From matwood@peruvian.cs.utah.edu Thu Feb 24 14:25:20 1994 Return-Path: Received: from cs.utah.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29205; Thu, 24 Feb 94 14:25:20 EST Received: from peruvian.cs.utah.edu by cs.utah.edu (5.65/utah-2.21-cs) id AA01848; Thu, 24 Feb 94 12:25:19 -0700 Received: by peruvian.cs.utah.edu (5.65/utah-2.15-leaf) id AA22238; Thu, 24 Feb 94 12:25:18 -0700 Message-Id: <9402241925.AA22238@peruvian.cs.utah.edu> To: cube-lovers@life.ai.mit.edu Subject: Book - "Simple Solution To Rubik's Cube" Date: Thu, 24 Feb 94 12:25:17 MST From: Mark Atwood (I sent my request to be added to this mailing list in to cube-lovers-request@ai.ai.mit.edu a few days ago and havnt heard back. Hope this works..) I was going thru my stuff a while back and found my two original Rubik's Cubes, one of which was given to me several years before they became wildly popular. The solution I learned was in the book "The Simple Solution To Rubik's Cube", which was a paperback of about 20-30 pages. I remember most of the solution steps outlined in the book, (my hands remember better than my head does), however, I can't find a copy of the book anywhere, to refresh my memory. Anyone got a copy or know where I can get one? ..Mark Atwood From anandrao@hk.super.net Fri Feb 25 03:32:04 1994 Return-Path: Received: from hk.super.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05974; Fri, 25 Feb 94 03:32:04 EST Received: by hk.super.net id AA21734 (5.65c/IDA-1.4.4 for cube-lovers@life.ai.mit.edu); Fri, 25 Feb 1994 16:31:42 +0800 Date: Fri, 25 Feb 1994 16:18:56 +0800 (HKT) From: "Mr. Anand Rao" Subject: Re: your mail To: Jan de Ruiter Cc: cube-lovers@life.ai.mit.edu In-Reply-To: <2038.9402181343@xirion.xirion.nl> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Fri, 18 Feb 1994, Jan de Ruiter wrote: > > Sorry about not reporting this earlier, but my search for solutions for > Rubiks Tangle 10x10 confirms the finding of Don Woods: no solutions! > [snip] > we could re-define the puzzle as follows: > find which four pieces to duplicate in order to find solutions for > the 10x10. > If the number of solutions varies depending on the choice, you could > even add a restriction: > find which four pieces to duplicate in order to find a set which has > the minimum number of solutions for the 10x10. ^^^^^^^ The kind Mr. Rubik has already done that - the minimum is - ZERO! The revised problem can be solved fairly easily using your program ( I don't know, though, how long it takes to run to completion for the 10*10 case) - try to place only 99 tiles out of the 100 given tiles. You may have several sub-solutions. It is then easy to determine for each of these sub-solutions which tile you need to complete the 10*10 mosaic. If this pattern has already been duplicated, i.e. you need THREE numbers of this tile to find the complete solution, this sub-solution will not work and so examine the next sub-solution .... Hopefully you find the solution this way. After running the program for the 99 tiles, the additional time required to solve the problem defined by you should not be significant because that would be a linear process. Anand. From sage@world.std.com Sun Feb 27 20:08:15 1994 Return-Path: Received: from news.std.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14131; Sun, 27 Feb 94 20:08:15 EST Received: from world.std.com by news.std.com (5.65c/Spike-2.1) id AA25026; Sun, 27 Feb 1994 20:08:13 -0500 Received: by world.std.com (5.65c/Spike-2.0) id AA05694; Sun, 27 Feb 1994 20:08:08 -0500 Date: Sun, 27 Feb 1994 20:08:08 -0500 (EST) From: Meisha n Thompson Subject: Puzzle To: cube-lovers@life.ai.mit.edu Cc: Meisha n Thompson Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Please add me to your mailing list. Thank You Meisha Thompson From xirion!jandr@relay.nl.net Mon Mar 7 05:38:41 1994 Return-Path: Received: from sun4nl.NL.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15552; Mon, 7 Mar 94 05:38:41 EST Received: from xirion by sun4nl.NL.net via EUnet id AA19748 (5.65b/CWI-3.3); Mon, 7 Mar 1994 11:38:38 +0100 Received: by xirion.xirion.nl id AA08619 (5.61/UK-2.1); Mon, 7 Mar 94 11:37:25 +0100 From: Jan de Ruiter Date: Mon, 7 Mar 94 11:37:25 +0100 Message-Id: <8619.9403071037@xirion.xirion.nl> X-Organization: Xirion Unix Software & Consultancy bv Burgemeester Verderlaan 15 X 3454 PE De Meern The Netherlands X-Phone: +31 3406 61990 X-Fax: +31 3406 61981 To: cube-lovers@life.ai.mit.edu To anandrao@HK.Super.NET Subject: Re: your mail Cc: cube-lovers@life.ai.mit.edu In-Reply-To: > >On Fri, 18 Feb 1994, Jan de Ruiter wrote: >> >> Sorry about not reporting this earlier, but my search for solutions for >> Rubiks Tangle 10x10 confirms the finding of Don Woods: no solutions! >> >[snip] >> we could re-define the puzzle as follows: >> find which four pieces to duplicate in order to find solutions for >> the 10x10. >> If the number of solutions varies depending on the choice, you could >> even add a restriction: >> find which four pieces to duplicate in order to find a set which has >> the minimum number of solutions for the 10x10. > ^^^^^^^ >The kind Mr. Rubik has already done that - the minimum is - ZERO! Correct, but I think you know what I mean: minimum >= 1 >The revised problem can be solved fairly easily using your program ( I >don't know, though, how long it takes to run to completion for the 10*10 >case) More than a week > - try to place only 99 tiles out of the 100 given tiles. You may >have several sub-solutions. It is then easy to determine for each of these >sub-solutions which tile you need to complete the 10*10 mosaic. I am sorry, but I have to disagree on this. It is not that simple. If you managed to place 99 pieces, you have already placed three or even all four of the duplicated pieces (depending on which one is left over) If you placed three, there are tree possibilities for the piece we need: - it is nonexistent (illegal colour combinations): no solution - it is one of the duplicated pieces: this means two of the four puzzles will be identical which is OK, but not so nice, or - it is any other piece: we found a good solution If you placed all four duplicated pieces already, any solution you find will not satisfy the conditions of the puzzle (i.e. precisely four duplicated pieces). And in both cases you have not solved: which four pieces to duplicate in order to find solutions for the 10x10. but: which four pieces to duplicate in order to find solutions for the 10x10, with the restriction that three of them must be identical to any three taken from the set of four duplicates given by Rubik. Solving the puzzle without this restriction requires a different approach. I was thinking of starting the program with 5 of each (120 pieces), and after placing the 5th duplicated piece remove the rest of the duplicates from the remaining pieces, and see if this leads to a solution. As soon as back- tracking removes the 5th duplicate, all other duplicates must be made accessible again. Jan From phiscock@ee.ryerson.ca Mon Mar 14 14:14:44 1994 Return-Path: Received: from ee.ryerson.ca (eccles.ee.ryerson.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27518; Mon, 14 Mar 94 14:14:44 EST Received: from eccles (eccles.ee.ryerson.ca) by ee.ryerson.ca (4.1/SMI-4.1) id AA24815; Mon, 14 Mar 94 14:09:20 EST From: phiscock@ee.ryerson.ca (Peter Hiscocks) Received: by eccles (4.1//ident-1.0) id AA24812; Mon, 14 Mar 94 14:09:19 EST Message-Id: <9403141909.AA24812@eccles> Subject: Anyone solved Rubik's Tangle? To: Cube-Lovers@ai.mit.edu Date: Mon, 14 Mar 94 14:09:18 EST X-Mailer: ELM [version 2.3 PL11] For those who haven't seen it, Rubik's Tangle is a new puzzle to drive us all nuts, break up our families, and divert us from the things we should be working on. It consists of 25 tiles, which form a 5x5 pattern. On each tile is a pattern of coloured ropes, the ends of which must match the ends of the ropes on the adjacent tiles. Certain clues are evident: the shape of each rope pattern is the same, there are equal numbers of each colour, and each tile given a letter label on the back. Before I waste my life on this, has anyone solved the problem? Peter -- Peter Hiscocks Phone: (416) 979-5000 Ext 6109 Department of Electrical Engineering Fax: (416) 979-5280 Ryerson Polytechnical University, Toronto, Canada From Don.Woods@eng.sun.com Mon Mar 14 16:14:51 1994 Return-Path: Received: from Sun.COM by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04648; Mon, 14 Mar 94 16:14:51 EST Received: from Eng.Sun.COM (zigzag.Eng.Sun.COM) by Sun.COM (sun-barr.Sun.COM) id AA18187; Mon, 14 Mar 94 13:14:25 PST Received: from colossal.Eng.Sun.COM by Eng.Sun.COM (4.1/SMI-4.1) id AA01909; Mon, 14 Mar 94 13:13:31 PST Received: by colossal.Eng.Sun.COM (5.0/SMI-SVR4) id AA18313; Mon, 14 Mar 94 13:15:06 PST Date: Mon, 14 Mar 94 13:15:06 PST From: Don.Woods@eng.sun.com (Don Woods) Message-Id: <9403142115.AA18313@colossal.Eng.Sun.COM> To: phiscock@ee.ryerson.ca Subject: Re: Anyone solved Rubik's Tangle? Cc: Cube-Lovers@ai.mit.edu X-Sun-Charset: US-ASCII Content-Length: 397 > Before I waste my life on this, has anyone solved the problem? Yes, it's been solved, and discussed at some length on this group. However, I haven't seen anyone who claims to have come up with an "insightful" solution, i.e. one in which you figure out a general approach that leads to a solution. All solutions I've heard of have been found by exhaustive search, often by computer. -- Don. From anandrao@hk.super.net Mon Mar 14 21:27:38 1994 Return-Path: Received: from hk.super.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20935; Mon, 14 Mar 94 21:27:38 EST Received: by hk.super.net id AA15765 (5.65c/IDA-1.4.4 for Cube-Lovers@ai.mit.edu); Tue, 15 Mar 1994 10:27:05 +0800 Date: Tue, 15 Mar 1994 10:15:06 +0800 (HKT) From: "Mr. Anand Rao" Subject: Re: Anyone solved Rubik's Tangle? To: Don Woods Cc: phiscock@ee.ryerson.ca, Cube-Lovers@ai.mit.edu In-Reply-To: <9403142115.AA18313@colossal.Eng.Sun.COM> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Mon, 14 Mar 1994, Don Woods wrote: > > Before I waste my life on this, has anyone solved the problem? > > Yes, it's been solved, and discussed at some length on this group. True. However, the 10*10 solution where you use all the four tangle puzzles to form a 10*10 pattern with matching edges, has been found to be impossible( Although the puzzle leaflet says that it is solvable). Once again there is no 'insightful' solution. Someone has posted that he has seen an intuitive solution which evades his memory for the time being but will try to recollect what it was ... reincarnation of Fermat's Last Problem :). There have been some interesting postings in this group on this topic in the last few weeks and you should read them . > However, I haven't seen anyone who claims to have come up with an > "insightful" solution, i.e. one in which you figure out a general > approach that leads to a solution. All solutions I've heard of > have been found by exhaustive search, often by computer. > > -- Don. > Anand Rao. From @mitvma.mit.edu:SHERE@SLACVM.BITNET Tue Mar 15 14:08:27 1994 Return-Path: <@mitvma.mit.edu:SHERE@SLACVM.BITNET> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25112; Tue, 15 Mar 94 14:08:27 EST Message-Id: <9403151908.AA25112@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 0921; Tue, 15 Mar 94 14:08:19 EST Received: from SLACVM.SLAC.STANFORD.EDU (NJE origin MAILER@SLACVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 7886; Tue, 15 Mar 1994 14:08:18 -0500 Received: by SLACVM (Mailer R2.08 R208004) id 1497; Tue, 15 Mar 94 11:07:04 PST Date: Tue, 15 Mar 1994 11:04 -0800 (PST) From: SHERE%SLACVM.BITNET@mitvma.mit.edu To: cube-lovers@life.ai.mit.edu Subject: Mailing List Hello, would you please add me.. shere@slac.stanford.edu .. to your mailing list? I've forgotten some of my key moves and am trying to brush up. I've downloaded your archived mail. Where might I find a GZ decompression utility? Anyway, thanks. Lee From @mail.uunet.ca:mark.longridge@canrem.com Sat Apr 2 21:21:55 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12740; Sat, 