered to above). At first it seemed a little unnatural, but once you get used to it, it seems super efficient. I suppose everyone feels that way about their own methods. The particular 8-fold mentioned above was my own invention, so I've always had a soft spot for it. Brent and I both started out on the cube the same way, which is I suppose standard. We spent some little time learning to do the base. Then we spent some considerably longer time learning to do the middle layer. We found later that we had both expected the same thing:- that when the middle and base layers were all successfully done, the top layer would automatically have to be right !! So of course, we were both temporarily devastated when it turned out otherwise; and we both realized that we were in the presence of a mighty puzzle, and were in for some great fun. So we went ahead and discovered all the usual group-theoretic things, one by one, over the months. I have anothger reminiscence to tell about my colleague. I once read of someone, (J.H.Conway ?), who was alleged to do the cube behind his back ! Well Brent practiced this trick also (unaware of anyone else having done it, if indeed it was done the same way, even). He invented the method after having discovered the only all-commuting position, i.e. with all edges flipped, corners all correct. He perfected a smooth method of doing this behind his back. The trick is, of course, merely to have a pre-prepared cube in this position. It doesn't QUITE look random, but if you ADD to it a couple of random twists, it now looks totally random; at first (and second) glance. He would show this "random" cube to us, let us hold it (very briefly!), then take it and do the "all-flips" behind his back. Keeping up a continuous patter, as he brought it back he would be saying "...so there's only a couple of twists to go", and then as it appeared he would do the last two twists by sight, without hesitation. As the two "randomizing" twists commute with the other position, he didn't have to memorize them; indeed he could even let the audience do them ! Of course this would mean he would have to have the pure "flipped" pattern to start with, which was easier to detect, alas. Well, one time, he was to give a talk to some school kids. He wanted to do the cube behind his back, as a piece-de-resistance. He decided to train himself up into being able to undo FOUR random twists by sight. He duly did this. Then when the talk came around, he had a cube prepared in "all-flip" position, with two twists added, to make it look quite random. Then, when the highlight of the talk came around, he would display it to the class, let one or two handle it briefly, to agree it was just another muddled up cube. Then, HE WOULD EVEN ASK two members of the audience to add an extra random twist each (just to prove the cube wasn't in a prepared position!) Then he would do the all-flips operation behind his back, keeping up his patter. He expected to be able to handle undoing the four random flips left over, by sight, as he was completing his patter. When the great event came along, everything went perfectly, without a hitch. BUT, amazingly, by a 144-to-1 chance, the two flips that the audience added exactly undid the two that he had put on himself ! So when he brought it from behind his back, it was already perfectly done. Without batting an eyelid, he brought his patter to a halt then and there. Needless to say, the kids were even more staggered than they would have been otherwise. He resisted all imploring entreaties to tell them how it was done (like all good conjurers); and I don't thimk he ever did the trick again! By great good luck, however, I have a vieotape of him doing this trick, from the demo itself. So if any of you are ever in New Zealand, you can look me up, and ask to see this amazing event ! Like Allan Wechsler, I would be delighted to hear anyone else's reminiscences, or cube anecdotes generally. There must be tons, so, don't be shy! Cheers, Bill Taylor. From reid@math.berkeley.edu Sun Jun 21 13:11:03 1992 Return-Path: Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) id AA17637; Sun, 21 Jun 92 13:11:03 EDT Received: from phnom-penh.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math)) id AA08266; Sun, 21 Jun 92 10:11:00 PDT Date: Sun, 21 Jun 92 10:11:00 PDT From: reid@math.berkeley.edu (michael reid) Message-Id: <9206211711.AA08266@math.berkeley.edu> To: Cube-Lovers@life.ai.mit.edu Subject: Re: reminiscences another call for reminiscences ... ) From: sjfc!ggww@cci632.cci.com (Gerry Wildenberg) ) ) Please remove me from the mailing list. yeah, remember back in those days when people were actually asking to be ADDED to the mailing list? :-) btw, administrivia should be sent to "cube-lovers-request@ai.mit.edu". thank you for your cooperation. ^^^^^^^ >From: wft@math.canterbury.ac.nz (Bill Taylor) > This eightfold way is just a commutator of a face move and (a commutator of > two face moves)); so it turns out to be group-theoretically natural, as > commutators do "as little as possible". here's the way i describe this. if sigma is a permutation on n symbols, (say 1, 2, 3, ... , n), define the "support" of sigma to be those integers which are NOT fixed by sigma. if tau is another permutation on the same set, such that supp(sigma) and supp(tau) are disjoint, then sigma and tau commute (i.e. the commutator is the identity). if supp(sigma) intersect supp(tau) has just one element, then the commutator is a three-cycle. as a rule of thumb, the smaller the intersection of the supports, the smaller the support of the commutator. in bill's example, ( R~ U L U~ R U L~ U~ ) the two permutations are "R" and "U L U~", which only affect one corner in common. (actually, to consider the cube as a permutation group, each corner is really 3 objects, one for each orientation.) but the analogy works well. this idea is also helpful for creating three-cycles of corner-edge pairs as well. on the 5x5x5 cube, you can make three-cycles of large blocks. in fact, a larger cube is probably a better visual aid for understanding/ explaining this concept. another good commutator to try is with the two sequences "B1 D2 B3" and "R1 U2 R3", which affect two corners in common. (this is a fairly well-known maneuver.) > "I couldn't remember how to do it, but my fingers could !!", he said. > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ > This was my experience too, a few years ago. It's quite uncanny, like > starting to ride a bike again after decades of not doing; only more so. i've also experienced the exact same thing. it seems as though my hands remember a short sequence by which i've just conjugated some maneuver, which helps at the end of the conjugation. also in these commutators, after i've done A and working on B, it feels as though my hands are anxious to undo A, almost as if the cube is spring loaded and will just snap back. [lame story about a friend pulling some hoax deleted] actually, i think i FINALLY understand the story. the idea is that not only is "brent" going to solve the cube behind his back, but he's also going to do it WITHOUT first looking at it. actually, the story isn't nearly as lame as i first thought, after realizing this. when i visited mit in 1984(?), i saw joe killian do the real trick. i certainly would have complained if i hadn't been allowed to scramble the cube exactly as i wished. in fact, i may have even insisted that he use MY cube (not too sure, though), just to be certain that the surface hasn't been textured in any way. it was quite impressive. about 5 minutes of studying, then behind the back without peeking. he said that all it took was a good system of remembering where all the pieces are. but i don't know what his system was. by the way, bill, this "brent" wouldn't happen to be your friend who can do the cube in 0.87 seconds, would he? :-) and speaking of tall tales, let's see if anyone can top this one: back in the days when i was into speed, er, speed CUBING, i'd solve the cube maybe 200 times every day. for some reason, i got into the habit of scrambling it behind my back (probably from listening to too many complaints like "you're just watching all the moves you're doing!" yeah, such a good complaint deserves such a fine solution.) well, anyway, one time i stopped scrambling it, and as usual, i get 15 second to study it (standard racing rules). however, much to my surprise, the cube was quite UNscrambled! how could this possibly be? well, the only explanation is certainly that after scrambling the cube thousands of times, my hands began to get into a rhythm (maybe even a rut). they'd just do the same sequence over and over again. depending upon my concentraion level, i'd find that sometimes i needed to make a conscious effort to vary the sequence. in fact, at least once i got a pattern that i'd previously seen: it was 4 dots with 6 corners twisted (hence has order 6). so it's not too implausible. like bill, and unlike dik, i spent quite some time struggling with the cube before i finally solved it; probably about 6 or 7 weeks. in fact for some time, probably about 2 weeks, i was convinced that it couldn't be done, except by very dumb luck (as in story above). of course, in those days, i hadn't heard of cube-lovers, hadn't even seen a computer, didn't know the furst-hopcroft-luks algorithm, hadn't even heard of anyone who could solve it ... but i was just a high school freshman (age 14) at the time. i didn't even know what a group was! this was shortly before the big craze started here in the u.s. (late 1980). at school, some friends and i talked about it, but the main questions were: how was it made, and how many combinations did it REALLY have? i was truly convinced that trying to solve it would be futile. there was also a shortage of them at the time, so i didn't get one until xmas. in fact, i remember the tv commercial that ideal put out. they didn't even make it clear that it actually turned in all possible directions! we had all sorts of ridiculous diagrams and ideas of cables and magnets, but none of them quite worked. and how could it turn in all directions? i heard of a bookstore somewhere that had one on display (but were otherwise sold out), so i went to see it. i remember spending a few minutes twisting it to find an axis that wouldn't turn! in fact, i could keep turning the same face in the same direction, around and around and around ... and the cables inside never got caught! sometimes i'm amazed at just how stupid i can be when i try ... anyway, the story about how i finally figured out how to solve it isn't nearly as interesting. after i first heard about people that could do it, i started to work on it more seriously. the key ideas were: get all the corners, (here was something that you could do, and then still do more without destroying what's already done. but this was hard and usually took more dumb luck and/or persistence.) then two opposite layers. (again, the middle slice still can turn, even with half turns on the sides F, B, R, L.) it took several days to flip the last two edges on the middle layer. (i just kept picking a different pair of opposite layers to solve and stumbled across a U layer monoflip in the process. of course, it took months before i realized what was actually happening.) also figuring out how to take it apart (and finally seeing how it was made) was helpful, 'cause then i could experiment easily. well, i've droned on long enough. anyone else got any interesting stories? mike From STEVENS@macc.wisc.edu Tue Jun 23 07:59:22 1992 Return-Path: Received: from vms3.macc.wisc.edu by life.ai.mit.edu (4.1/AI-4.10) id AA05161; Tue, 23 Jun 92 07:59:22 EDT Received: from VMSmail by vms3.macc.wisc.edu; Mon, 22 Jun 92 09:19 CDT Message-Id: <22062209193919@vms3.macc.wisc.edu> Date: Mon, 22 Jun 92 09:19 CDT From: PAul STevens - MACC - 2-9618 Subject: Re: reminiscences To: CUBE-LOVERS@life.ai.mit.edu X-Vms-To: IN%"cube-lovers@life.ai.mit.edu",STEVENS reid@math.berkeley.edu writes: >well, i've droned on long enough. anyone else got any interesting stories? I think my solution may be considered cheating; but I was pretty proud of it. I had almost decided to give up on the thing. But I had just designed and built an 8080 'computer'. It had 2k bytes of 2102's and had to be programmed with binary switches and whenever the program clobbered itself the entire program had to be re-entered in binary. So I wrote a program to look for combinations of moves that left most of the cube alone and only moved a few cubelets. I studied the best of these at great length and managed to combine some into 'better' moves, eventually