2 Apr 94 21:21:55 EST Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <88146(1)>; Sat, 2 Apr 1994 21:21:12 -0500 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA03000; Sat, 2 Apr 94 21:20:05 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1993C2; Sat, 2 Apr 94 21:16:50 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Invariant Shifting From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.733.5834.0C1993C2@canrem.com> Date: Sat, 2 Apr 1994 20:12:00 -0500 Organization: CRS Online (Toronto, Ontario) Something new to stop the drought of cube posts... Example of Invariant Shifting ----------------------------- The resultant position generated by process p8 is invariant under shifting, specifically 2 X on the Left and Right sides. P8 2 x ORDER 2: shift 0 D2 F2 T2 F2 B2 T2 F2 T2 1 T2 D2 F2 T2 F2 B2 T2 F2 2 F2 T2 D2 F2 T2 F2 B2 T2 3 T2 F2 T2 D2 F2 T2 F2 B2 4 B2 T2 F2 T2 D2 F2 T2 F2 5 F2 B2 T2 F2 T2 D2 F2 T2 6 T2 F2 B2 T2 F2 T2 D2 F2 7 F2 T2 F2 B2 T2 F2 T2 D2 This is the longest process I've found so far. Certainly this property is not true of all squares group processes. I suspect there are no processes in the full group with this property (of any significant length). Perhaps the fact that the L and R faces never rotate will give some clue on how to generate processes with this property. Q: Is this the longest such process? Further Notes on Antipodes in the Square's Group ------------------------------------------------ I just realized some things about sq group antipodes which I should have seen before... The closest 2 antipodes can be is 2 square's group moves. Take the position produced by p66: p66 Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 D2 Any turn will reduce this to a position requiring 14 moves. Undoing this move will regenerate the antipode. No single move can change position p66 into another antipode, therefore the closest any 2 antipodes can be is 2 moves. Futhermore any antipode can not be made into a local maximum which is 14 moves deep with 1 half turn. I will conclude that there are no local maxima in the square's group that neighbour each other closer than 2 moves. -> Mark <- Email: mark.longridge@canrem.com From pbeck@pica.army.mil Mon Apr 4 08:24:04 1994 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09001; Mon, 4 Apr 94 08:24:04 EDT Date: Mon, 4 Apr 94 8:22:52 EDT From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Cc: pbeck@pica.army.mil Subject: puzzle boxes Message-Id: <9404040822.aa26168@COR6.PICA.ARMY.MIL> I found a shop in NYC's chinatown that stocks Japanese puzzle/trick boxes. TING'S GIFT SHOP 18 DOYERS STREET NY, NY 10013 212-962-1081 - 4-way $25 - 4-way w/music $36 - 10-way $42 - 12-way $46 - 20-way $52 NYC sales tax 8% good puzzling and good eating From ishius@ishius.com Mon Apr 4 14:11:59 1994 Return-Path: Received: from holonet.net (giskard.holonet.net) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA28695; Mon, 4 Apr 94 14:11:59 EDT Received: from DialupEudora (ishius@localhost) by holonet.net (Anton Dovydaitis) with SMTP id LAA14679; Mon, 4 Apr 1994 11:07:23 -0700 Date: Mon, 4 Apr 1994 11:07:23 -0700 Message-Id: <199404041807.LAA14679@holonet.net> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: cube-lovers@ai.mit.edu From: ishius@ishius.com (Ishi Press International) Subject: puzzle boxes Ishi Press International offers a variety of Japanese Trick Puzzle Boxes, from 4 moves to 66 moves. These are handcrafted, wood inlaid Okiyama trickboxes. We also have unique puzzle boxes by Kamei. For a free catalog of our PUZZLES please send us your postal mailing address and we will mail you one. Please specify that you are interested in PUZZLES. Always feel free to write me if you have any questions or comments. Anton Dovydaitis Customer Support ======================================================================== Ishi Press International 800/859-2086 voice, 408/944-9110 FAX 76 Bonaventura Drive ishius@ishius.com The Americas San Jose, CA 95134 ishi@cix.compulink.co.uk Europe From mmoss@panix.com Mon Apr 4 17:30:58 1994 Return-Path: Received: from panix.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10263; Mon, 4 Apr 94 17:30:58 EDT Received: by panix.com id AA12590 (5.65c/IDA-1.4.4 for cube-lovers@life.ai.mit.edu); Mon, 4 Apr 1994 17:30:47 -0400 From: Matthew Moss Message-Id: <199404042130.AA12590@panix.com> Subject: About Rubix's tetrahedron... To: cube-lovers@life.ai.mit.edu (Cube Mailing List) Date: Mon, 4 Apr 1994 17:30:46 -0400 (EDT) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 811 Here's a question for y'all..... It's about the tetrahedron puzzle from Rubix (I forget the real name). [the one with legal moves consisting of removing a 4-piece tetrahedron from a 10-piece and putting it back on in different orientation] Anyway, mine is pretty loosed up, and occassionally when I am working on it, one piece will come loose and go skitting across the floor. There's no way I can remember the orientation it had on there. Has any study been done or does someone know if I put that piece back on, will it still be solveable if it's orientation is wrong (ie, different than it was before it fell off)? Do you understand my question? (I hope I'm not too confusing...) I've been thinking about implementing this via computer, just to test this out, but I thought I'd ask y'all first. Thanx. From @mail.uunet.ca:mark.longridge@canrem.com Sun Apr 10 23:26:50 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25590; Sun, 10 Apr 94 23:26:50 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <89026(2)>; Sun, 10 Apr 1994 23:26:15 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA12263; Sun, 10 Apr 94 23:24:59 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 19A249; Sun, 10 Apr 94 21:41:27 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: More Sq Notes From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.738.5834.0C19A249@canrem.com> Date: Sun, 10 Apr 1994 17:37:00 -0400 Organization: CRS Online (Toronto, Ontario) Additional Notes on Squares Group Patterns ------------------------------------------ Note that p80a, p99a and p108a are 2 DOT patterns, all of the form U1 (swap edges & corners in U and D faces) D3 or U1 (swap edges & corners in U and D faces) D1 or D1 (swap edges within U and edges within D) D3 P66a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1 (13) p67a alternate method F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2 (13) p80a alternate method U1 F2 R2 L2 U2 D2 F2 U2 D3 (9) p99a alternate method U1 R2 F2 B2 U2 D2 R2 D1 (8) P100a alternate method F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1 (12) p108a alternate method R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1 (12) p130a alternate method F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3 (13) p133a alternate method R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2 (11) A) In general, any sequence of half turns which swaps edges and corners in the U and D faces can be sanwiched between a single quarter turn of U and a single quarter turn of D. Such a process would lead to a square's group position. B) Furthermore, any sequence of half turns which swaps edges within U and edges within D can be sanwiched between a single quarter turn of U or D and a single quarter turn of U or D. Once again, such a process would lead to a square's group position. Here is an example of a position which takes over twice as many half turns as full group moves: L2 T2 L2 T2 L2 T2 F2 L2 T2 F2 T2 R2 B2 (13) U1 F2 R2 L2 B2 D1 (6) As discussed in point A above, sequences which move all elements of the U face to the D face and also move all elements of the D face to the U face (excepting the centres naturally) appear as 2 DOT patterns on the cube. This makes sense, as the initial quarter turn in process p80a must be balanced by another quarter turn. Since all of the elements subjected to the quarter turn are now in the D face, we must turn that face a quarter turn to remain in the squares group. From @mail.uunet.ca:mark.longridge@canrem.com Sun Apr 10 23:27:13 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25591; Sun, 10 Apr 94 23:27:13 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <89406(1)>; Sun, 10 Apr 1994 23:26:36 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA12271; Sun, 10 Apr 94 23:25:00 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 19A24A; Sun, 10 Apr 94 21:41:28 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: More Shifting From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.739.5834.0C19A24A@canrem.com> Date: Sun, 10 Apr 1994 17:39:00 -0400 Organization: CRS Online (Toronto, Ontario) More Notes on Invariant Shifting -------------------------------- Let us define a process as "Shift Invariant" if it results in the same displacement even after a series of left or right shifts. That is, from a process of length N we can generate N-1 processes which result in the same displacement by shifting the process. Sometimes the processes generated are not all unique! e.g. P8 2 x ORDER 2: (symmetry level 3) D2 F2 T2 F2 B2 T2 F2 T2 (8) Q: Is this the longest such process? A: No. The following processes are also shift invariant: 2 Swap D2 R2 D2 R2 D2 R2 (6) (symmetry level 12, SI level 2) p21 2 H L2 R2 B2 L2 R2 F2 (6) (symmetry level 6, SI level 6) Amazingly, the process p3 (found using Dik Winter's program) is actually a series of 20 processes which all result in the same displacement! p3 12 flip R1 L1 D2 B3 L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 R2 U3 F2 D3 (20) (symmetry level 1, SI level 20) Since p3 is shift invariant, we can easily shift the 3 consecutive half turns to the beginning without fear of altering the end result: L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 + R2 U3 F2 D3 R1 L1 D2 B3 From bagleyd@source.asset.com Thu Apr 14 11:21:48 1994 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16479; Thu, 14 Apr 94 11:21:48 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA10500; Thu, 14 Apr 1994 11:00:13 -0400 Date: Thu, 14 Apr 1994 11:00:13 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9404141500.AA10500@source.asset.com> To: cube-lovers@life.ai.mit.edu Subject: Mailing List Hi I was wondering if you can add me to your mailing list. I put some motif puzzles at ftp.x.org in /contrib/motif_puzzles. They are: rubik: a (nxnxn) rubik's cube pyramid: a (nxnxn) pyraminx with period 2 and period 3 cuts oct: an (nxnxn) octahedron with period 3 and period 4 cuts skewb: a diagonal cut rubik's cube cubes, triangles, & hexagons: sliding block puzzles There are no self-solvers provided with these. Also you may want to check out the tetris games which only use X. altetris: polyomino version of tetris alweltris: polyomino version of welltris altertris: polyiamond version of tetris (they bounce of the walls) alhextris: polyhexes version of tetris (again, they bounce off the walls) These are found in ftp.x.org in /contrib I recently heard that 10x10 tangle has no solution. I was trying to solve that one too. What a waste! (90 days of compute time, wow thats one efficient program, fast machine , or both!) Oh, unfortuately the motif puzzles need motif to compile. Is there a good public domain substitute for Motif? Have fun David From bagleyd@source.asset.com Thu Apr 14 13:21:45 1994 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23729; Thu, 14 Apr 94 13:21:45 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA64031; Thu, 14 Apr 1994 12:55:50 -0400 Date: Thu, 14 Apr 1994 12:55:50 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9404141655.AA64031@source.asset.com> To: cube-lovers@ai.mit.edu Subject: Mailing List Hi, I was wondering if you can add me to your mailing list. I put some motif puzzles at ftp.x.org in /contrib/motif_puzzles. They are: rubik: a (nxnxn) rubik's cube pyramid: a (nxnxn) pyraminx with period 2 and period 3 cuts oct: an (nxnxn) octahedron with period 3 and period 4 cuts skewb: a diagonal cut rubik's cube cubes, triangles, & hexagons: sliding block puzzles There are no self-solvers provided with these. A record keeps track of how many move it takes you to solve them. A record of 32767 means I never did it. (I do not follow any standard notation for a move (for example, on the rubik's cube a move is any 1/4 turn)). Unfortuately the motif puzzles need motif to compile. Is there a good public domain substitute for Motif? Also you may want to check out the tetris games which only use X. altetris: polyomino version of tetris alweltris: polyomino version of welltris altertris: polyiamond version of tetris (they bounce off the walls) alhextris: polyhexes version of tetris (again, they bounce off the walls) These are found in ftp.x.org in /contrib I recently heard that 10x10 tangle has no solution. I was trying to solve that one too. What a waste! (90 days of compute time, wow thats one efficient program, fast machine , or both!) Have fun David From iavlang@cs.vu.nl Tue Apr 19 03:52:25 1994 Return-Path: Received: from top.cs.vu.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA07113; Tue, 19 Apr 94 03:52:25 EDT Received: from iavlang.cs.vu.nl by top.cs.vu.nl id aa24220; 19 Apr 94 8:01 MET DST X-Sender: iavlang@top.cs.vu.nl Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Organisation: Vrije Universiteit Amsterdam X-Department: Faculteit der Wiskunde en Informatica X-Address: De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands X-Telephone: +31-20-5485572 X-Fax: +31-20-6427705 Date: Tue, 19 Apr 1994 08:01:19 +0200 To: cube-lovers@life.ai.mit.edu From: Izak van Langevelde Subject: Cubism for Fun Message-Id: <9404190801.aa24220@top.cs.vu.nl> In the archive of this mailing list I found the following: >CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch >Cubists Club). It appears a bit irregular, but a few times a year. >Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to >approximately $ 15.-. Institutional membership is also possible. >Information is available from the editor Gerald Maurice Unfortunaty, the abovementioned editor didn't respond to my email. Does anyone know whether CFF still exists? Who is the current editor? Thanks, Izak van Langevelde From iavlang@cs.vu.nl Tue Apr 19 09:32:59 1994 Return-Path: Received: from top.cs.vu.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14894; Tue, 19 Apr 94 09:32:59 EDT Received: from iavlang.cs.vu.nl by top.cs.vu.nl id aa25902; 19 Apr 94 10:10 MET DST X-Sender: iavlang@top.cs.vu.nl Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Organisation: Vrije Universiteit Amsterdam X-Department: Faculteit der Wiskunde en Informatica X-Address: De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands X-Telephone: +31-20-5485572 X-Fax: +31-20-6427705 Date: Tue, 19 Apr 1994 10:10:25 +0200 To: cube-lovers@ai.mit.edu From: Izak van Langevelde Subject: Cubism for Fun Message-Id: <9404191010.aa25902@top.cs.vu.nl> In the archive of this mailing list I found the following: >CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch >Cubists Club). It appears a bit irregular, but a few times a year. >Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to >approximately $ 15.-. Institutional membership is also possible. >Information is available from the editor Gerald Maurice Unfortunaty, the abovementioned editor didn't respond to my email. Does anyone know whether CFF still exists? Who is the current editor? Thanks, Izak van Langevelde From dik@cwi.nl Tue Apr 19 18:17:58 1994 Return-Path: Received: from meermin.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16561; Tue, 19 Apr 94 18:17:58 EDT Received: from boring.cwi.nl by meermin.cwi.nl with SMTP id AA23023 (5.65b/%I%/CWI-Amsterdam); Wed, 20 Apr 1994 00:16:33 +0200 Received: by boring.cwi.nl id AA05363 (5.65b/3.8/CWI-Amsterdam); Wed, 20 Apr 1994 00:15:17 +0200 Date: Wed, 20 Apr 1994 00:15:17 +0200 From: Dik.Winter@cwi.nl Message-Id: <9404192215.AA05363=dik@boring.cwi.nl> To: cube-lovers@ai.mit.edu, iavlang@cs.vu.nl Subject: Re: Cubism for Fun > In the archive of this mailing list I found the following: > >CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch > >Cubists Club). It appears a bit irregular, but a few times a year. > >Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to > >approximately $ 15.-. Institutional membership is also possible. > >Information is available from the editor Gerald Maurice > Unfortunaty, the abovementioned editor didn't respond to my email. > Does anyone know whether CFF still exists? Who is the current editor? It ought to work. Perhaps mail got lost? Just today I received CFF 33, a summary of the contents will be forthcoming. dik From dik@cwi.nl Tue Apr 19 18:47:54 1994 Return-Path: Received: from meermin.cwi.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18104; Tue, 19 Apr 94 18:47:54 EDT Received: from boring.cwi.nl by meermin.cwi.nl with SMTP id AA23196 (5.65b/%I%/CWI-Amsterdam); Wed, 20 Apr 1994 00:47:50 +0200 Received: by boring.cwi.nl id AA05401 (5.65b/3.8/CWI-Amsterdam); Wed, 20 Apr 1994 00:46:34 +0200 Date: Wed, 20 Apr 1994 00:46:34 +0200 From: Dik.Winter@cwi.nl Message-Id: <9404192246.AA05401=dik@boring.cwi.nl> To: cube-lovers@life.ai.mit.edu Subject: CFF33 Cubism For Fun number 33. I just received it. It is dated February 1994, so it is a bit late ;-). Here a summary of the contents. 1. Dr. Dragon's Polycons by Bernhard Wiezorke and Jacques Haubrich A new, apparently interesting, puzzle from Japan, a lot like polyonimos. Given a rectangular grid you can make pieces from the horizontal and vertical lines connecting the points. The writers coin the term 'monocon' for the piece consisting of a single line segment, 'dicon' for the two different pieces that consist of two connected line segment (one is angled, the other not). Similarly there are 5 'tricons' and 16 'tetracons'. The puzzle consists of 10 of the 16 'tetracons' that must be put on a 5x5 rectangular grid. The authors also look at extensions of the puzzle. 2. The Hollow Pyramid by Jan de Ruiter. In a previous issue there was a puzzle about a hollow pyramid made up of balls that must be constructed by the 25 different pieces that consist of 4 connected balls. Jan de Ruiter was the first to solve the puzzle (with a computer). Here he explains the program. 3. Junior Polycubes by Jacques Haubrich. Pieces consist of 1 to 4 connected cubes that must be put on a 6x6 square. Not so much a puzzle, more like Tangram: create forms. 4. Folding Puzzles by Leo Links. About puzzles where some intricate folding is needed to solve. 5. Cross Pattern Piling by Dieter Gebhardt. A puzzle where you put counters on a square and its four neighbours. The goal is to pile up to a common height for all the squares. The article also discusses a modified version where counting is done mod 2. Associated with it comes the 24th CFF contest. 6. Gouge Packing Puzzle by Gaetan Gouge. Description of and some elaborations about a packing puzzle. 7. Spots Puzzle by Harold Cataquet. Elaborations about a puzzle from A.L.Hoffman, Puzzles old and new, New York, 1920. 8. Arrow-Minded by Ivan Moscovich. Start with a fully-connected hexagon. Put random arrows on all edges. Next add nodes on all intersections. This gives 19 nodes in all. The problem is to find a Hamiltonian path along the nodes, minding the arrows (not always possible). An original puzzle by the writer. 9. Prime Pentacube Pakcings by Frits Gobel. Start with the pentacube consisting of a square base of four cubes and one cube on top of it in a corner. Is it possible to pack a 5x5x5 cube with 25 such pentacubes? What other figures can be packed? 10. Contest 25 by Ekkehard Kuenzell. Pack a figure with all 29 different pentacubes. 11. Rubik's Rabbits by Luc de Smet. Discussion of this latest by Rubik. 12. Party Impressions by Gerald Maurice. Impressions of the Puzzle Party and Cube Day, Augustus 1993 in the Netherlands. Further results of contest 23, a book review by Mark Peters and the first of a series of columns by Edward Hordern. Cubism For Fun is a newsletter published by Nederlandse Kubus Club NKC (Dutch Cubists Club). Applications for membership to the treasurer: Lucien Matthijsse Loenapad 12 3402 EP IJsselstein The Netherlands Membership fee NLG 25.- (about US$ 15.-; add transaction costs). dik From SCHMIDTG@beast.cs.hh.ab.com Tue Apr 19 19:38:21 1994 Return-Path: Received: from beast.cs.hh.ab.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21973; Tue, 19 Apr 94 19:38:21 EDT Date: Tue, 19 Apr 1994 19:38:17 -0400 (EDT) From: SCHMIDTG@beast.cs.hh.ab.com To: cube-lovers@ai.mit.edu Message-Id: <940419193817.20407598@iccgcc.cs.hh.ab.com> Subject: Re: Cubism for Fun and Games > > In the archive of this mailing list I found the following: > > >CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch > > >Cubists Club). It appears a bit irregular, but a few times a year. > > >Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to > > >approximately $ 15.-. Institutional membership is also possible. > > >Information is available from the editor Gerald Maurice > > > Unfortunaty, the abovementioned editor didn't respond to my email. > > Does anyone know whether CFF still exists? Who is the current editor? > >It ought to work. Perhaps mail got lost? Just today I received CFF 33, >a summary of the contents will be forthcoming. > >dik Glad to see someone has had some luck with CFF. I sent some postage stamps to the address posted in rec.puzzles a few years ago and never heard back. Also, obtained no response from my email to Longridge. Geez, I thought at least I deserved some sort of reply! -- Greg Schmidt schmidtg@iccgcc.decnet.ab.com From SCHMIDTG@beast.cs.hh.ab.com Tue Apr 19 20:45:05 1994 Return-Path: Received: from beast.cs.hh.ab.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27117; Tue, 19 Apr 94 20:45:05 EDT Date: Tue, 19 Apr 1994 20:45:03 -0400 (EDT) From: SCHMIDTG@beast.cs.hh.ab.com To: cube-lovers@ai.mit.edu Message-Id: <940419204503.204073c2@iccgcc.cs.hh.ab.com> Subject: Re: Cubism for Fun and Games (Correction and Apologies!) >> > In the archive of this mailing list I found the following: >> > >CFF is a newsletter published by the Nederlandse Kubus Club NKC (Dutch >> > >Cubists Club). It appears a bit irregular, but a few times a year. >> > >Yearly membership fee is now NLG 25.- (Dutch Guilders) which amounts to >> > >approximately $ 15.-. Institutional membership is also possible. >> > >Information is available from the editor Gerald Maurice >> >> > Unfortunaty, the abovementioned editor didn't respond to my email. >> > Does anyone know whether CFF still exists? Who is the current editor? >> >>It ought to work. Perhaps mail got lost? Just today I received CFF 33, >>a summary of the contents will be forthcoming. >> >>dik > >Glad to see someone has had some luck with CFF. I sent some postage stamps >to the address posted in rec.puzzles a few years ago and never heard back. >Also, obtained no response from my email to Longridge. Geez, I thought at >least I deserved some sort of reply! > Whoops, I think I got my signals crossed here. I was actually referring to DOTC Newsletter (Domain of the Cube) not CFF. I apologize for any misunder- standing caused by this. -- Greg Schmidt schmidtg@iccgcc.decnet.ab.com From coxj@rpi.edu Mon Apr 25 11:13:35 1994 Return-Path: Received: from mail1.its.rpi.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18585; Mon, 25 Apr 94 11:13:35 EDT Received: from cox (cox.stu.rpi.edu [128.113.85.84]) by mail1.its.rpi.edu (8.6.7/8.6.4) with SMTP id LAA15800 for ; Mon, 25 Apr 1994 11:13:32 -0400 Message-Id: <199404251513.LAA15800@mail1.its.rpi.edu> X-Sender: coxj@mail1.its.rpi.edu Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Mon, 25 Apr 1994 11:12:43 -0400 To: cube-lovers@life.ai.mit.edu From: coxj@rpi.edu (Jeffrey M. Cox) Subject: Unsubscribe X-Mailer: I would like to be taken off the mailing list. | _____ _ _ | Jeff "The Master" Cox | | / __ / ) / ) | coxj@rpi.edu | | / / /__) (_ (_ | | | \/ \__ / / | "There's a fine line between clever and stupid" | From ronnie@cisco.com Mon Apr 25 12:14:29 1994 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22257; Mon, 25 Apr 94 12:14:29 EDT Received: from localhost.cisco.com by lager.cisco.com (8.6.8+c/CISCO.SERVER.1.1) with SMTP id JAA03034; Mon, 25 Apr 1994 09:14:06 -0700 Message-Id: <199404251614.JAA03034@lager.cisco.com> X-Authentication-Warning: lager.cisco.com: Host localhost.cisco.com didn't use HELO protocol To: coxj@rpi.edu (Jeffrey M. Cox) Cc: cube-lovers@life.ai.mit.edu Subject: Re: Unsubscribe In-Reply-To: Your message of "Mon, 25 Apr 1994 11:12:43 EDT." <199404251513.LAA15800@mail1.its.rpi.edu> Date: Mon, 25 Apr 1994 09:14:05 -0700 From: "Ronnie B. Kon" > I would like to be taken off the mailing list. >| _____ _ _ | Jeff "The Master" Cox >| / __ / ) / ) | coxj@rpi.edu >| / / /__) (_ (_ | >| \/ \__ / / | "There's a fine line between clever and stupid" Would that be the line between cube-lovers and cube-lovers-request? Ronnie From alan@parsley.lcs.mit.edu Mon Apr 25 21:41:35 1994 Return-Path: Received: from parsley.lcs.