finding some that moved only three or so cubelets. These were then combined into a solution. A rather god-awful solution I think. But my fingers learned the moves and I have never abandoned them for fear of becoming totally confused. The same ugly solution has been passed on for at least one generation and perhaps will persist for hundreds of years. I still don't know what a group or commutator or ... is except what I have deduced from reading mail from this group. I get the front face corners exactly right, the back corners in the proper position, and then the back corners rotated properly. Finally the edges go where they belong one at a time, first on the front and back and finally the four on the sides/top/bottom. I have noticed a lack of discussion of cubes that have pictures on them such that the entire cube can be right except that a single center can be upside-down. I have also painted a 4x4x4 so that the center 4 squares on each face have to be in the proper position. Every time I solve this cube I have to rediscover how it is done. My fingers refuse to learn it for me. Behind the back? You gotta be kidding! PAul From MONET01@mizzou1.missouri.edu Tue Jun 23 13:31:40 1992 Return-Path: Received: from MIZZOU1.missouri.edu ([128.206.2.2]) by life.ai.mit.edu (4.1/AI-4.10) id AA17944; Tue, 23 Jun 92 13:31:40 EDT Message-Id: <9206231731.AA17944@life.ai.mit.edu> Received: from MIZZOU1 by MIZZOU1.missouri.edu (IBM VM SMTP V2R1) with BSMTP id 4884; Tue, 23 Jun 92 12:31:20 CDT Received: by MIZZOU1 (Mailer R2.08) id 2332; Tue, 23 Jun 92 12:31:19 CDT Date: Tue, 23 Jun 92 12:21:27 CDT From: MONET01@mizzou1.missouri.edu To: cube-lovers@life.ai.mit.edu Subject: Ultimate cube The recent posting about cubes with photos has prompted me to post about my favorite cube. I picked this one up around the end of the BIG cube craze and have kept it in my desk every since. The cube looks like someone took a knife to a normal solved cube and cut a diagonal 'x' through each face and folded the flaps back down the sides. This leads to a cube where opposing centers have an 'x' that has four colors in a mirror image. (It is hard to describe, sorry.) This cube has to be solved and then the centers oriented properly. The slick thing about the cube is that part way through the solution (fairly early on), you may have to swap top for bottom and start over. I like to fiddle with it because at first glance it looks impossible to determine which cubelet is which to a novice and to a semi-experienced cubist it is not as easy as it looks. The cube was made by ULTRACO and is called ULTIMATE CUBE (copyrighted 1982). Unfortunately, when I went back a few weeks later to buy a couple more cubes, they were all gone and the sales people had no idea what I was talking about. I think I got this cube at a Mall toy store. If anyone knows where I can get a replacement, I would be interested as the printing on a few squares has faded just like my youth. From hoey@aic.nrl.navy.mil Wed Jun 24 15:45:56 1992 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA23541; Wed, 24 Jun 92 15:45:56 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA09989; Wed, 24 Jun 92 15:45:51 EDT Date: Wed, 24 Jun 92 15:45:51 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9206241945.AA09989@Sun0.AIC.NRL.Navy.Mil> To: MONET01@mizzou1.missouri.edu, cube-lovers@life.ai.mit.edu Subject: Re: Ultimate cube MONET01@mizzou1.missouri.edu writes of a cube that ``looks like someone took a knife to a normal solved cube and cut a diagonal 'x' through each face and folded the flaps back down the sides. This leads to a cube where opposing centers have an 'x' that has four colors in a mirror image. (It is hard to describe, sorry.)'' I would appreciate a few more details. I think the color scheme of each face you describe is something like +-----+-----+-----+ |.1111|11111|1111.| |44.11|11111|11.22| |4444.|11111|.2222| +-----+-----+-----+ |44444|.111.|22222| |44444|44.22|22222| |44444|.333.|22222| +-----+-----+-----+ |4444.|33333|.2222| |44.33|33333|33.22| |.3333|33333|3333.| +-----+-----+-----+ where 1,2,3, and 4 are distinct colors, but there are still several ways to make the colors on different faces match up. Look at a corner, where the colors are +-------+ /a.bbbbb/c\ /aaa.bbb/ccc\ /aaaaa.b/ccccc\ +-------+.......+ \fffff.e\ddddd/ \fff.eee\ddd/ \f.eeeee\d/ +-------+ That is, one corner is colored a/b, another c/d, and the third e/f, where I expect some of a,b,c,d,e,f will be the same color. One possibility was pictured in Hofstatder's Scientific American article of February, 1981. It had b=c,d=e,f=a and used twelve colors. Jim Saxe and I were impressed by its wasteful use of color and its failure to exhibit edge orientation. From your remarks about turning it over, I suspect this isn't what you mean. You may be talking about the cube in which a=d,b=e,c=f which uses six colors. I would say it is as if you cut an 'x' on a cube and exchanged each triangle with the other triangle on the same edge of the cube. That is a reasonably good coloring. It isn't really necessary to solve it twice, though. To find out whether a given corner goes on the top or bottom, look at the two colors that the corner shares with the top face center. Either the corner will have the two colors in the same order as the top, or they will be reversed, and that determines whether that corner goes on the top or bottom. That tells you where the third color on that corner goes, and the last color is determined by elimination. There is an even more interesting coloring that uses only four colors. In this coloring a=c=e and the other three colors are distinct. Jim Saxe and I came up with this coloring in our discussions of Hofstatder's article. It isn't quite symmetric enough, since its reflection is a coloring in which b=d=f, a slightly different pattern. Our discussions then led to the Tartan coloring we talked about in our article of 16 February 1981. The only cube in the archives called the Ultimate Cube is the one that has ``over 43 quintillion solutions.'' It has all six sides colored the same. Dan Hoey Hoey@AIC.NRL.Navy.Mil From pbeck@pica.army.mil Fri Jun 26 13:39:29 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA04500; Fri, 26 Jun 92 13:39:29 EDT Date: Fri, 26 Jun 92 13:36:42 EDT From: Peter Beck (BATDD) To: cube-lovers@ai.mit.edu Subject: ultimate cube Message-Id: <9206261336.aa13556@COR4.PICA.ARMY.MIL> name "ultimate cube" is a brand name or TM my description of its coloring is: still has six solid colors if we visualize a standard cube as a box with 6 stickers, 1 to each flat face the the ultimate cube has these 6 stickers translated and rotated so that 4 of these squares come together on a center cubie with the diagonal of each square laying along the edge of the cube. From mouse@lightning.mcrcim.mcgill.edu Fri Jun 26 20:06:15 1992 Return-Path: Received: from Lightning.McRCIM.McGill.EDU by life.ai.mit.edu (4.1/AI-4.10) id AA17340; Fri, 26 Jun 92 20:06:15 EDT Received: by Lightning.McRCIM.McGill.EDU (5.65) id <9206270006.AA06976@Lightning.McRCIM.McGill.EDU>; Fri, 26 Jun 92 20:06:06 -0400 Date: Fri, 26 Jun 92 20:06:06 -0400 From: der Mouse Message-Id: <9206270006.AA06976@Lightning.McRCIM.McGill.EDU> To: cube-lovers@ai.mit.edu Subject: Re: ultimate cube > my description of ["ultimate" cube's] coloring is: > still has six solid colors > if we visualize a standard cube as a box with 6 stickers, 1 to each > flat face the the ultimate cube has these 6 stickers translated and > rotated so that 4 of these squares come together on a center cubie > with the diagonal of each square laying along the edge of the cube. Hm, there are 12 edges on a cube. That leaves half of them unaccounted for. What do they get? (Note that you also have to shrink the 6 stickers, beacuse the face diagonal is longer than an edge.) der Mouse mouse@larry.mcrcim.mcgill.edu From pbeck@pica.army.mil Tue Jun 30 10:05:29 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA10327; Tue, 30 Jun 92 10:05:29 EDT Received: by COR4.PICA.ARMY.MIL id aa22053; 30 Jun 92 7:56 EDT Date: Tue, 30 Jun 92 7:46:56 EDT From: Peter Beck (BATDD) To: cube-lovers@ai.mit.edu Subject: ultimate cube, correction Message-Id: <9206300746.aa20380@COR4.PICA.ARMY.MIL> ULTIMATE CUBE - sorry for my past misinformation this time I will try and describe it by observation. The cube is covered with 24 (4 to a face) 45, 45, 90 deg triangles. These triangles have there hypotenuse along the edge of the cube and their 90 deg apex at the center of the center cubie. Each opposite face has the same coloring except that the rotation of the colors is opposite. For example if the front face has a green, orange, yellow and red triangle in clockwise order then in order for the rear face to correspond it has a color rotation that is counter clockwise. The coloring scheme is that the top and bottom triangles on the side faces (ie, front, right, left, back) are the same. Six colors ( green, orange, yellow , red for the front and rear, and green, white,yellow,blue for the sides and white, orange, blue, red for the top and bottom) are used and there are 4 triangles of each color. I think this is an accurate description, if there are questions please ask I have my cube at my desk. From tjj@rolf.helsinki.fi Tue Jun 30 15:36:42 1992 Return-Path: Received: from funet.fi by life.ai.mit.edu (4.1/AI-4.10) id AA22260; Tue, 30 Jun 92 15:36:42 EDT Received: from rolf.Helsinki.FI by funet.fi with SMTP (PP) id <03302-0@funet.fi>; Tue, 30 Jun 1992 22:35:13 +0300 Received: by rolf.helsinki.fi (5.57/Ultrix3.0-C) id AA10177; Tue, 30 Jun 92 22:34:53 +0300 Date: Tue, 30 Jun 92 22:34:53 +0300 From: tjj@rolf.helsinki.fi (Timo Jokitalo) Message-Id: <9206301934.AA10177@rolf.helsinki.fi> To: cube-lovers@ai.mit.edu Subject: Please, quick, I need the address of the new puzzle shop in Amsterdam I believe it was with the Dutch Cubists' Club newletter that I got an advertisement of a new puzzle shop in Amsterdam. I tried to find this advertisement, not, but could not. I'm leaving Finland on Thursday afternoon, and will be passing through Amsterdam, so I would be forever grateful to any kind soul who would send me the address!!! Thanks, Timo (tjj@rolf.helsinki.fi) From tjj@rolf.helsinki.fi Tue Jun 30 21:15:05 1992 Return-Path: Received: from funet.fi by life.ai.mit.edu (4.1/AI-4.10) id AA02935; Tue, 30 Jun 92 21:15:05 EDT Received: from rolf.Helsinki.FI by funet.fi with SMTP (PP) id <03311-0@funet.fi>; Tue, 30 Jun 1992 22:36:55 +0300 Received: by rolf.helsinki.fi (5.57/Ultrix3.0-C) id AA10181; Tue, 30 Jun 92 22:36:37 +0300 Date: Tue, 30 Jun 92 22:36:37 +0300 From: tjj@rolf.helsinki.fi (Timo Jokitalo) Message-Id: <9206301936.AA10181@rolf.helsinki.fi> To: cube-lovers@ai.mit.edu In the mail I just sent, there were a couple of serious typing errors, but I think the gist of the message should be clear... sorry! Timo From news@cco.caltech.edu Thu Jul 9 19:28:24 1992 Return-Path: Received: from gap.cco.caltech.edu by life.ai.mit.edu (4.1/AI-4.10) id AA03141; Thu, 9 Jul 92 19:28:24 EDT Received: by gap.cco.caltech.edu (4.1/1.34.1) id AA17055; Thu, 9 Jul 92 13:17:25 PDT Newsgroups: mlist.cube-lovers Path: nntp-server.caltech.edu!ph From: ph@vortex.ama.caltech.edu (Paul Hardy) Subject: Re: Name query. In-Reply-To: ACW@riverside.scrc.symbolics.com's message of Thu, 1 Jan 1970 00: 00:00 GMT Message-Id: Sender: news@cco.caltech.edu Nntp-Posting-Host: ama.caltech.edu Organization: California Institute of Technology References: <9206112052.AA18593@strident.think.com> <19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM> Distribution: mlist Date: Thu, 9 Jul 1992 21:11:57 GMT Apparently-To: mlist-cube-lovers@nntp-server.caltech.edu In article <19920611213942.5.ACW@PALLANDO.SCRC.Symbolics.COM> ACW@riverside.scrc.symbolics.com (Allan C. Wechsler) writes: > While I'm reminiscing, I should confess that my standard corner operator > is still the same as it was then: (FUR)^5, which exchanges two corners, > leaves the rest of the corners alone, and fucks the edges completely. > (Prudes, do not hassle me. This has been a technical term in cubing > around MIT since The Beginning.) Because of this property of "furry > five", I have to home and orient all the corners first, before I touch > the edges. It's the kind of quirky algorithm you don't see among > younger cubers, because everybody these days learns how to solve the > thing from a book. In the Beginning, there were no books, and I proudly > state that I solved the cube from scratch, by brainpower. Later I > discovered that there were easier ways to do things than (FR)^105! I > had pages and pages covered with little cube diagrams with arrows > showing how the stickers were permuted by a particular sequence. > > I'm interested in hearing other reminiscences from people who actually > solved the cube -- you're disqualified if you learned how to solve it > from somebody else, or from a book. I also solved the cube alone at first. I solved the top and middle first, then spent some time pondering the final face. I realized that manipulating the corners was trickier than the edges because there were three faces rather than two, so I solved the bottom corners and then got the bottom edges in place. I eventually got Singmaster's book, and found that my method of solving two layers was faster than his. I don't quite remember now, but I think it was because I had found a quick method for flipping a piece on the middle edge around if necessary (i.e., if it was in the correct position but flipped the wrong way) without disturbing anything else on the top or middle of the cube. Still, Singmaster's book had many patterns that were fun to go through and see evolve. I've long since lost my copy of Singmaster's book (one move too many); is it still available? --Paul -- This is my address: ph@ama.caltech.edu This is UUCP: ...!