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23830; Mon, 25 Apr 94 21:41:35 EDT Received: by parsley.lcs.mit.edu id AA14007; Mon, 25 Apr 94 21:40:40 -0400 Date: Mon, 25 Apr 94 21:40:40 -0400 Message-Id: <9404260140.AA14007@parsley.lcs.mit.edu> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu To: coxj@rpi.edu Cc: cube-lovers@life.ai.mit.edu In-Reply-To: Jeffrey M. Cox's message of Mon, 25 Apr 1994 11:12:43 -0400 <199404251513.LAA15800@mail1.its.rpi.edu> Subject: Unsubscribe Date: Mon, 25 Apr 1994 11:12:43 -0400 From: coxj@rpi.edu (Jeffrey M. Cox) I would like to be taken off the mailing list. | _____ _ _ | Jeff "The Master" Cox | / __ / ) / ) | coxj@rpi.edu | / / /__) (_ (_ | | \/ \__ / / | "There's a fine line between clever and stupid" Despite the fact that you stupidly mailed your administrative request to the entire mailing list, I have removed you from Cube-Lovers. I thought I would also take this opportunity to remind the rest of you of the contents of the greeting message I send all new subscribers. Old-timers who've seen previous versions of this file can amuse themselves by noticing that this winter we had some of our heaviest traffic ever. ------- Begin Standard Greeting ------- Don't expect to receive any mail anytime soon. Cube-Lovers is mostly quiet these days. Our addresses are Cube-Lovers@AI.MIT.EDU for submissions and Cube-Lovers-Request@AI.MIT.EDU for administrivia. Please note that Cube-Lovers-Request is processed by a human being, not a computer program (such as LISTSERV or Majordomo). If your request is not instantly processed, it is because I don't spend my entire life reading my electronic mail. I do know how to interpret many of the commands typically sent to such programs, but I would prefer it if instead you can remember to address me in complete sentences. 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 twelve (compressed) files "cube-mail-0.gz" through "cube-mail-11.gz". 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 cube-mail-9 28 May 92 7 Jan 93 155755 cube-mail-10 20 Mar 93 6 Dec 93 171881 cube-mail-11 6 Dec 93 18 Feb 94 349772 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 ------- End Standard Greeting ------- From 70410.1050@compuserve.com Fri Apr 29 13:28:11 1994 Return-Path: <70410.1050@compuserve.com> Received: from arl-img-1.compuserve.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25900; Fri, 29 Apr 94 13:28:11 EDT Received: from localhost by arl-img-1.compuserve.com (8.6.4/5.940406sam) id NAA28174; Fri, 29 Apr 1994 13:28:12 -0400 Date: 29 Apr 94 13:24:23 EDT From: Jerry Slocum <70410.1050@compuserve.com> To: "Cube Lovers @ MIT" Subject: Mechanical Puzzles Message-Id: <940429172422_70410.1050_CHV84-1@CompuServe.COM> Announcements An exhibition of "MAZES and PUZZLES" opens May 27 and closes Sept.5 at the Museum of Science and Industry in Chicago. It includes a "people maze", 80 hands-on mechanical puzzles of many types to challenge visitors and 640 mechanical puzzles of all types and ages that are displayed in 21 cases. I will send a flyer with details to anyone upon request. A Directory of Puzzle Collectors (232), Mail Order puzzle sellers (96), Puzzle periodicals (6), and Retail puzzle stores (147), worldwide, has just been published by the non-profit Slocum Puzzle Foundation. Is is available for $10. postpaid. The Slocum Puzzle Foundation The Slocum Puzzle Foundation was established on August 10, 1993 as a nonprofit public benefit Corporation. It has been approved by the State of California and the U.S. Government as a charitable and educational Foundation. The purpose of the Foundation is to educate the public on puzzles, their history, development, and use in various cultures of the world. The Foundation will actively support the use of puzzles for education. The Foundation will educate the public on puzzles through: Puzzle exhibitions at museums, libraries, universities, and primary and secondary schools, with emphasis on interactive, hands-on puzzles. Publication of books, compendiums, and research papers on puzzles, and a Directory of puzzle collectors. Supporting and encouraging study and research of the history, development, and use of puzzles in various cultures of the world. Supporting and encouraging communication among puzzle experts, educators, historians, and the public. Building and maintaining a collection of puzzles and a library to support these activities and to be available for puzzle exhibitions, education, research and study. The first project of the Foundation is to support a Maze and Puzzle exhibition at the Museum of Science and Industry in Chicago, Illinois. The Directory is the first publication of the Slocum Puzzle Foundation. We are interested in suggestions of projects to support exhibitions, publications, research and educational activities. We would welcome volunteers to help with the activities of the Foundation. We would welcome donations of puzzles and puzzle literature and financial support. All gifts are deductible from California and Federal income taxes. Jerry Slocum Internet:70410.1050@compuserve.com Address: 257 South Palm Drive, Beverly Hills, CA 90212 Phone:310-273-2270 Fax:310-274-3644 From e0f2m2wm@credit.erin.utoronto.ca Sat Apr 30 23:59:34 1994 Return-Path: Received: from credit.erin.utoronto.ca by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10571; Sat, 30 Apr 94 23:59:34 EDT Received: by credit.erin.utoronto.ca id <34069>; Sun, 1 May 1994 00:02:31 -0400 From: Do Anh Vu To: cube-lovers@life.ai.mit.edu Subject: Request to unsubscribe from Cube-Lovers list Message-Id: <94May1.000231edt.34069@credit.erin.utoronto.ca> Date: Sun, 1 May 1994 00:02:25 -0400 Hi, I would like to be removed from the Cube-Lovers list. ...I'm staying away from my mail for now. Thanks in advance. Do Anh From bagleyd@source.asset.com Sun May 1 00:00:57 1994 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10807; Sun, 1 May 94 00:00:57 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA38537; Sat, 30 Apr 1994 23:38:49 -0400 Date: Sat, 30 Apr 1994 23:38:49 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9405010338.AA38537@source.asset.com> To: 70410.1050@compuserve.com, Cube-Lovers@ai.mit.edu Subject: Re: Mechanical Puzzles Mr Jerry Slocum I just got your Directory, great job! The last order seems mixed up. I ordered: Jug w/ <>'s 1989 Puzzle Calendar I only received Jug w/ Diamonds. (The directory was from a previous order). I included $29 , I believe. Also , thanks for the info on cube-lovers. If this is garbled its because my modem is not working so well. David Bagley 58 Winsor Place Glen Ridge NJ O7028 From pbeck@pica.army.mil Thu May 19 09:09:47 1994 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08289; Thu, 19 May 94 09:09:47 EDT Date: Thu, 19 May 94 9:09:30 EDT From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Subject: rubik's 94 items Message-Id: <9405190909.aa08091@COR6.PICA.ARMY.MIL> NEWer RUBIK'S PUZZLES There is a line of Rubik's puzzles, currently available that are distributed by Western Publishing in their line of GOLDEN GAMES. This line was to have two new items for 1994. They are available in europe but Western has decided not to make them available in the USA. 1 - Rubik's Maze: six connected cubes that lay in a plane and have lines drawn on them. The object is turn the blocks until a continuous path is constructed. 2 - Rubik's Rabbits: This looks a magician's top hat. Looking down at the hat it is divided into 8 wedges. The layers(5) are turned until a rabbit appears in each wedge or in no wedges at all. PRE-1994 ITEMS 1 - Rubik's Cube 4: standard cube with rubik's likeness on a center sticker. 2 - RUBIK's Fifteen: a plunger type sequential motion puzzle 3 - Rubik's Dice: a hollow cube with holes where the die spots go. Internally there are colored sheets of plastic that by flipping the cube can be made to cover up the holes 4 - Rubik's Tangle: comes in 4 versions, discussed at length on cube lovers MY PRICES: - Rubik's Cube 4: $10 - RUBIK's Fifteen: $10 - Rubik's Dice: $12 - Rubik's Tangle: $8 each or $30 for a set of all 4 CONTINENTAL USA POSTAGE: $2 for 1 item, $3 for 2 or more outside of CONTINENTAL USA 20% surface , 40% 1st class. PETER BECK 54 RICHWOOD PLACE DENVILLE , NJ 07834 DAYS 201-724-4812 From pbeck@pica.army.mil Thu May 19 09:12:09 1994 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08406; Thu, 19 May 94 09:12:09 EDT Date: Thu, 19 May 94 9:11:55 EDT From: Peter Beck (BATDD) To: cube-lovers@life.ai.mit.edu Subject: PIONEER PUZZLES Message-Id: <9405190911.aa08959@COR6.PICA.ARMY.MIL> I was wondering if anybody has bought puzzles from: PIONEER PUZZLES POB 183 CHEROKEE, TEXAS 76832 1800-441-1796. They make wire disentanglement puzzles. thanks pete From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Mon May 23 16:51:56 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14699; Mon, 23 May 94 16:51:56 EDT Message-Id: <9405232051.AA14699@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4733; Mon, 23 May 94 11:00:32 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 1173; Mon, 23 May 1994 11:00:32 -0400 X-Acknowledge-To: Date: Mon, 23 May 1994 11:00:24 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: Modelling Centerless Cubes On 13 Feb 1994, I proposed a way to model centerless cubes which would (in Dan Hoey's words) retain the symmetrical nature of the problem. I need to post a partial correction/retraction. The conventional model for centerless cubes loses the symmetrical nature of the problem. For example, for a corners-only cube, seven cubies are modeled rather than eight, and for an edges-only cube, eleven cubies are modeled rather than twelve. My proposal in February was to use cosets of the form xC to model centerless cubes, where x is a cube and where C is the set of twenty-four whole cube rotations. This proposal in turn requires an interpretation of C such that C is a subset of G, the entire cube group. C is a group, but normally it is not considered to be a subset of G, hence it is not normally considered to be a subgroup of G. That is, C moves the centers of the faces, but G does not. The required interpretation is obtained by removing the centers of each face, and defining rotational orientation by convention so that the cube is solved only when the Up color is Up, the Front Color is Front, and so forth. Under this interpretation, C is indeed a subset (and hence a subgroup) of G. More correctly, C[even] is a subset of G, C is a subset of GC (the corners only cube), C is a subset of GE (the edges only cube), and C is a subset of GS, where GS= (Q is the set of quarter turns and S is the set of slice moves). That is, when you start talking about C as a subset of G, you have to worry about odd and even permutations. Hence, you have to say C is a subset of GS or C[even] is a subset of G in order not to violate parity rules. All of the above was posted in February, and I am still comfortable with it. However, I went on to say that GS/C, G/C[even], GC/C, and GE/C were all groups under the operation xC * yC = (xy)C. I find that I must retract this claim. In my note in February, I did not give a proof, but rather appealed to a proof in Frey and Singmaster's _Handbook of Cubik Math_. I now find that