{decwrl,uunet}! This is my address on UUCP: ...!{decwrl,uunet}!caltech.edu!ama!ph Any questions? "Does Emacs have the Buddha nature?" --Paul Hardy "Yow!" --Zippy From alan@ai.mit.edu Sun Jul 19 10:51:55 1992 Return-Path: Received: from august (august.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA08560; Sun, 19 Jul 92 10:51:55 EDT Received: by august (4.1/AI-4.10) id AA14998; Sun, 19 Jul 92 10:52:45 EDT Date: Sun, 19 Jul 92 10:52:45 EDT Message-Id: <9207191452.AA14998@august> From: Alan Bawden Sender: Alan@lcs.mit.edu To: Cube-Lovers Subject: [nick@lcs.mit.edu: In-sol-u-bil-i-tyyyy!] Date: Thu, 16 Jul 92 13:11 EDT From: nick@lcs.mit.edu Reply-To: nick@lcs.mit.edu Subject: In-sol-u-bil-i-tyyyy! To: qotd@ghoti.lcs.mit.edu In a wonderful article about Claude Shannon in the April 92 IEEE Spectrum, a few lines from his poem called "a Rubric on Rubik's Cubics" (to the tune of Ta-ra-ra-boom-de-ay): Respect your cube and keep it clean, Lube your cube with Vaseline. Beware the dreaded cubist's thumb, the calloused hands and fingers numb. No borrower nor lender be, Rude folk might switch two tabs on thee. The most unkindest switch of all, Into Insolubility. [Chorus] In-sol-u-bil-i-ty! The strangest place to be However you persist Solutions don't exist! From @mail.uunet.ca:mark.longridge@canrem.com Mon Aug 3 02:57:03 1992 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA13101; Mon, 3 Aug 92 02:57:03 EDT Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <9679>; Mon, 3 Aug 1992 02:46:15 -0400 Received: from canrem.com by unixbox.canrem.COM id aa15064; Mon, 3 Aug 92 2:39:20 EDT Received: by canrem.com (PCB/Usenet Gateway) Path-id <19923.104.88888@dosgate>; 3 Aug 92 (02:30) Message-Id: <19923.104.88888@dosgate> From: Mark Longridge Date: Sun, 2 Aug 1992 20:00:00 -0400 To: cube-lovers@ai.mit.edu Subject: cube theory I've been doing some research to try and figure out some things about the cube. I've also tried (unsuccessfully) to develop a sort of CRC or checksum for the cube. With this cube "signature" I could then find out which depth each pattern requires. I'm puzzled how some computer types managed to find God's Algorithm for the 3x3x3 squares group. How do you keep track of all the patterns without repeating yourself? If it is by holding all patterns in an array the array must become huge. Maximum Depth (using q and h moves) ------------- 2x2x2 sq group 4 (24 total states) Pyraminx 11 (or 14 including the 3 tips) 2x2x2 11 (14 using q turns only) 3x3x3 corners only 11 3x3x3 sq group 15 (half turns only, don't know if using q improves this) 3x3x2 domino 18 (for 1 solution) A local maxima is a state where any possible move will bring you closer to a solution. This can occur on the 2x2x2 at depth 4 and on the 3x3x3 at depth 3. Note that all possible patterns at maximum depth are local maxima, however it is surprising that local maxima may occur in patterns much closer to the surface. To date, no work has been done to determine the depth of the dodecahedron (megaminx) or square 1. Some questions: What pattern is an example of local maxima? e.g. 3x3x3 at depth 3 -> 12-flip, 12-flip 8-twist q+h Depth Patterns 2x2x2 1 9 3x3x3 1 18 Dodecahedron 1 48 Analysis of the full cube group ------------------------------- Moves Deep arrangements (q+h) arrangements (q only) * 0 1 1 1 18 12 2 243 114 3 3,240 1,068 4 > 48,600 10,011 * Work by Zoltan Kaufmann Notes: At 1 move deep each of the 6 sides can turn 3 ways (+ - 2) giving 18 distinct patterns At 2 moves deep it is redundant to turn the same side again so 5 sides can turn 3 ways so 18x15=270 However, with opposite turns order is not significant, e.g. T,D = D,T F,B = B,F L,R = R,L since each of these can occur in 9 different ways there are 27 redundancies so 270 - 27 = 243 At 3 moves deep with the first 2 moves on opposite faces don't turn the face used in move one since: T,D,T = T2,D F,B,F = F2,B L,R,L = L2,R This can occur in 3x3x3=27 ways for each case so 81 are dropped (Remember the first 2 moves have already been weeded of redundancy!) Also when the 2nd and 3rd moves are of opposite faces e.g. T,R,L = T,L,R B,R,L = B,L,R F,R,L = F,L,R D,R,L = D,L,R T,B,F = T,F,B D,B,F = D,F,B L,B,F = L,F,B R,B,F = R,F,B F,T,D = F,D,T B,T,D = B,D,T L,T,D = L,D,T R,T,D = R,D,T since each of these can occur 27 different ways in each of the cases this gives 27x12 = 324 redundancies Thus 243x15 = 3645, removing the redundancies gives 3645-81-324=3240 At 4 moves deep.... still working on this one! Zoltan Kaufmann has done 4 moves deep using quarter turns, but has anyone calculated farther using q and h turns? I'd be interested in the source code of any programs people have written on finding path-lengths. Also what is an example of a local maxima close to the surface, e.g. 4 moves. I believe Jim Saxe and Dan Hoey have done some work in this regard. One more question: What is the maximum number of moves required if you do the 3x3x3 one face last? The best results I've seen are 19 q and h moves. -> Mark Longridge -- Canada Remote Systems - Toronto, Ontario/Detroit, MI World's Largest PCBOARD System - 416-629-7000/629-7044 From hoey@aic.nrl.navy.mil Mon Aug 3 11:10:35 1992 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA00431; Mon, 3 Aug 92 11:10:35 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA22147; Mon, 3 Aug 92 11:10:30 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Mon, 3 Aug 92 11:10:29 EDT Date: Mon, 3 Aug 92 11:10:29 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9208031510.AA03178@sun13.aic.nrl.navy.mil> O: Mark Longridge Cc: Cube-lovers@life.ai.mit.edu Subject: Re: cube theory Dear Mark, I sent you email on 8 June. Did you receive it? Are you interested in acquiring a Tartan cube? Did you ever get Minh Thai's book on Rubik's Revenge? With respect to the 3x3x3 squares group, > How do you keep track of all the patterns without repeating yourself? there are several ways. The most general is to use the Sims table (aka the FHL table) for the subgroup, which gives a mixed-base enumeration of the positions. See my message of 1 February 1981. > A local maxima is a state where any possible move will bring you > closer to a solution. That's ``local maximum''. ``Maxima'' is the plural of ``maximum''. > Note that all possible patterns at maximum depth are local > maxima, .... We call such positions ``global maxima'' because they are at the overall maximum depth. The statement is then that every global maximum is a local maximum. > To date, no work has been done to determine the depth of the > dodecahedron (megaminx).... Well, I've done some looking at it. Since my initial remarks on 23 September 1982, I've figured out a way to generate a recurrence for it, but it seems I haven't put it down anywhere. Are you interested? (Do I have to tell anyone to answer that question only to Hoey@AIC.NRL.Navy.Mil, not the list?) > What pattern is an example of local maxima? e.g. 3x3x3 at depth 3 > -> 12-flip, 12-flip 8-twist Jim Saxe and I listed the 25 symmetric local maxima in our message on Symmetry and Local Maxima, dated 14 December 1980. We verified Jim's conjecture that the four-spot is a local maximum, but not on the grounds of symmetry, and reported that on 22 March 1981. Do you have access to the cube-lovers archives? > Moves Deep arrangements (q+h) arrangements (q only) * > 0 1 1 > 1 18 12 > 2 243 114 > 3 3,240 1,068 > 4 > 48,600 10,011 This is in the archives, too 5 93,840 (22 March 1981) 6 878,880 (14 August 1981) 7 8,221,632 (7 December 1981) David C. Plummer and I had hoped to use his program (which counted the 7 QT positions) to extend this to 8 QT, but we got busy. I still have hopes.... > At 2 moves deep it is redundant to turn the same side again.... > However, with opposite turns order is not significant, e.g. T,D = D,T.... This approach appeared on 9 January 1981. It showed how to follow the argument to 25 QT, and to get what are still the best known lower bounds for the ordinary cube and for the supergroup. > Also what is an example of a local maxima close to the surface, e.g. > 4 moves. I believe Jim Saxe and Dan Hoey have done some work in this > regard. It's known there are no local maxima at 7 QT or less. The shortest known local maxima are Pons Asinorum and the four-spot, both at 12 QT. I know of no results between 8 and 11 QT. Dan Hoey Hoey@AIC.NRL.Navy.Mil From pbeck@pica.army.mil Mon Aug 10 14:13:31 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA14995; Mon, 10 Aug 92 14:13:31 EDT Date: Mon, 10 Aug 92 14:11:51 EDT From: Peter Beck (BATDD) To: CUBE-LOVERS@life.ai.mit.edu Subject: smart alex - new puzzle Message-Id: <9208101411.aa10693@COR4.PICA.ARMY.MIL> SMART ALEX about $14 retail from 2 MCH FUN 777-108th Avenue N.E. #2340 Bellevue, WA 98004 206-453-5659 purchase source PUZZLETTS MIKE GREEN 24843 144th Place S.E. KENT, WA 98042 206-630-1432 or from myself DESCRIPTION: This puzzle is similar to the Hungarian puzzle UFO but a little more complex. The puzzle has a cube at its center which is cut in half and these two pieces can rotate with respect to each other. In the plane of this cut there are four hubs that can rotate, 2 on the x-axis and 2 on the y-axis. These hubs have a hexagonal cross section and are divided into six equilateral pie shaped wedges. By rotating the hubs and then by rotating the center cube 3 hub pieces on each axle are moved at a time. The hub pieces have 2 colors each one perpendicular to the axis and the other on its edge. There are 2 wedges with the same coloring. The center cube is also colored. The object is to arrange the hub pieces so that the edge pieces align with the center cube and that the perpendicular sides are solid colored and aligned. From pbeck@pica.army.mil Mon Aug 10 16:22:55 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA19839; Mon, 10 Aug 92 16:22:55 EDT Date: Mon, 10 Aug 92 14:12:42 EDT From: Peter Beck (BATDD) To: CUBE-LOVERS@life.ai.mit.edu Subject: mazeland Message-Id: <9208101412.aa10952@COR4.PICA.ARMY.MIL> MAZELAND GARDENS and DISCOVERY CENTER POB 573 Alexandria Bay, NY 13607 315-482-LOST 800-585-FUNN Alexandria Bay is in the Thousand Islands region near where I-81 crosses into Canada. Mazeland Gardens is an entertainment center (the property was formerly a miniature golf course) of mazes. It has 2 mazes that are constructed with arborvitae hedges, one a 1/2 acre in size and the other a full acre. In addition it has 2 mazes that are constructed with stakes and colored tape to mark the walls. I went without kids and had fun. The families with kids looked like they were having a good time. If you are in the area you might want to give it a try. PS The fellow working the desk said that there is are 2 other mazes on the east coast that he is aware of: one is in North Carolina somewhere and the other is in Daytona beach Florida/ Pete Beck From hoey@aic.nrl.navy.mil Thu Aug 20 13:51:38 1992 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA16858; Thu, 20 Aug 92 13:51:38 EDT Received: from sun30.