I mis-applied their proof. In order to show the nature of the problem, I find it useful to go through an attempted proof and determine the point at which it fails. Note that the proposed group elements are not cubes, they are cosets. We proceed as follows: 1. Associativity: (xC * yC) * zC = (xy)C * zC = ((xy)z)C = (x(yz))C = xC * (yz)C xC * (yC * zC) Note that the associativity of the proposed group G/C derives directly from the associativity of G. 2. Identity: we propose that the identity is iC iC * xC = (ix)C = xC xC * iC = (xi)C = xC Note that the identity of the proposed group G/C derives directly from the identity i of G. Further note that the identity iC of the proposed group G/C is C, which is precisely what you would want for the identity of a centerless cube. 3. Inverse: we propose that (xC)'=x'C xC * x'C = (xx')C = iC x'C * xC = (x'x)C = iC Note that the inverse of xC in the proposed group G/C derives from the inverse of x in G. 4. Closure: This is where we have our problem. We require that if xC * yC = (xy)C, then (xy)C must be a coset of C. But the representation of xC and yC is not unique. That is, xC=(xd)C, where d is in C, and yC=(ye)C where e is in C. It is the case that (x(ye))C = (xy)C, but in general it is not the case that ((xd)y)C = (xy)C. Hence, we can have xC=(xd)C, but have it be the case that xC * yC is not equal (xd)C * yC. Hence, we do not have closure. Strictly speaking, this same problem afflicts our "proof" for the inverse, but I deferred discussing the problem until I got to closure. If the problem is repaired for closure, it is also repaired for inverses (see the next paragraph for a discussion of normal subgroups). Cosets of a subgroup H are said to be normal if xH = Hx for all x. I was implicitly and incorrectly assuming that C is a normal subgroup of G, but it is not. For normal subgroups, closure of coset multiplication is readily proven. Frey and Singmaster's proof is for normal subgroups only, and I was applying it to C, which is not normal. It is instructive to consider briefly what xC vs. Cx means for cubes. We can interpret the left coset xC as simply holding a cube in your hands and rotating it any way you wish in space without performing any twists. The right coset Cx is a little trickier. The cube x must be pre-multiplied by each c in C to form Cx. If you have cube x in your hands, there is no obvious thing you can do to form Cx. The thing that is most intuitive to me personally is to think in terms of "rotation by color", which is the way I described pre-multiplication when I first posted some of the results of my work back in December. That is, think of holding the cube still, but recoloring it by permuting the colors. The elements of the coset Cx look the "same" but with the colors permuted. It is not possible to perform this operation on a real cube (short of pulling off the stickers and putting them back on), but the operation can be readily performed on a computer model. Having said all this, I keep thinking that there must be a way to define an operation on the cosets xC so that they form a group. However, I have been unsuccessful in doing so. I would welcome any advice from the net. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From ishius@ishius.com Thu May 26 14:32:42 1994 Return-Path: Received: from holonet.net ([198.207.169.7]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29144; Thu, 26 May 94 14:32:42 EDT Received: from DialupEudora (ishius@localhost) by holonet.net (Anton Dovydaitis) with SMTP id LAA17753; Thu, 26 May 1994 11:27:13 -0700 Message-Id: <199405261827.LAA17753@holonet.net> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 26 May 1994 11:28:21 -0800 To: cube-lovers@ai.mit.edu From: ishius@ishius.com (ishius@holonet.net) Subject: ISHI PRESS MAILING LIST! ISHI PRESS MAILING LIST! Earlier, I have intruded on this fine discussion to solicit e-mail addresses for Ishi Press's retail puzzle e-mail list. Due to some unforseen hardware problems, however, we were cut off from the net for a couple weeks, and have lost all of our e-mail lists. If you would like to receive e-mailings of Ishi Press's impressive line of mechanical puzzles, send us your e-mail address. If you would like a full color catalog, including puzzle reference guide, send us your postal address as well. PLEASE INDICATE THAT YOU ARE INTERESTED IN PUZZLES AND/OR GO (an ancient oriental strategy game). Retail E-mailings are sent about once per month, so we won't be stuffing your e-mail box with more junk to slog through. While some e-mailings duplicate our paper sales literature, there are often descriptions, reviews, and offers that we would never include in a general mailing (such as damaged or second merchandise, unique items, out of print books). Always feel free to write me if you have any questions or comments. Anton Dovydaitis Customer Support =========================================================================== Ishi Press International 408/944-9900 vc, 408/944--9110 FAX 76 Bonaventura Drive 800/859-2086 Toll Free Order Line San Jose, CA 95134 ishius@ishius.com (or @holonet.net) From bagleyd@source.asset.com Fri May 27 13:46:56 1994 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA28346; Fri, 27 May 94 13:46:56 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA64537; Fri, 27 May 1994 13:42:15 -0400 Date: Fri, 27 May 1994 13:42:15 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9405271742.AA64537@source.asset.com> To: cube-lovers@ai.mit.edu Subject: xrubik Hi I just finished up "xrubik", a UNIX-X Rubik's cube. It has been tested on Linux, SunOS, and HP-UX. It currently resides on ftp.x.org at /contrib/games/puzzles. Here's the blurb from the README in that directory: ------------------------------------------------------------- xrubik has been converted from xmrubik. New features: hold down control key to move whole cube letters that represent colors can now be changed in mono-mode Bug fix: when xmrubik did not recognize when the cube was solved nontrivially (i.e. the number of cubes on an edge > 1 OOPS.) The /R5contrib/xmpuzzles: xmpyramid xmoct xmskewb xmcubes xmtriangles xmhexagons are currently being changed to exclude Motif dependencies. The Motif versions will no longer be maintained. The proposed collection includes: SLIDING BLOCK PUZZLES xcubes: expanded 15 puzzle xtriangles: same complexity as 15 puzzle xhexagons: 2 modes: one ridiculously easy, one harder than 15 puzzle ROTATIONAL 3D PUZZLES xrubik: a nxnxn rubik's cube xpyramid: a nxnxn tetrahedron (a nxnxn pyraminx) with Period 2, Period 3, and Combined cut modes xoct: a nxnxn octahedron with Period 3, Period 4, and Combined cut modes xskewb: a cube with diagonal cuts The rest of the platonic solids (the dodecahedron and the icosahedron) seem too hard for me. These programs do not have self-solvers like "magiccube" (Motif) or "puzzle" (X). ----------------------------- Have fun David (the newbie) From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Mon May 30 22:48:07 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22299; Mon, 30 May 94 22:48:07 EDT Message-Id: <9405310248.AA22299@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2575; Mon, 30 May 94 21:36:09 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 6359; Mon, 30 May 1994 21:36:08 -0400 X-Acknowledge-To: Date: Mon, 30 May 1994 21:36:07 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: Branching Factors and God's Algorithm Search Trees At various times, there have been discussions about what the maximum distance from Start might be in God's algorithm. One argument is made with respect to worst/best case branching factors. For example, a simple calculation is that the first move has at most twelve possibilities and that each subsequent move has eleven possibilities, when dealing with Q-turns only. For Q-turns plus H-turns, the same argument would be eighteen possibilities for the first move and seventeen possibilities for each subsequent move. My experience is that search trees tend to develop relatively constant branching factors after some sort of variable startup. I expect Rubik's cube to be no different. I just wonder if anyone has calculated some number of levels for the full Rubik's cube, enough levels for the hypothesized steady state branching factor to be achieved. I have not done so, but if anyone has, it might shed considerable light on the question of the maximum distance from Start. Subsets of the cube such as corners only and edges only have been calculated. It is suggestive to examine branching factors for the cases which have already been calculated. The question of "average branching factor" is subject to interpretation because it is not necessarily clear when the distribution has achieved its steady state. I am including a number of tables giving branching factors for the cases which have been calculated already. I will preface the tables with the following comments: 1. The distributions for edges-only cubes have a variable branching factor during a startup phase, then have a relatively constant branching factor for several levels. and finally the distribution has sort of a tail. 2. The distributions for corners-only cubes have a variable branching factor during a startup phase, and almost immediately the distribution has a tail. The number of cases simply is not large enough to support an extended constant branching factor in the middle of the distribution. It's sort of like a very short airplane flight where it is time to descend about the time the ascent is completed. 3. I would expect the distributions for a full cube to have an even longer period with a constant branching factor than the distributions for edges-only cubes because the number of cases is so much larger. There should be plenty of time for a plateau between the startup phase and any tail of the distribution. 4. There are an equal number of odd and even permutations. For the cases where you restrict yourself to Q-turns, there are therefore equal numbers of states an even distance from Start and an odd distance from Start. Hence, the distribution tends either to have two adjacent levels with approximately equal numbers of states, or else tends to have one dominant level with a level on each side of the dominant level with about half the number of states in the dominant level. 5. For the cases where you allow both Q-turns and H-turns, there tends to be one dominant level which contains most of the of the states. 6. Those of you who followed all the traffic on this list in December and January will recall that my work with God's algorithm exploits symmetric conjugates in order to reduce the size of the problem. It turns out that using conjugates does not change the average branching factor once you get past the startup portion of the distribution. This effect can be a bit hard to see for corners-only cubes because the steady state portion of the distribution is so short, but the effect is very striking for edges-only cubes. I would expect the effect to be very striking, as well, for the case of the full cube. ------------------------------------------------------------------ 2x2x2 Cube using Q-turns and H-turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 9 9.00 2 2.00 4.50 2 54 6.00 5 2.50 10.80 3 321 5.94 19 3.80 16.89 4 1847 5.75 68 3.58 27.16 5 9992 5.41 271 3.99 36.87 6 50136 5.02 1148 4.24 43.67 7 227536 4.54 4915 4.28 46.29 8 870072 3.82 18364 3.74 47.38 9 1887748 2.17 39707 2.16 47.54 10 623800 0.33 13225 0.33 47.17 11 2644 0.00 77 0.01 34.34 Total/Avg 3674160 ? 4.83 77802 ? 3.54 47.22 ------------------------------------------------------------------ 2x2x2 Cube using Q-turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 6 6.00 1 1.00 6.00 2 27 4.50 3 3.00 9.00 3 120 4.44 6 2.00 20.00 4 534 4.45 17 2.83 31.41 5 2256 4.22 59 3.47 38.24 6 8969 3.98 217 3.68 41.33 7 33058 3.69 738 3.40 44.79 8 114149 3.45 2465 3.34 46.31 9 360508 3.16 7646 3.10 47.15 10 930588 2.58 19641 2.57 47.38 11 1350852 1.45 28475 1.45 47.44 12 782536 0.58 16547 0.58 47.29 13 90280 0.12 1976 0.12 45.69 14 276 0.00 10 0.01 27.60 Total/Avg 3674160 ? 