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA19935; Thu, 20 Aug 92 13:51:25 EDT Return-Path: Received: by sun30.aic.nrl.navy.mil; Thu, 20 Aug 92 13:51:25 EDT Date: Thu, 20 Aug 92 13:51:25 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9208201751.AA14111@sun30.aic.nrl.navy.mil> To: Allan C Wechsler , wft@math.canterbury.ac.nz (Bill Taylor) Cc: Cube-Lovers@ai.mit.edu Subject: Re: subgroups On 14 Jan 1992, Allan C. Wechsler posted >Regarding the meta-approach of descending through a series of subgroups, >how much leverage does properly selecting the chain give you? It seems >like the jump from to is pretty large. >There are probably other paths through the subgroup lattice. >Does anyone have a table of subgroups? As far as selecting the chain goes, I have been meaning to look into that a bit. Of course, since Bill posted, the results of Hans Kloosterman, Michael Reid, and Dik Winter have shown that you indeed get a lot of leverage. I would like to get some idea of the possible group towers, for a more general approach to selecting which towers give you leverage. But what I haven't been able to figure out is how to figure out which coset of G1 wrt G2 you're in. I've been able to figure it out for specific groups, but if we wanted to do this for a lot of chains, we would need to do coset identification given G1 and G2 as a table of strong generators. We could in fact ensure that the strong generators of G1 form a subset of those of G2. Is that a hard thing to do? More to the point, I've heard that the FHL algorithm should more properly be called Sims's algorithm and that Furst, Hopcroft, and Luks mostly analyzed the performance. I haven't read anything by Sims on it, though. Is there a good reference that treats this sort of algorithm in a more general setting? I have toyed with implementing the Jerrum improvements to FHL, but it is a mighty complicated beast. Also, a talk announced in the archives mentioned 1987 work by Akos Seress that was supposed to be an improvement, but I don't know whether it got published. Anyway, if not, do you know if there is a good general way of finding out which coset a given position is in. On 29 Jan 1992, wft@math.canterbury.ac.nz (Bill Taylor) posted > There hasn't been any response to this, seemingly, which is a pity. For some reason, I never saw Bill's message. I just noticed it when comparing my files against the archives. [ Archives seekers note: the archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in directory /pub/cube-lovers. ] > In any event, I would like to know of any other well-known subgroups. > There are the slice group; double-slice group; U group; square group; > anti-slice group. How many others are there not mentioned here, that > people know of ? There were some tables in Singmaster with more examples, and there are the stuck-faces groups that I wrote about on 21 July 1981. I seem to recall there was some non-obvious equivalence between two groups, perhaps the slice group and the antislice group. But a general list of popular subgroups would be interesting. Of course a list of *all* the subgroups would have, um, over three beelion of them. I suspect it has more than 4.3x10^19. Does anyone know a good way of counting how many subgroups there are? Or even a way of estimating the number? By comparison, the symmetries of the cube form a 48-element group with 98 subgroups. Dan Hoey Hoey@AIC.NRL.Navy.Mil From ACW@riverside.scrc.symbolics.com Thu Aug 20 16:24:45 1992 Return-Path: Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA22502; Thu, 20 Aug 92 16:24:45 EDT Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 888810; 20 Aug 1992 16:25:53-0400 Date: Thu, 20 Aug 1992 16:25-0400 From: Allan C. Wechsler Subject: Re: subgroups To: hoey@aic.nrl.navy.mil, ACW@riverside.scrc.symbolics.com, wft@math.canterbury.ac.nz Cc: Cube-Lovers@ai.mit.edu In-Reply-To: <9208201751.AA14111@sun30.aic.nrl.navy.mil> Message-Id: <19920820202540.7.ACW@PALLANDO.SCRC.Symbolics.COM> Date: Thu, 20 Aug 1992 13:51 EDT From: hoey@aic.nrl.navy.mil [...] Of course a list of *all* the subgroups would have, um, over three beelion of them. I suspect it has more than 4.3x10^19. Does anyone know a good way of counting how many subgroups there are? Or even a way of estimating the number? By comparison, the symmetries of the cube form a 48-element group with 98 subgroups. All we should really be interested in are conjugate classes of subgroups. I think. From alan@ai.mit.edu Thu Aug 20 20:03:23 1992 Return-Path: Received: from transit (transit.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA29289; Thu, 20 Aug 92 20:03:23 EDT Received: by transit (4.1/AI-4.10) id AA15643; Thu, 20 Aug 92 20:06:49 EDT Date: Thu, 20 Aug 92 20:06:49 EDT Message-Id: <9208210006.AA15643@transit> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: Cube-Lovers Subject: Archives Date: Thu, 20 Aug 92 13:51:25 EDT From: hoey@aic.nrl.navy.mil ... [ Archives seekers note: the archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in directory /pub/cube-lovers. ] ... No, that isn't right. The correct address is FTP.AI.MIT.EDU (which is at 128.52.32.11 -- at least this week). Here is the text I currently send to people who are new subscribers or who express interest in the archives: Using FTP, connect to FTP.AI.MIT.EDU, login as "anonymous" (any password), and in the directory "pub/cube-lovers" you will find the nine (compressed) files "cube-mail-0.Z" through "cube-mail-8.Z". Archive vital statistics (when uncompressed): File From To Size (bytes) ---- ---- -- ------------ cube-mail-0 12 Jul 80 23 Oct 80 185037 cube-mail-1 3 Nov 80 9 Jan 81 135719 cube-mail-2 10 Jan 81 3 Aug 81 138566 cube-mail-3 3 Aug 81 3 May 82 137753 cube-mail-4 4 May 81 11 Dec 82 139660 cube-mail-5 11 Dec 82 6 Jan 87 173364 cube-mail-6 10 Jan 87 13 Apr 90 216733 cube-mail-7 12 Oct 90 9 Sep 91 137508 cube-mail-8 1 Nov 91 25 May 92 171205 In addition, the file "recent-mail" contains a copy of the currently active section of the archive. (Unfortunately, due to the way mail works here at the AI Lab, it is not possible to have new mail accumulate directly into this file, so there may be some delay before a new message arrives here.) - Alan From reid@math.berkeley.edu Thu Aug 20 20:10:13 1992 Return-Path: Received: from math.berkeley.edu by life.ai.mit.edu (4.1/AI-4.10) id AA29497; Thu, 20 Aug 92 20:10:13 EDT Received: from phnom-penh.berkeley.edu.berkeley.edu by math.berkeley.edu (4.1/1.33(math)) id AA19361; Thu, 20 Aug 92 17:03:31 PDT Date: Thu, 20 Aug 92 17:03:31 PDT From: reid@math.berkeley.edu (michael reid) Message-Id: <9208210003.AA19361@math.berkeley.edu> To: ACW@riverside.scrc.symbolics.com, hoey@aic.nrl.navy.mil, wft@math.canterbury.ac.nz Subject: Re: subgroups Cc: Cube-Lovers@ai.mit.edu dan hoey writes: > On 14 Jan 1992, Allan C. Wechsler posted > >Regarding the meta-approach of descending through a series of subgroups, > >how much leverage does properly selecting the chain give you? It seems > >like the jump from to is pretty large. > >There are probably other paths through the subgroup lattice. > >Does anyone have a table of subgroups? well, i don't know ALL the subgroups, but i did some investigation before devising my three stage algorithm. one of the great advantages of thistlethwaite's four stage method is that since each subgroup restricts the motion of various faces, it is routine to exhaustively search the cosets spaces at each stage, since we only make twists that leave us in the given space. so i looked at all possible ways of restricting various faces, up to symmetry. there are three possible restrictions for a face: no restriction, half turns only, no turns. our problem is then coloring the faces of the cube with 3 colors, up to symmetry (rigid and non-rigid). the polya polynomial for the faces of the cube under this group of symmetries is: ( x^6 + 3 x^5 + 9 x^4 + 13 x^3 + 14 x^2 + 8 x ) / 48 so there are 56 different ways to three-color the faces. i spent the better part of an evening and most of the night calculating (by hand) the orders of these subgroups. shortly thereafter, i saw an announcement for the group theory package GAP, which specifically mentions calculating the order of the rubik's cube group. so i used the package to verify my answers. here's the list (i don't see a canonical way of ordering them): 1. |<>| = 1 2. || = 2 = 2 3. || = 2^2 = 4 4. || = 2^2 = 4 5. || = 2^3 = 8 6. || = 2^4 = 16 7. || = 2 3 = 12 8. || = 2^6 3^2 5^2 = 14400 9. || = 2^6 3^8 5^2 7 = 73483200 10. || = 2^5 3 = 96 11. || = 2^12 3^4 5^2 7 = 58060800 12. || = 2^12 3^4 5^2 7 = 58060800 13. || = 2^14 3^4 5^2 7^2 = 1625702400 14. || = 2^14 3^11 5^2 7^2 = 3555411148800 15. || = 2^14 3^13 5^3 7^2 = 159993501696000 16. || = 2^5 3^4 = 2592 17. || = 2^8 3^5 5^2 7 = 10886400 18. || = 2^10 3^12 5^2 7^2 = 666639590400 19. || = 2^18 3^12 5^2 7^2 = 170659735142400 20. || = 2^6 3 = 192 21. || = 2^13 3^4 5^2 7 = 116121600 22. || = 2^15 3^4 5^2 7^2 = 3251404800 23. || = 2^15 3^11 5^2 7^2 = 7110822297600 24. || = 2^15 3^13 5^3 7^2 = 319987003392000 25. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 26. || = 2^11 3^4 = 165888 27. || = 2^13 3^5 5^2 7^2 = 2438553600 28. || = 2^14 3^5 5^2 7^2 = 4877107200 29. || = 2^14 3^5 5^2 7^2 = 4877107200 30. || = 2^14 3^13 5^3 7^2 11 = 1759928518656000 31. || = 2^14 3^13 5^3 7^2 11 = 1759928518656000 32. || = 2^14 3^13 5^3 7^2 11 = 1759928518656000 33. || = 2^24 3^13 5^3 7^2 11 = 1802166803103744000 34. || = 2^24 3^13 5^3 7^2 11 = 1802166803103744000 35. || = 2^13 3^4 = 663552 36. || = 2^16 3^5 5^2 7^2 = 19508428800 37. || = 2^16 3^5 5^2 7^2 = 19508428800 38. || = 2^16 3^5 5^2 7^2 = 19508428800 39. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 40. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 41. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 42. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 43. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 44. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 45. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 46. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 47. || = 2^13 3^4 = 663552 48. || = 2^16 3^5 5^2 7^2 = 19508428800 49. || = 2^16 3^5 5^2 7^2 = 19508428800 50. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 51. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 52. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 53. || = 2^16 3^14 5^3 7^2 11 = 21119142223872000 54. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 55. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 56. || = 2^27 3^14 5^3 7^2 11 = 43252003274489856000 subgroups with the same order are equal (possibly after necessary rotation of the cube) with the following exceptions: (3, 4), (11, 12), (30, 31) and (30, 32). equality of various pairs of subgroups can be obtained from the three maneuvers: R L F2 R2 F B L F2 B2 R2 F2 B2 L F B3 R3 L3 ~ U2 , so that = , F2 U2 L2 F2 R2 U2 F2 R F2 U2 R2 F2 L2 U2 F2 ~ L , so that = and R2 F2 B2 L2 U2 L2 F2 B2 R2 ~ D2 , so that = . thistlethwaite's filtration is 56 --> 53 --> 49 --> 47 --> 1. kloosterman replaced 47 by a subgroup not on this list (one not obtained by restricting face turns). call this 56 --> 53 --> 49 --> kl --> 1. (in his final stage, kloosterman allows all twists available in the subgroup 49.) my filtration is 56 --> 19 --> 17 --> 1 , which was chosen precisely because it had the smallest size of the largest coset space amongst all three stage filtrations with subgroups from the above. winter's filtration is 56 --> 49 --> kl --> 1. it may be the case that this can be improved by replacing kl with 17 , and allowing all face turns available in the subgroup 49. i haven't had the time to look into this yet. using subgroups on the list above, we see that the only reasonable two stage filtrations are: 56 --> 29 --> 1 with coset sizes 8868372480 and 4877107200 56 --> 22 --> 1 with coset sizes 13302558720 and 3251404800 56 --> 27 --> 1 with coset sizes 17736744960 and 2438553600 56 --> 49 --> 1 with coset sizes 2217093120 and 19508428800 56 --> 13 --> 1 with coset sizes 26605117440 and 1625702400 of these, the best seems to be 56 --> 49 --> 1 , since it has the most symmetries (16). the number of symmetries the others have is 4 (for 29), 8 (for 22), 2 (for 27) and 2 (for 13). furthermore, aside from subgroup 49, the other intermediate groups seem to have too much restriction to be efficient. also, of course, dik winter has already calculated that the stage 56 --> 49 can always be accomplished in 12 face turns. > On 29 Jan 1992, wft@math.canterbury.ac.nz (Bill Taylor) posted > > There hasn't been any response to this, seemingly, which is a pity. > For some reason, I never saw Bill's message. I just noticed it when > comparing my files against the archives. [ Archives seekers note: the > archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in > directory /pub/cube-lovers. ] i also seem to have missed both allen's post as well as bill's reply. perhaps 'twas the twilight zone between the start of my subscription to cube-lovers and the time it takes recent messages to reach the archives. however, i don't find the archives on ftp.lcs , but rather on ftp.ai.mit.edu. also i see we've spawned cube-mail-8.Z. > > In any event, I would like to know of any other well-known subgroups. > > There are the slice group; double-slice group; U group; square group; > > anti-slice group. How many others are there not mentioned here, that > > people know of ? in addition to those listed above there are subgroups generated by combinations of face turns and slice turns, e.g. , , , etc. i haven't looked at these at all. there's a lot of work to be done here. mike From ronnie@cisco.com Thu Aug 20 23:32:46 1992 Return-Path: Received: from ale.