3.05 77802 ? 2.92 47.22 ------------------------------------------------------------------ Corners of 3x3x3 Cube using Q-turns and H-turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 18 18.00 2 2.00 9.00 2 243 13.50 9 4.50 27.00 3 2874 11.83 71 7.89 40.48 4 28000 9.74 637 8.97 43.96 5 205416 7.34 4449 6.98 46.17 6 1168516 5.69 24629 5.54 47.44 7 5402628 4.62 113049 4.59 47.79 8 20776176 3.85 433611 3.84 47.91 9 45391616 2.18 947208 2.18 47.92 10 15139616 0.33 316823 0.33 47.79 11 64736 0.00 1481 0.00 43.71 Total/Avg 88179840 ? 4.74 1841970 ? 4.63 47.87 ------------------------------------------------------------------ Corners of 3x3x3 Cube using Q-turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 12 12.00 1 1.00 12.00 2 114 9.50 5 5.00 22.80 3 924 8.11 24 4.80 38.50 4 6539 7.08 149 6.21 43.89 5 39528 6.04 850 5.70 46.50 6 199926 5.06 4257 5.01 46.96 7 806136 4.03 16937 3.98 47.60 8 2761740 3.43 57800 3.41 47.78 9 8656152 3.13 180639 3.13 47.92 10 22334112 2.58 466052 2.58 47.92 11 32420448 1.45 676790 1.45 47.90 12 18780864 0.58 392558 0.58 47.84 13 2166720 0.12 45744 0.12 47.37 14 6624 0.00 163 0.00 40.64 Total/Avg 88179840 ? 4.48 1841970 ? 4.29 47.87 ------------------------------------------------------------------ Edges of 3x3x3 Cube Without Centers using Q-turns and H-Turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 18 18.00 2 2.00 9.00 2 243 13.50 9 4.50 27.00 3 3240 13.33 75 8.33 43.20 4 42359 13.07 919 12.25 46.09 5 538034 12.70 11344 12.34 47.43 6 6666501 12.39 139325 12.28 47.85 7 79820832 11.97 1664347 11.95 47.96 8 888915100 11.14 18524022 11.13 47.99 9 8056929021 9.06 167864679 9.06 48.00 10 27958086888 3.47 582489607 3.47 48.00 11 3883792136 0.14 80930364 0.14 47.99 12 8827 0.00 314 0.00 28.11 Total/Avg 40874803200 ? 12.26 851625008 ? 11.99 48.00 ------------------------------------------------------------------ Edges of 3x3x3 Cube Without Centers Using Q-turns Distance Number of Branching Number of Branching Ratio of from Cubes Factor M Factor Cubes to Start Conjugates Conjugates 0 1 1 1.00 1 12 12.00 1 1.00 12.00 2 114 9.50 5 5.00 22.80 3 1068 9.37 25 5.00 42.72 4 9759 9.14 215 8.60 45.39 5 88144 9.03 1860 8.65 47.39 6 786500 8.92 16481 8.86 47.72 7 6916192 8.79 144334 8.76 47.92 8 59623239 8.62 1242992 8.61 47.97 9 495496593 8.31 10324847 8.31 47.99 10 3695351994 7.46 76993295 7.46 48.00 11 17853871137 4.83 371975385 4.83 48.00 12 18367613703 1.03 382690120 1.03 48.00 13 395043663 0.02 8235392 0.02 47.97 14 1080 0.00 54 0.00 20.00 15 1 0.00 1 0.02 1.00 Total/Avg 40874803200 ? 8.80 851625008 ? 8.63 48.00 ------------------------------------------------------------------ = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From Joel.Franklin@altosax.reed.edu Thu Jun 9 15:09:41 1994 Return-Path: Received: from dharma.reed.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA07939; Thu, 9 Jun 94 15:09:41 EDT Received: from 134.10.2.28 by dharma.reed.edu (/\==/\ Smail3.1.25.1 #25.21) id ; Thu, 9 Jun 94 12:09 PDT Message-Id: <93428@altosax.reed.EDU> Date: 09 Jun 94 12:07:54 PDT From: Joel.Franklin@altosax.reed.edu (Joel Franklin) Subject: To: CUBE-LOVERS@life.ai.mit.edu How do I subscribe to this list? From rprakash@cdotp.ernet.in Wed Jun 22 08:37:49 1994 Return-Path: Received: from sangam.ncst.ernet.in by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA01414; Wed, 22 Jun 94 08:37:49 EDT Received: (from uucp@localhost) by sangam.ncst.ernet.in (8.6.8.1/8.6.6) with UUCP id SAA03435 for cube-lovers@ai.ai.mit.edu; Wed, 22 Jun 1994 18:07:29 +0530 Received: from cdotp.UUCP by doe.ernet.in (4.1/SMI-4.1-MHS-7.0) id AA09375; Wed, 22 Jun 94 17:18:53+0530 Received: by cdotp.ernet.in (4.1/SMI-4.1) id AA07212; Wed, 22 Jun 94 17:14:48+050 Date: Wed, 22 Jun 94 17:14:48+050 From: rprakash@cdotp.ernet.in (PRAKASH R.) Message-Id: <9406221214.AA07212@cdotp.ernet.in> To: cube-lovers@life.ai.mit.edu Dear cube-lovers-request, please send me some information & problems about the cube. i haven't been able to solve the cube fully yet. i can get about 60-70% . i need some suggestions about how to solve the cube also. thanking you, -love prakash r. From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Wed Jun 29 14:11:23 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20966; Wed, 29 Jun 94 14:11:23 EDT Message-Id: <9406291811.AA20966@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4561; Wed, 29 Jun 94 13:45:43 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 1210; Wed, 29 Jun 1994 13:45:43 -0400 X-Acknowledge-To: Date: Wed, 29 Jun 1994 13:45:42 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: Comments on Cube Lengths (Long, 1 of 2) As you all know, the length of a cube X is defined as the shortest process P such that XP = I, and is denoted as |X|. One definition of God's Algorithm is simply that God's algorithm is the knowledge of |X| for all cubes. I wish to make some observations about |X| as related to various models of the cubes and their symmetries. The set C is the set of 24 rotations of the cube. The set M is the set of 48 rotations and reflections of the cube, where half of M is C and the other half of M is C reflected. The first (obvious) observation is that |X| = |m'Xm| for all m in M. That is, the set of all M-conjugates of a cube have the same length. Another way to say the same thing is that |m'Xm| = |n'Xn| for all m and n in M. Actually, there is an even more obvious observation that probably should be made. The set G is the set of all cubes, where G is generated as G=, where Q is the set of 12 quarter-turns of the cube. The *really* obvious observation is that if X is in G, then m'Xm is in G for all m in M. Furthermore, if GC is the set of corners-only cubes, then X in GC implies m'Xm in GC, and if GE is the set of edges-only cubes, then X in GE implies m'Xm in GE. Finally, the observation that |X| = |m'Xm| remains true whether X is in G, in GC, or in GE. The process of forming M-conjugates in G (or in GC or in GE) induces a partition which is an equivalence relation. Hence, the set {m'Xm} for all m in M is an equivalence class. Since, |m'Xm| = |n'Xn| for all m and n in M, it is meaningful to speak of the length of {m'Xm}, namely |{m'Xm}| = |Y|, where Y is any element of {m'Xm}. Now, consider cubes of the form Xc where X is in G and c is in C. We first observe that Xc is in G if and only if c is even. Half the elements in C are even, and half are odd. An odd permutation in C is even on the corners but is odd on the edges; hence, Xc is not in G when c is odd. On the other hand, Xc is in GC for all X in GC, and Xc is in GE for all X in GE. The fact that Xc is in G only for even elements of C is why I thought it was important to make the "really obvious" observation that m'Xm is in G for all X in G and all m in M. The two cases m'Xm and Xc are similar on the surface, but different when you dig a little bit deeper. With respect to lengths, we can observe that |Xc| >= |X| whenever Xc is well-defined (that is, whenever c is even for G, or for all c for GC and GE). The process of performing rotations in G (even rotations in G, or any rotation in GC or in GE) induces a partition which is an equivalence relation. Hence, the set {Xc} for all (or even, as appropriate) c in C is an equivalence class. The collection of all sets of the form {Xc} can serve as a model for cubes without centers. However, it is not the case that |Xc| = |Xd| for all c and d in C. Nonetheless, it is meaningful to speak of |{Xc}|. Namely, |{Xc}| = min{|Xd|} for all (or even) d in C. Hence, we have |{Xc}| <= |Xd| for all (or even) d in C. The definition |{Xc}| = min{|Xd|} probably requires a bit of justification. For a cube without centers, the solved or Start state is {Ic} for all (or even) c in C. Hence, Start is C (or C[even]), and we need the shortest process P such that XP is in C in order to calculate |{Xc}|. Consider the set {P[1], P[2], ... P[24]} where P[n] is the shortest process for which (Xc[n])P[n] = I. Observe, that XP[n] is in C for all n in 1..24. This immediately gives us |P| <= |P[n]| for all n in 1..24. Conversely, if XP is in C, then there exists some c[n] in C such that Xc[n]P = I. This gives us |P[n]| <= |P| for some n in 1..24. Therefore, |P| = min{|P[n]|} for n in 1..24. Note that we have |{Xc}| <= |X| <= |Xd|. On its face, this may seem somewhat paradoxical, but I believe that it is entirely correct. There is a huge difference is speaking of |{Xc}| as opposed to speaking of |Xd|. Xd is an (atomic) element of G; {Xc} is a set. Elements of {Xc} are also in G, but the *set* {Xc} is not in G. My model for cubes without centers is really {m'Xmc} rather than {Xc}. However, the results from above are readily combined. That is, we can speak of |{m'Xmc}|, namely |{m'Xmc}| = min{|(m'Xm)d|} for all (or even) d in C. As before, we have |{m'Xmc}| <= |m'Xm| <= |m'Xmd|. Note that in the middle of this last string of inequalities we could insert any of |X| = |m'Xm| = |{m'Xm}|. In my God's algorithm data base for cubes without centers, I store ordered pairs of the form (Y,|{m'Xmc}|), where Y is a representative element of the set {m'Xmc}. Note that Y is in G (or GC or GE, as appropriate). It is a picky point, but the data base does *not* consist of ordered pairs of the form (Y,|Y|). Remember that |Y| >= |{m'Xmc}|. My God's algorithm data base for cubes with centers nominally consists of ordered pairs of the form (Z,|{m'Xm}|), where Z is a representative element of the set {m'Xm}. Unlike the case without centers, we do have |Z| = |{m'Xm}|, so we could also say the data base elements are of the form (Z,|Z|). However, the data representation is really a bit different to take advantage of the relationship between sets of the form {m'Xmc} and sets of the form {m'Xm}. A set of the form {m'Xmc} can be partitioned into (up to) twenty-four sets of the form {m'Xm}, where the (up to) twenty-four sets are indexed by C. Let Y=Repr({m'Xmc}). Then, the data base is ordered pairs of the form (Yc[i],|Yc[i]|) for i in 1..24. Note that Yc[i] is in G, but can be said to be a representative element for sets of the form {m'(Yc[i])m}, which in turn is a set of the form {m'Xm} for some X in G. Finally, there is no real need to store the Yc[i]; it is only necessary to store the lengths. Hence, a data base element for cubes with centers is really, really of the form: (Y,|{m'Xmc}|,|Yc[1]|,|Yc[2]|, .... |Yc[24]|), where Y is a representative element of {m'Xmc}. Note that this is a very compressed format. The representative element Y is stored only once for the 24 different values for c. Note also that the solution for cubes without centers is stored in the same data base as the solution for cubes with centers. Finally, since |m'Yc[i]m| = |Yc[i]|, we have stored the length of all cubes by storing the length of only one cube for each M-conjugancy class. It is not really necessary explicitly to store the solution for cubes without centers to have the solution for cubes without centers in the same data base. That is, |{m'Xmc}|=min(|Yc[i]|) for i in 1..24. But it takes very little space to do so and is convenient for certain calculations. (to be continued) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Wed Jun 29 15:00:05 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23837; Wed, 29 Jun 94 15:00:05 EDT Message-Id: <9406291900.AA23837@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4592; Wed, 29 Jun 94 13:56:03 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 1494; Wed, 29 Jun 1994 13:56:03 -0400 X-Acknowledge-To: Date: Wed, 29 Jun 1994 13:56:02 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: Comments on Cube Lengths (Long, 2 of 2) (continuation) I have described this data base structure once before (a little less formally before), but I wanted to describe it again because there is some interesting (to me, at least) analysis that can be derived from the data base, over and above God's algorithm. First, it is interesting to compare |{m'Xmc}| to the various |Yc[i]|. Recall that |Yc[i]| >= |{m'Xmc}| for all i in 1..24. Also, by the definition of |{m'Xmc}|, there is at least