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA04659; Thu, 20 Aug 92 23:32:46 EDT Received: by ale.cisco.com; Thu, 20 Aug 92 17:52:40 -0700 Date: Thu, 20 Aug 92 17:52:40 -0700 From: Ronnie B. Kon Message-Id: <9208210052.AA00306@ale.cisco.com> To: Cube-Lovers@life.ai.mit.edu Subject: Back to the 16th century In the Cube Lovers' archives we have files: From To Size Time Bytes/Month ---- -- ---- ------ ----------- 12 Jul 80 23 Oct 80 185037 3 months 61679 3 Nov 80 9 Jan 81 135719 2 months 67860 10 Jan 81 3 Aug 81 138566 6 months 23094 3 Aug 81 3 May 82 137753 9 months 15306 4 May 81 11 Dec 82 139660 19 months 7351 11 Dec 82 6 Jan 87 173364 48 months 3612 10 Jan 87 13 Apr 90 216733 39 months 5557 12 Oct 90 9 Sep 91 137508 12 months 11459 1 Nov 91 25 May 92 171205 7 months 24458 Ladies and Gentlemen, I believe we are witnessing a Renaissance of Cubing! Ronnie From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 1 22:09:24 1992 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA14825; Tue, 1 Sep 92 22:09:24 EDT Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10871>; Tue, 1 Sep 1992 22:09:02 -0400 Received: from canrem.com by unixbox.canrem.COM id aa23354; Tue, 1 Sep 92 20:43:35 EDT Received: by canrem.com (PCB/Usenet Gateway) Path-id <1992Sep1.104.99639@dosgate>; 1 Sep 92 (19:39) Message-Id: <1992Sep1.104.99639@dosgate> From: Mark Longridge Date: Mon, 31 Aug 1992 20:00:00 -0400 To: cube-lovers@ai.mit.edu Subject: rare variants Hello fellow cube lovers... Although this is not a strict Rubik's cube type post I think it should be of some interest to the other subscribers. I collect cube variants, and although the variations in cube colours are interesting my great interest is variants in mechanisms (which require different solving techniques). I've been corresponding with other cube buffs around the world in an effort to record ALL the significant cube variants and I use the following classification system: M = Manufactured in quantity, readily available S = Produced in small quantities only R = Rare, a few prototypes exist, difficult to get P = Prototype, the inventor has the only one! C = Exists only as a computer simulation and/or cardboard mockup I = Intellectual idea only (perhaps on pencil and paper) In my opinion, Square 1 is the most interesting cube variant in recent years, and it gave me the most trouble! Here are some of the tough ones to get, and if anyone knows of any others email me and I'll maintain a list of them... Trajber's Octahedron (R) Evidently Greg Stevens owns one Octahedral puzzle with rotating faces Extended Missing Link (S) Missing Link with 6 tiers Master Pyraminx (P??) Looks like a normal pyraminx BUT it's edges can turn (just the strip) 180 degrees and 2 vertices can be swapped Space Grenade (P??) Other weird one from Uwe Meffert. Mike Green of Puzzletts showed me a picture of this, still not sure how it moves Pyraminx Disc Chess (S) Planar puzzle with 6 rotating discs, similar to Raba's Rotoscope Masterball (S) This seems to be a recent one, it's like a VIP sphere but it has 8 vertical cuts instead of one (like a tangerine) and 4 hortizontal sections Halpern's Tetrahedron (P) Also called Pyraminx Tetrahedron Like a pyraminx BUT it has face centres which are small triangles and it's faces rotate. Very rare. Pyraminx Hexagon (C) Jerry Slocum says he got a cardboard mockup of this from Meffert. I wrote a computer simulation of it. Imagine a Rubik's cube with an N-prism shape, thus the top and bottom are hexagons, and there are 6 (rather than 4) adjacent sides. The top and bottom can rotate 60 degrees and the adjacent sides can only rotate 180 degrees. Twist Torus (I) My own concept. Imagine a torus segmented 4 ways length-wise so it can slide around. Additionally there are 12 rings around the circumference which can rotate at right angles to the segments. Still thinking of a good colour arrangement for this one. Super Skewb (I) Another idea of mine. It's a skewb and a 2x2x2 cube! A compound of two mechanisms. Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus (Anyone got multiples of these) Imagine my disappointment when I found out the Mach Ball, Skewb and Moody Ball all have the same basic mechanism! Anyway if anyone has a rare variant or puzzle idea please post here or email me... Mark Longridge 259 Thornton Rd N Oshawa Ontario Canada L1J 6T2 Email: mark.longridge@canrem.com -- Canada Remote Systems - Toronto, Ontario/Detroit, MI World's Largest PCBOARD System - 416-629-7000/629-7044 From diamond@jit081.enet.dec.com Tue Sep 1 23:58:01 1992 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA19112; Tue, 1 Sep 92 23:58:01 EDT Received: by enet-gw.pa.dec.com; id AA21950; Tue, 1 Sep 92 20:57:51 -0700 Message-Id: <9209020357.AA21950@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Tue, 1 Sep 92 20:58:00 PDT Date: Tue, 1 Sep 92 20:58:00 PDT From: 02-Sep-1992 1249 To: cube-lovers@ai.mit.edu Apparently-To: cube-lovers@ai.mit.edu Subject: Re: rare variants Mr. Longridge! Please post the addresses of where to buy that stuff! Oh, I'm drooling all over my keyboard. Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name of the city but it's near Nurnberg, Germany) makes lots of variations on cubes and skewbs etc., but the underlying mechanisms all seem to be standard ones. > Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus > (Anyone got multiples of these) Hmmmm. I might confess to owning two Trick Hauses in exchange for something of sufficient persuasion, such as some of the others listed in Mr. Longridge's post. In hopes of being able to exchange for some neat stuff, I'd better not mention that Mr. Constantin can sell Trick Hauses for something around DM 20. >Imagine my disappointment when I found out the Mach Ball, Skewb and >Moody Ball all have the same basic mechanism! Yeah, but with Mach Ball you have to orient the square-like pieces. Haven't seen Moody Ball. -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From ronnie@cisco.com Wed Sep 2 00:50:42 1992 Return-Path: Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA20803; Wed, 2 Sep 92 00:50:42 EDT Received: from lager.cisco.com by wolf.cisco.com with TCP; Tue, 1 Sep 92 21:50:36 -0700 Message-Id: <9209020450.AA01685@wolf.cisco.com> To: 02-Sep-1992 1249 Cc: cube-lovers@ai.mit.edu Subject: Re: rare variants In-Reply-To: Your message of "Tue, 01 Sep 92 20:58:00 PDT." <9209020357.AA21950@enet-gw.pa.dec.com> Date: Tue, 01 Sep 92 21:50:35 PDT From: "Ronnie B. Kon" > Mr. Longridge! Please post the addresses of where to buy that stuff! > Oh, I'm drooling all over my keyboard. > > Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name > of the city but it's near Nurnberg, Germany) makes lots of variations on > cubes and skewbs etc., but the underlying mechanisms all seem to be standard > ones. > > > Honourable mention: Pyraminx Star, Puck, Ufo, Trick Haus > > (Anyone got multiples of these) > > Hmmmm. I might confess to owning two Trick Hauses in exchange for something > of sufficient persuasion, such as some of the others listed in Mr. Longridge' s > post. In hopes of being able to exchange for some neat stuff, I'd better not > mention that Mr. Constantin can sell Trick Hauses for something around DM 20. Unless the the Bundespost will deliver mail to "a city near Nurnberg" I think you're pretty safe. Unless of course you'll post the full address? Ronnie From diamond@jit081.enet.dec.com Wed Sep 2 00:59:23 1992 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA20846; Wed, 2 Sep 92 00:59:23 EDT Received: by enet-gw.pa.dec.com; id AA24605; Tue, 1 Sep 92 21:59:05 -0700 Message-Id: <9209020459.AA24605@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Tue, 1 Sep 92 21:59:19 PDT Date: Tue, 1 Sep 92 21:59:19 PDT From: 02-Sep-1992 1356 To: cube-lovers@ai.mit.edu Cc: ronnie@cisco.com Apparently-To: ronnie@cisco.com, cube-lovers@ai.mit.edu Subject: Re: rare variants >> Meanwhile, Jean-Claude Constantin (Pirkach 14, D-8535 ah I forgot the name >> of the city but it's near Nurnberg, Germany) makes lots of variations on ronnie@cisco.com writes: >Unless the the Bundespost will deliver mail to "a city near Nurnberg" I think >you're pretty safe. Unless of course you'll post the full address? I was telling the truth -- I forgot the name of the city while in the middle of typing. Meanwhile, I had posted the full address on rec.puzzles a few months ago. Just now, I have recalled the name of the city, and here is the full address (if I don't forget again while typing :-) Constantin Geduldspiele Pirkach 14 D-8535 Emskirchen Germany -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From Don.Woods@eng.sun.com Sun Sep 6 14:40:52 1992 Return-Path: Received: from Sun.COM by life.ai.mit.edu (4.1/AI-4.10) id AA03843; Sun, 6 Sep 92 14:40:52 EDT Received: from Eng.Sun.COM (zigzag-bb.Corp.Sun.COM) by Sun.COM (4.1/SMI-4.1) id AA15176; Sun, 6 Sep 92 11:40:50 PDT Received: from colossal.Eng.Sun.COM by Eng.Sun.COM (4.1/SMI-4.1) id AA20580; Sun, 6 Sep 92 11:40:53 PDT Received: by colossal.Eng.Sun.COM (4.1/SMI-4.1) id AA28174; Sun, 6 Sep 92 11:42:34 PDT Date: Sun, 6 Sep 92 11:42:34 PDT From: Don.Woods@eng.sun.com (Don Woods) Message-Id: <9209061842.AA28174@colossal.Eng.Sun.COM> To: cube-lovers@ai.mit.edu, mark.longridge@canrem.com Subject: Re: rare variants I have a puzzle not in your list; it's rare enough that I've never seen it in a store or catalog. I got mine from a friend. He calls it "The Barrel". Imagine a transparent cylinder divided into 6 circular slices. Slices 2-5 each have 5 pockets equally spaced around the circumference, just below the surface (see left figure below). Slices 1 and 6 have three pockets in positions corresponding to 3 of the 5 (see right figure). ******* ******* **** **** **** **** ***** ***** ***** ***** ****** ****** ****** ****** ***************** ***************** ******************* ******************* ******************* ******************* * ************* * ********************* * ************* * ********************* * ************* * ********************* ********************* ********************* ********************* ********************* ********************* ********************* ********************* ********************* ******************* ******************* *** ******* *** *** ******* *** * ******* * * ******* * * ********* * * ********* * ************* ************* *********** *********** ******* ******* Through the center of the cylinder runs a piece with a cap on each end. The caps each have 3 prongs poking into the cylinder, lined up on the 3 openings in slices 1 and 6. However, the central piece is long enough that if the prongs are pushed into slice 6, the prongs at the other end are lifted out of slice 1, and vice versa. So, at any given time, three of the end pockets are filled by one of the endcaps. The other 23 pockets contain colored balls. Originally, the 3 balls in the unpronged endcap are black, and the balls in the other slices are lined up by color; i.e. 4 blue balls lined up above one another in slices 2-5, 4 green balls, likewise lined up, etc. The possible moves are: 1) Slide the endcaps up and down. E.g., from the starting position, this would push three balls of different colors into slice 6, and push the 3 black balls from slice 1 into slice 2 (and also push various balls of the same color down, but that has no visible effect). 2) Turn slices 2 and 3 together; they do not move separately. 3) Turn slices 4 and 5 together; ditto. Slices 1 and 6 are fixed, so they always line up with the end cap prongs and with each other. That's all there is to it. It certainly has the "cubish" feel to me, in that it's impossible to make single moves that affect only a small portion of the puzzle. -- Don. From diamond@jit081.enet.dec.com Sun Sep 6 20:11:45 1992 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA08950; Sun, 6 Sep 92 20:11:45 EDT Received: by enet-gw.pa.dec.com; id AA29557; Sun, 6 Sep 92 17:11:40 -0700 Message-Id: <9209070011.AA29557@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Sun, 6 Sep 92 17:11:44 PDT Date: Sun, 6 Sep 92 17:11:44 PDT From: 07-Sep-1992 0906 To: cube-lovers@ai.mit.edu Apparently-To: cube-lovers@ai.mit.edu Subject: Re: rare variants Don.Woods@eng.sun.com writes about the barrel puzzle. If I recall correctly, this was made by Nintendo before they switched to computer games. I also believe the name was "Ten Billion Barrel." (Mine is buried somewhere and I couldn't solve it, but I didn't want to buy the book that was published at one time, sigh... I didn't need a book for the cube, so why should I cheat for a piddly little barrel... sigh.) Anyway, there are still a few available. But I have to warn, if anyone wants one, it will cost more for postage and for my train fare going to the store, than to buy the thing. If anyone wants one, we can arrange it by e-mail. But I'd really prefer to trade for some of those wonderful things that Mr. Longridge described. (I'm drooling all over my keyboard again, just remembering them.) -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 15 17:14:06 1992 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA11012; Tue, 15 Sep 92 17:14:06 EDT Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10252>; Tue, 15 Sep 1992 17:13:51 -0400 Received: from canrem.com by unixbox.canrem.COM id aa27586; Tue, 15 Sep 92 17:06:45 EDT Received: by canrem.com (PCB/Usenet Gateway) Path-id <199215.104.106005@dosgate>; 15 Sep 92 (16:56) Message-Id: <199215.104.106005@dosgate> From: Mark Longridge Date: Mon, 14 Sep 1992 20:00:00 -0400 To: cube-lovers@ai.mit.edu Subject: pyraminx revisited Notes on the Pyraminx --------------------- According to Dr. Ronald Turner-Smith there are 5 different Pyraminx puzzles, all of different complexity. The following are listed in order from easiest to hardest (to the best of my knowledge): Pyraminx Star: Easiest of all pyraminx?? A simplification of the popular pyraminx because of the little uni-coloured (usually grey or silver) tetrahedrons on the 3 middle pieces of each face. Effectively all middle pieces on this pyraminx are the same colour! Snub Pyraminx: Same as standard pyraminx with tips removed Popular Pyraminx: The standard pyraminx of which appeared in vast quanities after the cube caught on. Senior Pyraminx: This is a mystery puzzle. No one seems to know anything about it, yet Turner-Smith's book refers to it and gives the maximum number of moves for it! It is between the Popular Pyraminx and Master Pyraminx in difficulty. Master Pyraminx: All the moves of the standard pyraminx plus 180 degree turns of the edges (just the strip, not the whole face) 446,965,972,992,000 combinations. Interestingly in -- Canada Remote Systems - Toronto, Ontario, Canadas World's Largest PCBOARD System - 416-629-7000/629-7044 From @mail.uunet.ca:mark.longridge@canrem.com Tue Sep 15 18:37:59 1992 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA14355; Tue, 15 Sep 92 18:37:59 EDT Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10307>; Tue, 15 Sep 1992 18:37:53 -0400 Received: from canrem.com by unixbox.canrem.COM id aa00746; Tue, 15 Sep 92 18:30:29 EDT Received: by canrem.com (PCB/Usenet Gateway) Path-id <199215.104.106150@dosgate>; 15 Sep 92 (18:27) Message-Id: <199215.104.106150@dosgate> From: Mark Longridge Date: Mon, 14 Sep 1992 20:00:00 -0400 To: cube-lovers@ai.mit.edu Subject: pyraminx revisited Notes on the Pyraminx --------------------- According to Dr. Ronald Turner-Smith there are 5 different Pyraminx puzzles, all of different complexity. The following are listed in order from easiest to hardest (to the best of my knowledge): Pyraminx Star: Easiest of all pyraminx?? A simplification of the popular pyraminx because of the little uni-coloured (usually grey or silver) tetrahedrons on the 3 middle pieces of each face. Effectively all middle pieces on this pyraminx are the same colour! Snub Pyraminx: Same as standard pyraminx with tips removed Popular Pyraminx: The standard pyraminx of which appeared in vast quanities after the cube caught on. Senior Pyraminx: This is a mystery puzzle. No one seems to know anything about it, yet Turner-Smith's book refers to it and gives the maximum number of moves for it! It is between the Popular Pyraminx and Master Pyraminx in difficulty. Master Pyraminx: All the moves of the standard pyraminx plus 180 degree turns of the edges (just the strip, not the whole face) 446,965,972,992,000 combinations. Interestingly in the ads for Dr. Ronald Turner-Smith's book "The Amazing Pyraminx" the Master Pyraminx is cited as a puzzle superior to Rubik's Cube because there are no centre pieces and it's harder! (Both points debatable IMHO) Also Turner-Smith gives the following maximum number of moves for each type of Pyraminx puzzle: (The popular pyraminx is now known to be 11 moves at most or 14 moves if the tips are included) Type 1 24 moves ?? Type 2 28 moves ?? Type 3 38 moves ?? Type 4 215 moves (Senior Pyraminx) Type 5 255 moves (Master Pyraminx) Also it is known that transparent pyraminx puzzles were made. This would be a good idea for the cube as well. Meffert also considered a textured pyraminx for the blind, and ones with leather and wood finishes. All the post-cube puzzles compare themselves to the cube, such as the Master Pyraminx, and more recently Smart Alex. It seems that Rubik's Cube is the benchmark for all others to compare with. Alas, Uwe Meffert's puzzle club was a bust. Barring unique prototypes (perhaps Singmaster or Hofstadter have a Master Pyraminx, I'll check) none of the following were produced: (most of these are documented in the extremely rare "Pyraminx The Exciting new 1982 range" or the even more obscure 1983 edition booklet. Both of these have full colour pages with photos of cardboard mockups of all the variants.) 1982: Pyraminx Star (exists in small quanities, in Constantin's catalog) Pyraminx Pentagon, Pyraminx Hexagon (Computer Simulation/Mockup only) One can also imagine Septagons, Octagons etc... Pyraminx Barrel Junior, Pyraminx Barrel Senior (Mockup only) Pyraminx Disc Chess (Prototypes exist) Pyraminx Ultimate (as shown in July 82 Scientific American, Mockup) Pyraminx Crystal, Pyraminx Ball (Mockups, July 82 S.A) I'd really like to see the mechanism for a working crystal! Pyraminx Assembly Puzzles, 4 types (They exist) Pyraminx Octahedron (An octahedral skewb, I believe Braun & Bandelow made some) Gerd Braun is the inventor of the Moody Ball (rare but exists) Pyraminx Tetrahedron (Ben Halpern made a prototype) 1983: Space Grenade (???) Crystal Ball (Looks like an orb, definitely not the same though) However.... Just a few days ago I got Constantin's catalog. Surprisingly there is a picture of Josef Trajber's Octahedron inside. Also there is a picture of what appears to be a Pyraminx Ball. Other ideas he includes are a 2x2x2 siamese cube, new variants on Fisher's cube, e.g. Fisher's Domino, and a Pyraminx Ultimate for 180 DM! ...and so the search for new cube variants continues. Please send me your comments (Does anyone actually own a working Master Pyraminx??) I'm also interested in exchanging full cube lists with other collectors. Mark Longridge Email: mark.longridge@canrem.com 259 Thornton Rd N Oshawa Ontario Canada L1J 6T2 -- Canada Remote Systems - Toronto, Ontario, Canadas World's Largest PCBOARD System - 416-629-7000/629-7044 From tom@goat.clipper.ingr.com Tue Sep 15 21:28:48 1992 Return-Path: Received: from ingr.ingr.com by life.ai.mit.edu (4.1/AI-4.10) id AA19739; Tue, 15 Sep 92 21:28:48 EDT Received: from clipper.clipper.ingr.com by ingr.ingr.com (5.65c/1.920611) id AA01312; Tue, 15 Sep 1992 20:34:24 -0500 Received: from goat by clipper.clipper.ingr.com (5.61/1.910401) id AA28058; Tue, 15 Sep 92 16:21:41 -0700 Received: by goat.clipper.ingr.com (5.61/1.910201) id AA00258; Tue, 15 Sep 92 15:56:26 -0700 Subject: Re: pyraminx revisited To: cube-lovers@ai.mit.edu Date: Tue, 15 Sep 92 15:56:22 PDT In-Reply-To: <199215.104.106150@dosgate>; from "Mark Longridge" at Sep 14, 92 8:00 pm X-Mailer: ELM [version 05.00.01.20] Message-Id: <9209151556.AA00256@goat.UUCP> From: tom@goat.clipper.ingr.com (Tom Granvold) > Also it is known that transparent pyraminx puzzles were made. This > would be a good idea for the cube as well. Meffert also considered > a textured pyraminx for the blind, and ones with leather and wood > finishes. I have one of the textured pyraminx. I had to mail order it from Hong Kong in the early '80s. > Just a few days ago I got Constantin's catalog. Where can one get a copy of Constantin's catalog? > ...and so the search for new cube variants continues. Please send me > your > comments (Does anyone actually own a working Master Pyraminx??) I'm > also interested in exchanging full cube lists with other collectors. I wish I did :-) Tom Granvold ------------------------------------------------------ Mail: 2400 Geng Rd., Palo Alto, Calif., 94303 Email: tom@clipper.ingr.com ------------------------------------------------------ From @mail.uunet.ca:mark.longridge@CANREM.COM Mon Sep 21 00:05:34 1992 Return-Path: <@mail.uunet.ca:mark.longridge@CANREM.COM> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA04313; Mon, 21 Sep 92 00:05:34 EDT Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10241>; Mon, 21 Sep 1992 00:05:23 -0400 Received: from canrem.com by unixbox.canrem.COM id aa02771; Sun, 20 Sep 92 23:58:15 EDT Received: by canrem.com (PCB/Usenet Gateway) Path-id <199220.104.108416@dosgate>; 20 Sep 92 (23:51) Message-Id: <199220.104.108416@dosgate> From: Mark Longridge Date: Sat, 19 Sep 1992 20:00:00 -0400 To: cube-lovers@ai.mit.edu Subject: pyraminx revisted Notes on the Pyraminx --------------------- According to Dr. Ronald Turner-Smith there are 5 different Pyraminx puzzles, all of different complexity. The following are listed in order from easiest to hardest (to the best of my knowledge): Pyraminx Star: Easiest of all pyraminx?? A simplification of the popular pyraminx because of the little uni-coloured (usually grey or silver) tetrahedrons on the 3 middle pieces of each face. Effectively all middle pieces on this pyraminx are the same colour! Snub Pyraminx: Same as standard pyraminx with tips removed Popular Pyraminx: The standard pyraminx of which appeared in vast quanities after the cube caught on. Senior Pyraminx: This is a mystery puzzle. No one seems to know anything about it, yet Turner-Smith's book refers to it and gives the maximum number of moves for it! It is between the Popular Pyraminx and Master Pyraminx in difficulty. Master Pyraminx: All the moves of the standard pyraminx plus 180 degree turns of the edges (just the strip, not the whole face) 446,965,972,992,000 combinations. Interestingly in the ads for Dr. Ronald Turner-Smith's book "The Amazing Pyraminx" the Master Pyraminx is cited as a puzzle superior to Rubik's Cube because there are no centre pieces and it's harder! (Both points debatable IMHO) Also Turner-Smith gives the following maximum number of moves for each type of Pyraminx puzzle: (The popular pyraminx is now known to be 11 moves at most or 14 moves if the tips are included) Type 1 24 moves ?? Type 2 28 moves ?? Type 3 38 moves ?? Type 4 215 moves (Senior Pyraminx) Type 5 255 moves (Master Pyraminx) Also it is known that transparent pyraminx puzzles were made. This would be a good idea for the cube as well. Meffert also considered a textured pyraminx for the blind, and ones with leather and wood finishes. All the post-cube puzzles compare themselves to the cube, such as the Master Pyraminx, and more recently Smart Alex. It seems that Rubik's Cube is the benchmark for all others to compare with. Alas, Uwe Meffert's puzzle club was a bust. Barring unique prototypes (perhaps Singmaster or Hofstadter have a Master Pyraminx, I'll check) none of the following were produced: (most of these are documented in the extremely rare "Pyraminx The Exciting new 1982 range" or the even more obscure 1983 edition booklet. Both of these have full colour pages with photos of cardboard mockups of all the variants.) 