one i in 1..24 such that |Yc[i]| = |{m'Xmc}|. I posted a comparison of |{m'Xmc}| to |Yc[i]| for corners-only cubes on 4 December 1993. (I have the "without centers" part of the edges-only data base done, but it will take many more months to complete the "with centers" part. So corners-only is the only complete data base we have to work with.) At the time, I received a couple of comments to the effect that people didn't understand what I was comparing to what. I hope this note clarifies the situation. For example (and referring to my previous note with respect to corners-only cubes), there is only one element of the form {m'Xmc} for which |{m'Xmc}| = 0. The only element for which the length is 0 is Start, but in a corners only cube without centers, any rotation of Start is still at Start and still has length 0. Here is a brief excerpt from my note of 4 December 1993. Corresponding Distances from Start Using Only q-turns Without With Centers Centers Number Distance from Distance from of Nodes Start Start 0 0 1 2 1 4 2 6 1 What this means is as follows. First, we have |{m'Imc}| = 0. Let Y=Repr({m'Imc}). Then, there is 1 element of the form Yc for which |Yc|=0, 1 element of the form Yc for which |Yc|=2, 2 elements of the form Yc for which |Yc|=4, and 1 element of the form Yc for which |Yc|=6. In words, suppose you have a corners-only cube (peel off the edge tabs, but keep the corners and centers). Then, suppose the corners look "solved" if you ignore the centers. The corners will be rotated relative to the centers. In all, there are 24 different ways they can be rotated, including the identity, where they are not rotated. But under M-conjugancy, some of the 24 rotations are equivalent. Under M-conjugancy, there is one way they can be 0 moves from Start, one way they can be two moves from Start (RL' is equivalent to DU', for example ), two ways they can be four moves from Start, and one way they can be six moves from Start. Among other things, this says that any rotation of the corners (ignoring the edges) can be accomplished in no more than six quarter turns. This example illustrates why a set of the form {m'Xmc} may be partitioned into "up to" twenty-four elements of the form {m'Xm}, rather than "exactly" twenty-four elements. Normally, a set of the form {m'Xmc} contains 1152 elements, where 1152=24*48. It can in turn be partitioned into twenty-four elements of the form {m'Xm} which contain forty-eight elements each. But cubes which are "symmetric" reduce the number because various M-conjugates are equivalent. I normally think of the God's algorithm data base as a matrix, with the rows indexed by the representative elements Y, and the columns indexed by C (or more simply, by 1..24). Because of M-conjugate symmetry, there are always a few empty cells in the matrix. M-conjugate symmetry did not cause me any computational difficulty when I was working with cubes without centers. That is, suppose {m'(X1)mc} and {m'(X2)mc} are the same set for X1 not equal X2. My "representative element calculator" would calculate the same representative element Y in both cases. But in the case of cubes with centers, the "representative element calculator" had to calculate both a representative element Y and an associated rotation index Cind in 1..24. When a set {m'Xmc} had exactly 1152 elements (most of the time), the calculation of Cind was correct. But when a set {m'Xmc} had fewer than 1152 elements, I would get a different Cind depending on which element of the set I started with. That is, the loops in the program actually calculate 1152 elements in any case, but if the set really has less than 1152 elements, then some of the elements are generated multiple times. (The loops have no way of knowing ahead of time how many elements are going to be in the set.) The generation of the same set elements multiple times severely messed up the calculation of Cind until I figured out what was going on. I want to finish by getting back to what I started with, the lengths of cubes. As I said, the God's algorithm results for edges without centers are complete (posted to the list back in December), but the God's algorithm calculations for edges with centers are still work in progress. However, I noticed something striking about the partial edges with centers results when I compared them with the completed edges without centers results. For example, here is a table which compares the results when using q-turns only. Distance Number of Branching Number of Branching from M-Conjugate Factor M-Conjugate Factor Start Classes Classes Without With Centers Centers 0 1 1 1 1 1.00 1 1.00 2 5 5.00 5 5.00 3 25 5.00 25 5.00 4 215 8.60 215 8.60 5 1860 8.65 1886 8.77 6 16481 8.86 16902 8.96 7 144334 8.76 150442 8.90 8 1242992 8.61 1326326 8.81 9 10324847 8.31 11505339 8.67 10 76993295 7.46 96755918 8.40 11 371975385 4.83 750089528 7.75 12 382690120 1.03 .... 13 8235392 0.02 work 14 54 0.00 in 15 1 0.02 progress Total 851625008 As you can see, with or without centers, there are the same number of cubes (actually, equivalence classes) at each distance from Start from level 0 through level 4. From level 5 on, there are more cubes with centers than without. Why is the number the same through level 4, and what happens at level 5 to make the numbers different? Actually, overall there are about twenty-four times more cubes with centers than without, so it is not surprising to find more cubes with centers than without at fairly low levels in the search tree. So fundamentally, the question is, why does the divergence occur at level 5? Well, I can't explain why it is level 5 exactly, but I can explain what is going on. Consider level 0. There is one row in the data base where |{m'Xmc}|=0. There are twenty-four cells in the same row for |Yc[i]|, corresponding to the twenty-four rotations of the representative element Y. For exactly one of these cells, we have |Yc[i]|=0. The remainder of the cells are either undefined (meaning the cell represents a rotation which is M-conjugate equivalent with another rotation), or else we have |Yc[i]|>=5. Hence, any rotation of the edges of the cube requires at least 5 q-turns to accomplish. After the data base is complete, we can determine exactly how many q-turns are required to accomplish each rotation of the edges, just as we can already do with the corners. Similar comments apply to level 1 through 4. There is exactly one rotation of the representative element that has the same length as representative element. All the other rotations of the representative element are either M-conjugate equivalent to the representative element, or else have a length greater than or equal to 5. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From jkato@tmastb.eec.toshiba.co.jp Tue Jul 5 20:56:33 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16890; Tue, 5 Jul 94 20:56:33 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA29268; Wed, 6 Jul 94 09:56:25 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA05152; Wed, 6 Jul 94 09:56:34 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA25603; Wed, 6 Jul 94 09:59:08 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA27918; Wed, 6 Jul 94 09:51:13 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA03691; Wed, 6 Jul 94 09:56:14 JST Date: Wed, 6 Jul 94 09:56:14 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407060056.AA03691@tmastb.eec.toshiba.co.jp> To: cube-lovers@life.ai.mit.edu Subject: Cube-Lovers ML I am a member of NKC(CFF) and Puzzle KONWAKAI(Academy of Recreational Mathematics, Japan). I knew your Mailing List from Jerry's Puzzle Collectors Directory. I would like to join your ML. In this summer I am going to 14th International Puzzle collectors Party in Seattle. ------ Thank you, Toshi(Junk) Kato 2-14-60 Hishinuma, Chigasaki 253 Japan Tel/Fax: +81-467-52-1447 E-mail: jkato@tmastb.eec.toshiba.co.jp From jkato@tmastb.eec.toshiba.co.jp Wed Jul 6 07:00:52 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06244; Wed, 6 Jul 94 07:00:52 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA29726; Wed, 6 Jul 94 20:00:23 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA29247; Wed, 6 Jul 94 20:00:31 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA25259; Wed, 6 Jul 94 20:02:58 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA06748; Wed, 6 Jul 94 19:55:10 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA05730; Wed, 6 Jul 94 20:00:14 JST Date: Wed, 6 Jul 94 20:00:14 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407061100.AA05730@tmastb.eec.toshiba.co.jp> To: Cube-Lovers@ai.mit.edu Subject: SBP "Magic sQ" Sliding Block Puzzle "Magic sQ" Fig.1 is incomplete. +---+---+---+ Can you complete a magic square | 2 | 9 | 4 | with minimum sliding steps? +---+---+---+ | 7 | 5 | 3 | You, very easy or not? +---+---+---+---+ | 1 | 6 | 8 | | Fig.1 +---+---+---+---+ ------ Toshi(Junk) Kato from Japan E-mail: jkato@tmastb.eec.toshiba.co.jp Tel/Fax: +81-467-52-1447 From mouse@collatz.mcrcim.mcgill.edu Fri Jul 8 15:24:40 1994 Return-Path: Received: from Collatz.McRCIM.McGill.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29240; Fri, 8 Jul 94 15:24:40 EDT Received: (root@localhost) by 8873 on Collatz.McRCIM.McGill.EDU (8.6.8.1 Mouse 1.0) id PAA08873; Fri, 8 Jul 1994 15:24:30 -0400 Date: Fri, 8 Jul 1994 15:24:30 -0400 From: der Mouse Message-Id: <199407081924.PAA08873@Collatz.McRCIM.McGill.EDU> To: Cube-Lovers@ai.mit.edu Subject: Re: SBP "Magic sQ" Cc: jkato@tmastb.eec.toshiba.co.jp > Sliding Block Puzzle "Magic sQ" > Fig.1 is incomplete. +---+---+---+ > Can you complete a magic square | 2 | 9 | 4 | > with minimum sliding steps? +---+---+---+ > | 7 | 5 | 3 | > You, very easy or not? +---+---+---+---+ > | 1 | 6 | 8 | | Fig.1 > +---+---+---+---+ Not hard, but cutely deceptive. The figure as supplied is a magic square with the 1 and 6 switched. It is not possible to switch two adjacent tiles in a quadrilateral sliding-block puzzle of this sort (there's an easy induction proof that only even permutations are possible). Thus, either it's not possible or the solution involves some other magic square. Since the 8 must clearly be in the lower right corner of the resulting magic square, there are only two squares possible. One is the magic square that is almost present already; the other is its reflection: 2 9 4 2 7 6 7 5 3 9 5 1 6 1 8 4 3 8 Fortunately, the "other" magic square is an even permutation from the provided start position. It's then just a straightforward tree search to find a solution. A simple brute-force "meet in the middle" breadth-first search finds a solution easily. Move the 8 aside, then move as follows (* represents the blank space): 2 9 4 2 9 4 2 9 4 2 9 4 2 9 4 2 9 4 2 9 4 2 9 4 7 5 3 -> 7 5 3 -> 7 5 3 -> * 5 3 -> 5 * 3 -> 5 3 * -> 5 3 6 -> 5 3 6 -> 1 6 * 1 * 6 * 1 6 7 1 6 7 1 6 7 1 6 7 1 * 7 * 1 +-------+ 2 9 4 2 * 4 2 4 * 2 4 6 2 4 6 2 4 6 | 2 4 6 | 2 4 6 5 * 6 -> 5 9 6 -> 5 9 6 -> 5 9 * -> 5 9 1 -> 5 9 1 -> 5 9 1 -> * 9 1 -> 7 3 1 7 3 1 7 3 1 7 3 1 7 3 * 7 * 3 | * 7 3 | 5 7 3 +-------+ 2 4 6 2 * 6 * 2 6 9 2 6 9 2 6 9 2 6 9 2 6 9 2 6 9 * 1 -> 9 4 1 -> 9 4 1 -> * 4 1 -> 4 * 1 -> 4 7 1 -> 4 7 1 -> * 7 1 -> 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 * 3 * 5 3 4 5 3 * 2 6 2 * 6 2 7 6 2 7 6 2 7 6 9 7 1 -> 9 7 1 -> 9 * 1 -> 9 5 1 -> 9 5 1 4 5 3 4 5 3 4 5 3 4 * 3 4 3 * Then put the 8 back, and you're done, in a total of 30 moves (28 shown, plus the two moves of the 8). For those who care about such things, the boxed position is the midpoint at which the two searches met. (This is fairly obvious - since the number of moves is even, the configuration on which the searches met must be the middle one.) This solution exhibits curious near-symmetries in portions of the path taken by the blank space. Anyone have any thoughts on this? Perhaps I should modify the program so it reports _all_ solutions of this length; there may be something interesting lurking here. der Mouse mouse@collatz.mcrcim.mcgill.edu From jkato@tmastb.eec.toshiba.co.jp Mon Jul 11 00:08:01 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12507; Mon, 11 Jul 94 00:08:01 