1982: Pyraminx Star (exists in small quanities, in Constantin's catalog) Pyraminx Pentagon, Pyraminx Hexagon (Computer Simulation/Mockup only) One can also imagine Septagons, Octagons etc... Pyraminx Barrel Junior, Pyraminx Barrel Senior (Mockup only) Pyraminx Disc Chess (Prototypes exist) Pyraminx Ultimate (as shown in July 82 Scientific American, Mockup) Pyraminx Crystal, Pyraminx Ball (Mockups, July 82 S.A) I'd really like to see the mechanism for a working crystal! Pyraminx Assembly Puzzles, 4 types (They exist) Pyraminx Octahedron (An octahedral skewb, I believe Braun & Bandelow made some) Gerd Braun is the inventor of the Moody Ball (rare but exists) Pyraminx Tetrahedron (Ben Halpern made a prototype) 1983: Space Grenade (???) Crystal Ball (Looks like an orb, definitely not the same though) However.... Just a few days ago I got Constantin's catalog. Surprisingly there is a picture of Josef Trajber's Octahedron inside. Also there is a picture of what appears to be a Pyraminx Ball. Other ideas he includes are a 2x2x2 siamese cube, new variants on Fisher's cube, e.g. Fisher's Domino, and a Pyraminx Ultimate for 180 DM! ...and so the search for new cube variants continues. Please send me your comments (Does anyone actually own a working Master Pyraminx??) I'm also interested in exchanging full cube lists with other collectors. Mark Longridge Email: mark.longridge@canrem.com 259 Thornton Rd N Oshawa Ontario Canada L1J 6T2 -- Canada Remote Systems - Toronto, Ontario, Canadas World's Largest PCBOARD System - 416-629-7000/629-7044 From mb8d+@andrew.cmu.edu Mon Sep 21 21:31:25 1992 Return-Path: Received: from po3.andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA08728; Mon, 21 Sep 92 21:31:25 EDT Received: by po3.andrew.cmu.edu (5.54/3.15) id for cube-lovers@ai.mit.edu; Mon, 21 Sep 92 21:31:22 EDT Received: via switchmail; Mon, 21 Sep 1992 21:31:21 -0400 (EDT) Received: from pcs6.andrew.cmu.edu via qmail ID ; Mon, 21 Sep 1992 19:47:17 -0400 (EDT) Received: from pcs6.andrew.cmu.edu via qmail ID ; Mon, 21 Sep 1992 19:46:44 -0400 (EDT) Received: from mms.0.1.873.MacMail.0.9.CUILIB.3.45.SNAP.NOT.LINKED.pcs6.andrew.cmu.edu.pmax.ul4 via MS.5.6.pcs6.andrew.cmu.edu.pmax_ul4; Mon, 21 Sep 1992 19:46:44 -0400 (EDT) Message-Id: <0ejZvYC00WBK48jY0m@andrew.cmu.edu> Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT) From: Matthew John Bushey To: cube-lovers@ai.mit.edu Subject: cubes are great Cc: Does anyone out there know what is the cubed root of 81? Just wondering.... From yekta@huey.jpl.nasa.gov Mon Sep 21 22:38:35 1992 Return-Path: Received: from huey.Jpl.Nasa.Gov by life.ai.mit.edu (4.1/AI-4.10) id AA10534; Mon, 21 Sep 92 22:38:35 EDT Received: from hercules.JPL.NASA.GOV ([128.149.68.28]) by huey.Jpl.Nasa.Gov (4.1/SMI-4.1+DXRm2.2) id AA00679; Mon, 21 Sep 92 19:34:07 PDT Date: Mon, 21 Sep 92 19:34:07 PDT From: yekta@huey.jpl.nasa.gov (Yekta Gursel) Message-Id: <9209220234.AA00679@huey.Jpl.Nasa.Gov> Received: by hercules.JPL.NASA.GOV (4.1/SMI-4.1) id AA09483; Mon, 21 Sep 92 19:38:22 PDT To: mb8d+@andrew.cmu.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu> Subject: cubes are great Little smaller than its square root. Are you having fun yet? --Yekta From gls@think.com Tue Sep 22 11:50:24 1992 Received: from mail.think.com by life.ai.mit.edu (4.1/AI-4.10) id AA26659; Tue, 22 Sep 92 11:50:24 EDT Return-Path: Received: from Strident.Think.COM by mail.think.com; Tue, 22 Sep 92 11:50:22 -0400 From: Guy Steele Received: by strident.think.com (4.1/Think-1.2) id AA24984; Tue, 22 Sep 92 11:50:22 EDT Date: Tue, 22 Sep 92 11:50:22 EDT Message-Id: <9209221550.AA24984@strident.think.com> To: mb8d+@andrew.cmu.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu> Subject: cubes are great Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT) From: Matthew John Bushey Does anyone out there know what is the cubed root of 81? Just wondering.... Well, the "root of 81" is 9 (recall that when you don't say what kind of root you want, the default is "square"), and 9 cubed is 729. ... Eh? Oh, you meant the "cube root", not the "cubed root"? Well, that's another kettle of fish entirely. The n'th root of x is equal to x raised to the power 1/n. I fed this to my friendly Common Lisp system: > (expt 81 1/3) 4.3267487109222245 If I were you, I wouldn't trust the last few digits of this approximation, but fifteen decimal places ought to hold you for now. Here's how you could estimate it in your head. Note that 81 = 3 to the fourth power, so 1/3 4 1/3 4/3 1/3 81 = ( 3 ) = 3 = 3 ( 3 ) Now, the cube root of 3 is surely between 1 and 2, because 1 cubed is 1 and 2 cubed is 8. So the cube root of 3 is 1 plus some smaller fractional amount x. 3 2 3 So 3 = (1 + x) = 1 + 3 x + 3 x + x (binomial expansion). 3 Let's ignore the x term, which is probably small because x is sort of small. Then 2 2 1 + 3 x + 3 x = 3 so x + x = 2/3 . 2 Hm... if x = 1/2, then x + x = 3/4, which is a bit 2 too big. So figure x is about 0.4; then x + x = .4 + .16 = .56 which is too small. So probably x is about 0,45 or so. So the cube root of 3 is about 1.45, and the cube root of 81 is 3 times that, or about 4.35 -- not a bad approximation. --Guy STeele From bosch@smiteo.esd.sgi.com Tue Sep 22 12:10:48 1992 Return-Path: Received: from sgi.sgi.com (SGI.COM) by life.ai.mit.edu (4.1/AI-4.10) id AA27072; Tue, 22 Sep 92 12:10:48 EDT Received: from [192.48.193.1] by sgi.sgi.com via SMTP (920330.SGI/910110.SGI) for cube-lovers@ai.mit.edu id AA11832; Tue, 22 Sep 92 09:10:45 -0700 Received: by smiteo.esd.sgi.com (911016.SGI/920502.SGI.AUTO) for @sgi.sgi.com:cube-lovers@ai.mit.edu id AA02225; Tue, 22 Sep 92 09:10:44 -0700 From: bosch@smiteo.esd.sgi.com (Derek Bosch) Message-Id: <9209221610.AA02225@smiteo.esd.sgi.com> Subject: Gaby Games address needed To: cube-lovers@ai.mit.edu Date: Tue, 22 Sep 92 9:10:44 PDT X-Mailer: ELM [version 2.3 PL4] Does anyone out there in cube-land know the address for Gaby Games? They are an Israeli manufacturer of interesting 3-d interlocking wooden puzzles. I have posted this to rec.puzzles, with no help so far. Derek Bosch bosch@sgi.com From ronnie@cisco.com Tue Sep 22 13:39:12 1992 Return-Path: Received: from wolf.cisco.com by life.ai.mit.edu (4.1/AI-4.10) id AA29713; Tue, 22 Sep 92 13:39:12 EDT Received: from lager.cisco.com by wolf.cisco.com with TCP; Tue, 22 Sep 92 10:37:51 -0700 Message-Id: <9209221737.AA25287@wolf.cisco.com> To: Matthew John Bushey Cc: cube-lovers@ai.mit.edu Subject: Re: cubes are great In-Reply-To: Your message of "Mon, 21 Sep 92 19:46:44 EDT." <0ejZvYC00WBK48jY0m@andrew.cmu.edu> Date: Tue, 22 Sep 92 10:37:50 PDT From: "Ronnie B. Kon" > > Does anyone out there know what is the cubed root of 81? > > Just wondering.... > Let's see: the root of 81 is 9. 9 cubed is 729. Ronnie From azimmerm@rnd.stern.nyu.edu Mon Oct 5 16:53:08 1992 Return-Path: Received: from rnd.stern.nyu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA17208; Mon, 5 Oct 92 16:53:08 EDT Received: by rnd.stern.nyu.edu (4.1/1.34) id AA28496; Mon, 5 Oct 92 16:51:57 EDT Date: Mon, 5 Oct 92 16:51:57 EDT From: Al Zimmermann To: Cube-Lovers@ai.mit.edu Subject: Reminiscences Message-Id: Is everybody ready for more reminiscences? I got my first cube at Harrad's in London in October of 1980 while I was on vacation there with my girl friend. I spent every non-touristy moment working out and recording moves until, on day 13, I got the final face. When we got back to the States, my girl friend and I broke up. Do you think there's a moral here? Al Zimmermann From diamond@jit081.enet.dec.com Mon Oct 5 20:54:41 1992 Return-Path: Received: from enet-gw.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) id AA22961; Mon, 5 Oct 92 20:54:41 EDT Received: by enet-gw.pa.dec.com; id AA18960; Mon, 5 Oct 92 17:54:31 -0700 Message-Id: <9210060054.AA18960@enet-gw.pa.dec.com> Received: from jit081.enet; by decwrl.enet; Mon, 5 Oct 92 17:54:38 PDT Date: Mon, 5 Oct 92 17:54:38 PDT From: 06-Oct-1992 0949 To: cube-lovers@ai.mit.edu Apparently-To: cube-lovers@ai.mit.edu Subject: Re: miniscences Al Zimmermann writes: >I spent every non-touristy moment >working out and recording moves until, on day 13, I got the final face. >When we got back to the States, my girl friend and I broke up. Do you think >there's a moral here? Yes, at least three: (1) Every non-touristy moment that you weren't recording moves, you should have spent with your girl friend instead of working out :-) (2) You should have given equal attention to the final face and to your girl friend's face :-) (3) You should have chosen a girl friend who could solve the cube, like I did. But I thank you for warning about the dangers of getting back to the States. Maybe I shouldn't go back after all :-) -- Norman Diamond diamond@jit081.enet.dec.com [Digital did not write this.] From imp@kolvir.solbourne.com Fri Oct 9 17:59:44 1992 Return-Path: Received: from Solbourne.COM by life.ai.mit.edu (4.1/AI-4.10) id AA25966; Fri, 9 Oct 92 17:59:44 EDT Received: from kolvir.Solbourne.COM by Solbourne.COM (4.1/Solbourne-4.1) id AA16499; Fri, 9 Oct 92 15:59:44 MDT Received: from localhost by kolvir.Solbourne.COM (4.1/SMI-4.1) id AA06857; Fri, 9 Oct 92 15:59:40 MDT Message-Id: <9210092159.AA06857@kolvir.Solbourne.COM> To: cube-lovers@ai.mit.edu Subject: Quick question.... Date: Fri, 09 Oct 1992 15:59:40 MDT From: Warner Losh I was wondering if there were any X programs out there that allowed one to play with a rubic's cube (3x3x3 ... nxnxn) on a workstation? Archie didn't seem to know of any. Warner From pbeck@pica.army.mil Fri Oct 23 07:49:03 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA12683; Fri, 23 Oct 92 07:49:03 EDT Date: Fri, 23 Oct 92 7:47:24 EDT From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Cc: pbeck@pica.army.mil Subject: 13th IPP (1993) Message-Id: <9210230747.aa12659@COR4.PICA.ARMY.MIL> ............................................................... <--> 13th International puzzle collector's party and fair " and the 1993 Dutch Cube Day party transcribed by pbeck, 10/23/92 ............................................................... WHEN ---- 8/20 - 8/22/93 WHERE ---- Amsterdam vicinity LODGING ---- about $90 per night at MOTEL BREUKELEN STATIONSWEG 91 3621 LK BREUKELEN NETHERLANDS TEL: 03462 - 65888 FAX: 03462 - 62894 *** INVITATIONS *** Admission by invitation only!!! Contact: Mr. W.G.H. STRIJBOS BREDEROSTRAAT 18 5921 BM VENLO NETHERLANDS TEL: +31 (0) 77 -826213 FAX: +31 (0) 4704 - 4656 AGENDA: 8/20 13:00 - 16:30 PUZZLE EXCHANGE 17:30 - 22:00 DINNER AND MAGIC SHOW 8/21 10:00 - 17:00 PUZZLE PARTY AND FAIR (SALES) COST FOR ABOVE 150 DUTCH GUILDERS & IT WILL BE HELD AT MOTEL BREUKELEN - 100 EXTRA FOR SAT SALES TABLE 8/22 10:00 - 17:00 CUBE DAY _ THIS WILL BE HELD AT CHESS & GO CENTER IN AMSTELVEEN AND PROBABLY HAS AN EXTRA COST. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ *** 4 NIGHT RESERVATION (8/19 - 8/23) AT " MOTEL BREUKELEN" IS 625 DUTCH GUILDERS AND INCLUDES BREAKFAST - ONE DOUBLE BED IN EACH ROOM. ---> RESERVATION REQUEST FOR THIS PACKAGE MUST BE MADE BY *** JAN 10 1993 *** TO STRIJBOS >>>>>>>>>>>>>>>>>> If I was unclear or if you have other questions ask them to the list since several members of the Dutch Cube Club (party hosts) are subscribers. From hirsh@cs.rutgers.edu Wed Nov 4 15:18:42 1992 Return-Path: Received: from pei.rutgers.edu by life.ai.mit.edu (4.1/AI-4.10) id AA20102; Wed, 4 Nov 92 15:18:42 EST Received: by pei.rutgers.edu (5.59/SMI4.0/RU1.5/3.08) id AA09079; Wed, 4 Nov 92 15:18:38 EST Sender: Haym Hirsh Date: Wed, 4 Nov 92 15:18:37 EST From: Haym Hirsh Reply-To: Haym Hirsh To: cube-lovers@ai.mit.edu Subject: masterball Cc: Haym Hirsh Message-Id: A friend just sent me email about a new (to him and to me) puzzle called "masterball". Anyone know anything about it? Is it worth getting? Haym > I saw a Rubik's cube variant today called "Masterball." Have you > seen it? It is a sphere with 32 faces. If you consider the sphere > to be a world globe, there are 8 longitudinal slices each going > through the axis of the globe, dividing the sphere into 8 segments > like a sliced orange (sorry for starting to mix my metaphors [actually, > I guess I was mixing similes, but I know *you* wouldn't bring up > such a trivial point]). > > Oooops I guess there are only 4 longitudinal slices, each through > the axis, to divide the globe into 8 segments. > > There are also 3 slices of latitude, one through the equator one > each in the northern and the southern hemisphere parallel to > the equator. > > Resultant 32 faces. Mechanism has some similarities to Square One. > > Two different versions of Masterball are available. One has eight > different colors, corresponding to 8 segments. The other has only > black and white. I don't remember the home pattern of the black > and white sphere, I presume it is a degenerate case of the 8 color > sphere with black and white alternating slices. > > Cost: $24.95 each. My source is the same store in San Francisco > (Stonestown mall) that provided the Rubiks Tangle, Rubiks Dice, etc. From azimmerm@rnd.stern.nyu.edu Wed Nov 4 17:30:12 1992 Return-Path: Received: from rnd.stern.nyu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA29389; Wed, 4 Nov 92 17:30:12 EST Received: by rnd.stern.nyu.edu (4.1/1.34) id AA20786; Wed, 4 Nov 92 17:12:04 EST Date: Wed, 4 Nov 92 17:12:04 EST From: Al Zimmermann To: Haym Hirsh Cc: cube-lovers@ai.mit.edu, Haym Hirsh Subject: Re: masterball In-Reply-To: Your message of Wed, 4 Nov 92 15:18:37 EST Message-Id: > A friend just sent me email about a new (to him and to me) puzzle > called "masterball". Anyone know anything about it? Is it worth > getting? > > Haym > > > I saw a Rubik's cube variant today called "Masterball." Have you > > seen it? It is a sphere with 32 faces. If you consider the sphere > > to be a world globe, there are 8 longitudinal slices each going > > through the axis of the globe, dividing the sphere into 8 segments > > like a sliced orange (sorry for starting to mix my metaphors [actually, > > I guess I was mixing similes, but I know *you* wouldn't bring up > > such a trivial point]). > > > > Oooops I guess there are only 4 longitudinal slices, each through > > the axis, to divide the globe into 8 segments. > > > > There are also 3 slices of latitude, one through the equator one > > each in the northern and the southern hemisphere parallel to > > the equator. > > > > Resultant 32 faces. Mechanism has some similarities to Square One. > > > > Two different versions of Masterball are available. One has eight > > different colors, corresponding to 8 segments. The other has only > > black and white. I don't remember the home pattern of the black > > and white sphere, I presume it is a degenerate case of the 8 color > > sphere with black and white alternating slices. > > > > Cost: $24.95 each. My source is the same store in San Francisco > > (Stonestown mall) that provided the Rubiks Tangle, Rubiks Dice, etc. > > Games magazine seems to like the puzzle. They included Mastermind Rainbow (the polychromatic version) in this year's "Games 100" listing. Their write-up isn't very informative, but there's a picture. It appears on page 59 of the Dec. '92 issue. By the way, they indicate that the puzzle is available from Baekgaard at 1-800-323-5413. From pbeck@pica.army.mil Thu Nov 5 21:56:13 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA15053; Thu, 5 Nov 92 21:56:13 EST Date: Thu, 5 Nov 92 11:11:51 EST From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Subject: RE: MASTERBALL Message-Id: <9211051111.aa20544@COR4.PICA.ARMY.MIL> Masterball is similar to the VIP Sphere and Orb puzzles. It is nicely made and has been available in europe for a couple of years. I believe ISHI is also selling it. If you don't have the ISHI x-mas flyer call them and get it. This flyer has several unique items for slae, eg, 5x5x5, skewb From yekta@huey.jpl.nasa.gov Fri Nov 6 11:46:52 1992 Return-Path: Received: from huey.Jpl.Nasa.Gov by life.ai.mit.edu (4.1/AI-4.10) id AA16339; Fri, 6 Nov 92 11:46:52 EST Received: from hercules.JPL.NASA.GOV ([128.149.68.28]) by huey.Jpl.Nasa.Gov (4.1/SMI-4.1+DXRm2.2) id AA25552; Fri, 6 Nov 92 08:41:53 PST Date: Fri, 6 Nov 92 08:41:53 PST From: yekta@huey.jpl.nasa.gov (Yekta Gursel) Message-Id: <9211061641.AA25552@huey.Jpl.Nasa.Gov> Received: by hercules.JPL.NASA.GOV (4.1/SMI-4.1) id AA07093; Fri, 6 Nov 92 08:40:56 PST To: cube-lovers@life.ai.mit.edu In-Reply-To: Peter Beck (BATDD)'s message of Thu, 5 Nov 92 11:11:51 EST <9211051111.aa20544@COR4.PICA.ARMY.MIL> Subject: MASTERBALL Could you post ISHI's phone number? --Yekta From dik@cwi.nl Sun Nov 8 16:16:16 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02772; Sun, 8 Nov 92 16:16:16 EST Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA20202 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 21:15:15 GMT Received: by boring.cwi.nl id AA21833 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 22:15:14 +0100 Date: Sun, 8 Nov 1992 22:15:14 +0100 From: Dik.Winter@cwi.nl Message-Id: <9211082115.AA21833.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Where are the archives? Can somebody tell me where the cube-lovers archives are maintained at this moment? Thanks, dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From dik@cwi.nl Sun Nov 8 16:15:20 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA02765; Sun, 8 Nov 92 16:15:20 EST Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA20202 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 21:15:15 GMT Received: by boring.cwi.nl id AA21833 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 22:15:14 +0100 Date: Sun, 8 Nov 1992 22:15:14 +0100 From: Dik.Winter@cwi.nl Message-Id: <9211082115.AA21833.dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: Where are the archives? Can somebody tell me where the cube-lovers archives are maintained at this moment? Thanks, dik -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl From alan@ai.mit.edu Sun Nov 8 17:00:39 1992 Return-Path: Received: from corn-pops (corn-pops.ai.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) id AA03727; Sun, 8 Nov 92 17:00:39 EST Received: by corn-pops (4.1/AI-4.10) id AA03768; Sun, 8 Nov 92 17:00:08 EST Date: Sun, 8 Nov 92 17:00:08 EST Message-Id: <9211082200.AA03768@corn-pops> From: Alan Bawden Sender: Alan@lcs.mit.edu To: Dik.Winter@cwi.nl Cc: Cube-Lovers@ai.mit.edu In-Reply-To: Dik.Winter@cwi.nl's message of Sun, 8 Nov 1992 22:15:14 +0100 <9211082115.AA21833.dik@boring.cwi.nl> Subject: Where are the archives? Date: Sun, 8 Nov 1992 22:15:14 +0100 From: Dik.Winter@cwi.nl Can somebody tell me where the cube-lovers archives are maintained at this moment? Such questions should be addressed to Cube-Lovers-Request@AI.MIT.EDU. But since you asked publicly, I might as well answer publicly as well. If you are interested in the archives of the Cube-Lovers mailing list: Using FTP, connect to FTP.AI.MIT.EDU, login as "anonymous" (any password), and in the directory "pub/cube-lovers" you will find the nine (compressed) files "cube-mail-0.Z" through "cube-mail-8.Z". Archive vital statistics (when uncompressed): File From To Size (bytes) ---- ---- -- ------------ cube-mail-0 12 Jul 80 23 Oct 80 185037 cube-mail-1 3 Nov 80 9 Jan 81 135719 cube-mail-2 10 Jan 81 3 Aug 81 138566 cube-mail-3 3 Aug 81 3 May 82 137753 cube-mail-4 4 May 81 11 Dec 82 139660 cube-mail-5 11 Dec 82 6 Jan 87 173364 cube-mail-6 10 Jan 87 13 Apr 90 216733 cube-mail-7 12 Oct 90 9 Sep 91 137508 cube-mail-8 1 Nov 91 25 May 92 171205 In addition, the file "recent-mail" contains a copy of the currently active section of the archive. (Unfortunately, due to the way mail works here at the AI Lab, it is not possible to have new mail accumulate directly into this file, so there may be some delay before a new message arrives here.) Finally, the file "README" contains the information you are currently reading. - Alan From dik@cwi.nl Sun Nov 8 17:35:25 1992 Return-Path: Received: from charon.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) id AA04853; Sun, 8 Nov 92 17:35:25 EST Received: from boring.cwi.nl by charon.cwi.nl with SMTP id AA20522 (5.65b/3.2/CWI-Amsterdam); Sun, 8 Nov 1992 22:35:18 GMT Received: by boring.cwi.nl id AA21966 (5.65b/2.10/CWI-Amsterdam); Sun, 8 Nov 1992 23:35:17 +0100 Date: Sun, 8 Nov 1992 23:35:17 +0100 From: Dik.Winter@cwi.nl Message-Id: <9211082235.AA21966.dik@boring.cwi.nl> To: cube-lovers@ai.mit.edu Subject: Reply to old message Going through the archives I found a message from Mike Reid, dated August 20, where he investigates the filtration through different subgroups. I missed the original, but he says: > winter's filtration is 56 --> 49 --> kl --> 1. it may be the case that > this can be improved by replacing kl with 17 , and allowing all face > turns available in the subgroup 49. i haven't had the time to look into > this yet. For the record, the filtration is from Herbert Kociemba, and it is: 56 --> 49 --> 1. So there is no intermediate stage between 49 and 1. dik From pbeck@pica.army.mil Mon Nov 9 11:22:43 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA12080; Mon, 9 Nov 92 11:22:43 EST Date: Mon, 9 Nov 92 9:04:24 EST From: Peter Beck (BATDD) To: yekta@huey.jpl.nasa.gov Cc: cube-lovers@life.ai.mit.edu Subject: ishi's phone number Message-Id: <9211090904.aa25819@COR4.PICA.ARMY.MIL> ISHI PRESS 408-944-9110 From gk1k+@andrew.cmu.edu Wed Nov 18 22:59:34 1992 Return-Path: Received: from po2.andrew.cmu.edu by life.ai.mit.edu (4.1/AI-4.10) id AA10302; Wed, 18 Nov 92 22:59:34 EST Received: by po2.andrew.cmu.edu (5.54/3.15) id for cube-lovers@ai.mit.edu; Wed, 18 Nov 92 22:59:30 EST Received: via switchmail; Wed, 18 Nov 1992 22:59:27 -0500 (EST) Received: from pcs16.andrew.cmu.edu via qmail ID ; Wed, 18 Nov 1992 22:57:46 -0500 (EST) Received: from pcs16.andrew.cmu.edu via qmail ID ; Wed, 18 Nov 1992 22:57:43 -0500 (EST) Received: from mms.0.1.873.MacMail.0.9.CUILIB.3.45.SNAP.NOT.LINKED.pcs16.andrew.cmu.edu.sun4c.411 via MS.5.6.pcs16.andrew.cmu.edu.sun4c_411; Wed, 18 Nov 1992 22:57:43 -0500 (EST) Message-Id: Date: Wed, 18 Nov 1992 22:57:43 -0500 (EST) From: George Cornelius Kuhl To: cube-lovers@ai.mit.edu Subject: cube question Cc: what is the cube root of 81? George From ACW@riverside.scrc.symbolics.com Thu Nov 19 09:31:59 1992 Return-Path: Received: from RIVERSIDE.SCRC.Symbolics.COM by life.ai.mit.edu (4.1/AI-4.10) id AA01606; Thu, 19 Nov 92 09:31:59 EST Received: from PALLANDO.SCRC.Symbolics.COM by RIVERSIDE.SCRC.Symbolics.COM via INTERNET with SMTP id 954415; 19 Nov 1992 09:33:00-0500 Date: Thu, 19 Nov 1992 09:32-0500 From: Allan C. Wechsler Subject: cube question To: gk1k+@andrew.cmu.edu, cube-lovers@ai.mit.edu In-Reply-To: Message-Id: <19921119143257.8.ACW@PALLANDO.SCRC.Symbolics.COM> Date: Wed, 18 Nov 1992 22:57 EST From: George Cornelius Kuhl what is the cube root of 81? George [4 /3 /16 /1 /1 /5 /1 /1 /2 /16 /1 /44 /1 /2 /1 /1 /1 /1 /1 /3 /12 ...] From pbeck@pica.army.mil Mon Nov 23 09:35:38 1992 Return-Path: Received: from COR4.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) id AA26950; Mon, 23 Nov 92 09:35:38 EST Date: Mon, 23 Nov 92 9:32:46 EST From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Subject: ATLANTA Puzzle Party & exhibition Message-Id: <9211230932.aa13919@COR4.PICA.ARMY.MIL> 1/14 - 4/10/93 there will be a puzzle exhibition at THE ATALANTA INTERNATIONAL MUSEUM OF ART AND DESIGN SPECIAL EVENTS FOR PUZZLERS: 1/14 opening and reception at 6PM 1/15 eve reception 1/16 open to puzzlers only 9am - 13:00 1/17 puzzlers only there will be puzzle trades and sales (for puzzlers) on either sat or sun POCs TOM RODGERS 404-351-7744 TYLER BARREET 404-998-7432 Please distribute to all interested parties. From @mail.uunet.ca:mark.longridge@canrem.com Fri Jan 8 00:57:44 1993 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from mail.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) id AA01786; Fri, 8 Jan 93 00:57:44 EST Received: from canrem.COM ([142.77.253.2]) by mail.uunet.ca with SMTP id <10049>; Fri, 8 Jan 1993 00:57:22 -0500 Received: from canrem.com by unixbox.canrem.COM id aa04957; Fri, 8 Jan 93 0:55:40 EST Received: by canrem.com (PCB/Usenet Gateway) Path-id <19938.104.159061@dosgate>; 8 Jan 93 (00:49) Message-Id: <19938.104.159061@dosgate> From: Mark Longridge Date: Thu, 7 Jan 1993 19:00:00 -0500 To: cube-lovers@ai.mit.edu Subject: computer cubing With thanks to Dan Hoey for getting me on the right track, I have finally got most of the squares group evaluted. The big breakthru was developing a checksum for a squares position. I know it's been done before, but I wanted to prove to myself I could do it on a mere 386 with 4 megs of memory. My latest program (rubik5.exe) took 24 hours to number the squares group up to 8 moves deep. The point of all this was to create a squares group database to aid in developing an optimal solver for the cube. Ultimately the database will have an entry for every squares group position, along with it's optimal solution. I would be interested in hearing from any others who have created such a database, and what type of compression or checksum was used for the arrangement. Also I've received a call from Richard Schneider. He is publishing a comprehensive book on square 1, plus a follow-up book on pretty patterns and shapes. This will be available in the States shortly. I haven't been seeing anything from cube-lovers in a while, I hope it's still up and running. To: Mike Reid --- Hope you see this! Any further progress been made on God's Algorithm? I'm still trying to catch up. I'm still interested in that code of yours to find improvements on some pretty patterns I've discovered. Anyways here is what my program has found so far: Squares group (u2, d2, l2, r2, f2, b2) Moves Deep Number of patterns ---------- ------------------ 0 1 1 6 2 27 3 120 4 519 5 1932 6 6484 7 20310 8 53000 (and counting) :-> Got to improve it's speed.... -> Mark <- -- Canada Remote Systems - Toronto, Ontario World's Largest PCBOARD System - 416-629-7000/629-7044