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA17248; Mon, 11 Jul 94 13:07:26 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA14931; Mon, 11 Jul 94 13:07:34 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA00693; Mon, 11 Jul 94 13:11:28 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA22006; Mon, 11 Jul 94 13:01:23 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA17550; Mon, 11 Jul 94 13:06:53 JST Date: Mon, 11 Jul 94 13:06:53 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407110406.AA17550@tmastb.eec.toshiba.co.jp> To: mouse@collatz.mcrcim.mcgill.edu Cc: Cube-Lovers@ai.mit.edu In-Reply-To: der Mouse's message of Fri, 8 Jul 1994 15:24:30 -0400 <199407081924.PAA08873@Collatz.McRCIM.McGill.EDU> Subject: Re: SBP "Magic sQ" Many thanks. Especially to Dan Hoey and der Mouse . I have recieved E-mail individually from Mr.Dan Hoey too, Date: Thu, 7 Jul 94 16:43:02 EDT From: hoey@AIC.NRL.Navy.Mil To: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Subject: Re: SBP "Magic sQ" A little hack tells me you can solve that by moving 1,7,3,6,1,9,4,1,7,5,9,4,2,9,4,7,5,9,2,5,8. That's 21 moves, or 30 if you count by the tile. It's optimal in both metrics. Do you know anything about Rubik's Cube? Dan Hoey Hoey@AIC.NRL.Navy.Mil I was surprised to recieve solutions both so quickly. Cubes-Lover is very smart team,　I think. Thanks again, Junk Kato jkato@tmastb.eec.toshiba.co.jp From @mail.uunet.ca:mark.longridge@canrem.com Fri Jul 15 03:19:03 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25039; Fri, 15 Jul 94 03:19:03 EDT Received: by mail.uunet.ca via suspension id <91049-1>; Fri, 15 Jul 1994 03:02:49 -0400 Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <92031-3>; Fri, 15 Jul 1994 02:30:57 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA02807; Fri, 15 Jul 94 01:45:11 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1A658F; Fri, 15 Jul 94 01:40:47 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: DOTC 1.4 is done From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.770.5834.0C1A658F@canrem.com> Date: Fri, 15 Jul 1994 01:32:00 -0400 Organization: CRS Online (Toronto, Ontario) Domain of the Cube 1.4 is finally done! --------------------------------------- I've finally finished the new issue of the DOTC newsletter, and I'm basically happy with it. I believe I owe my fellow cubists an apology for taking so long, especially Greg Schmidt and Dan Hoey. I've enjoyed using the cube since 1981 and I wish I had more time and energy to put into it. I was also rather ill earlier this year, and things at work seemed to always interrupt. Nevertheless, the first 20 copies are finally ready to mail. Despite the fact these initial copies are slightly flawed I am no longer willing to wait. This time the issues have beige covers and are stapled like a booklet, much the same as David Singmaster's Cubic Circular. I'm pleased with the printer's results, and I am mailing out the first issues tomorrow. I have considerable work done on issue 1.5 and I expect the next issue to be ready relatively soon. - Mark New Technique for Pattern Finding: Cycle a process until you find the identity, e.g. (F1 B1 R1 D1)^24 = I then bisect the process if the order is even, ( F1 B1 R1 D1 ) ^ 12 = Pattern, naturally this process is order 2. --------------------------------------------------------------------- Hmmmm, actually I have some questions that have been bugging me for some time. I while back a guy was watching me use my cube program and I explained that the reason I like studying group theory is because it provided greater insights into the cube. He then asked me: "What are other uses of group theory?" and "What are the practical uses of group theory" to which I haltingly replied (somewhat vaguely) that it helped show relationships between geometry and algebra. I felt this explanation unsatisfactory. I also mumbled about symmetry and architecture. I'm sure there is a better answer than that! Also why is it in math that |-11| means absolute value and can also be the order of G, e.g. Let G be a Group, and |G| means the order of G. Here is another tidbit for the cube archives: Rare 11-cycle of edges: ( L2 B1 R1 D3 L3 ) ^ 7 (35) alternately: F2 R3 U1 D3 B3 D1 L3 U3 D1 B1 L1 D1 B2 U2 D2 R2 B2 D1 (18) -> Mark <- From jkato@tmastb.eec.toshiba.co.jp Mon Jul 18 06:11:51 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17123; Mon, 18 Jul 94 06:11:51 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA04049; Mon, 18 Jul 94 19:11:40 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA09681; Mon, 18 Jul 94 19:11:50 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA02578; Mon, 18 Jul 94 19:15:59 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA17013; Mon, 18 Jul 94 19:05:14 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA16075; Mon, 18 Jul 94 19:11:24 JST Date: Mon, 18 Jul 94 19:11:24 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407181011.AA16075@tmastb.eec.toshiba.co.jp> To: Cube-Lovers@ai.mit.edu Subject: A real robot solve the Rubik's Cube but... A real robot which had artificial eyes and arms and computer brain was manufactured at Kawasaki Heavy Industry Co.,Ltd in Japan last year. He can solve the real commercial Rubik's Cube. As he has not so intelligent, it takes about 12 minutes and 120 steps average between starting and finishing the Cube to solve it. Are there any other live robot like him over the world? Do you know? And you can help him more clever with your solving algolithm, can't you? ------ Toshi(Junk) Kato from Japan E-mail: jkato@tmastb.eec.toshiba.co.jp Tel/Fax: +81-467-52-1447 From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Mon Jul 18 10:44:30 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25760; Mon, 18 Jul 94 10:44:30 EDT Message-Id: <9407181444.AA25760@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 6338; Mon, 18 Jul 94 10:41:50 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 7644; Mon, 18 Jul 1994 10:41:50 -0400 X-Acknowledge-To: Date: Mon, 18 Jul 1994 10:41:49 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: God's Algorithm, Additional Level The following is for Q-turns only, whole cubes (both corners and edges), and not considering M-conjugates. I think it would be possible to squeeze out another couple of levels by considering M-conjugates. The best previous result I have found in the archives was through level 7 (reported on 7 December 1981, and again on 3 August 1992). Distance Number Branching from of Factor Start Cubes 0 1 1 12 12.00 2 114 9.50 3 1,068 9.37 4 10,011 9.37 5 93,840 9.37 6 878,880 9.37 7 8,221,632 9.35 8 76,843,595 9.35 (new) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From newfield@vsl.ist.ucf.edu Mon Jul 18 11:56:37 1994 Return-Path: Received: from vsl.ist.ucf.edu (sparc1.vsl.ist.ucf.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00687; Mon, 18 Jul 94 11:56:37 EDT Received: from flix.vsl.ist.ucf.edu by vsl.ist.ucf.edu (4.1/SMI-4.1) id AA24280; Mon, 18 Jul 94 11:53:21 EDT From: newfield@vsl.ist.ucf.edu (Dale Newfield) Received: by flix.vsl.ist.ucf.edu (931110.SGI) id AA03650; Mon, 18 Jul 94 11:53:16 -0400 Date: Mon, 18 Jul 94 11:53:16 -0400 Message-Id: <9407181553.AA03650@flix.vsl.ist.ucf.edu> To: Cube-Lovers@ai.mit.edu Subject: Re: A reaal robot solve the Rubik's Cube but... Sorry to pick nits, but if it is autonomous(sp?), which I think you implied, wouldn't it be an android, instead of a robot? -Dale Dale Newfield I'd rather newfield@vsl.ist.ucf.edu be playing dn1l@{cs,andrew}.cmu.edu xlife. From jkato@tmastb.eec.toshiba.co.jp Mon Jul 18 22:40:44 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03334; Mon, 18 Jul 94 22:40:44 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA06478; Tue, 19 Jul 94 11:40:33 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA20180; Tue, 19 Jul 94 11:40:43 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA20317; Tue, 19 Jul 94 11:44:58 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA03398; Tue, 19 Jul 94 11:34:01 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA17895; Tue, 19 Jul 94 11:40:15 JST Date: Tue, 19 Jul 94 11:40:15 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407190240.AA17895@tmastb.eec.toshiba.co.jp> To: Cube-Lovers@ai.mit.edu Subject: Re: A real robot solve the Rubik's Cube but... Dale Newfield said: Received: from inet-gw-2.pa.dec.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05611; Mon, 18 Jul 94 23:11:47 EDT Received: from jrdmax.jrd.dec.com by inet-gw-2.pa.dec.com (5.65/27May94) id AA18406; Mon, 18 Jul 94 20:08:17 -0700 Message-Id: <9407190307.AA21918@jrdmax.jrd.dec.com> Received: from jrdv04.enet.dec.com by jrdmax.jrd.dec.com (5.65/JULT-4.3) id AA21918; Tue, 19 Jul 94 12:07:47 +0900 Received: from jrdv04.enet.dec.com; by jrdmax.enet.dec.com; Tue, 19 Jul 94 12:07:53 +0900 Date: Tue, 19 Jul 94 12:07:53 +0900 From: Norman Diamond 19-Jul-1994 1206 To: cube-lovers@ai.mit.edu Apparently-To: cube-lovers@ai.mit.edu Subject: Re: A real robot solve the Rubik's Cube but... Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=ISO-2022-JP Toshi Kato says: >Dale Newfield said: > >Pardon me. I wonder if I shoudn't use such words "real" and "live". Half right. It was a real dead robot :-) -- Norman Diamond diamond@jrdv04.enet.dec.com [Digital did not write this.] From jkato@tmastb.eec.toshiba.co.jp Tue Jul 19 00:24:58 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA13058; Tue, 19 Jul 94 00:24:58 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA12852; Tue, 19 Jul 94 13:24:54 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA29994; Tue, 19 Jul 94 13:25:04 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA24830; Tue, 19 Jul 94 13:29:20 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA04886; Tue, 19 Jul 94 13:18:24 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA18202; Tue, 19 Jul 94 13:24:38 JST Date: Tue, 19 Jul 94 13:24:38 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9407190424.AA18202@tmastb.eec.toshiba.co.jp> To: cube-lovers@ai.mit.edu Subject: Re: A real robot solve the Rubik's Cube but... Norman Diamond　said: > >Pardon me. I wonder if I shouldn't use such words "real" and "live". > >Half right. It was a real dead robot :-) Thanx. I think the robotic machine isn't alive now too. ----- Toshi(Junk) Kato E-mail:jkato@tmastb.eec.toshiba.co.jp From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Tue Jul 19 10:55:43 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00780; Tue, 19 Jul 94 10:55:43 EDT Message-Id: <9407191455.AA00780@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9928; Tue, 19 Jul 94 08:56:30 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 8616; Tue, 19 Jul 1994 08:56:30 -0400 X-Acknowledge-To: Date: Tue, 19 Jul 1994 08:56:28 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: More on Centerless Cubes On 13 Feb 1994, I proposed a model for centerless cubes which I claimed met two criteria: 1) it was a group, and 2) it maintained the symmetrical nature of the problem. On 23 May 1994, I retracted the claim that the proposed model was a group. I am now of the opinion that it is impossible to satisfy both criteria simultaneously. I can make a very small modification to the proposed model to make it a group, but the small modification costs the model its cubic symmetry. G is the full cube group, GC is the corners only cube group, and GE is the edges only cube group. The proposed model for centerless cubes consisted of partitioning any of G, GC, or GE into sets of the form {Xc} for all c in C, where C is the set of twenty-four rotations of the cube and X is a cube. The sets are the elements of the proposed group. The sets are called cosets and can also be denoted as xC. The partitions are denoted as G/C, GC/C, and GE/C, respectively. Originally, the proposed group operator was {Xc} * {Yc} = {XYc}. This operator fails to maintain closure, and hence fails to define a group. In order to illustrate the slight modification which will define a group, we will start by restricting ourselves to GC. An operator which works to define GC/C as a group is {Xc} * {Yc} = {VWc}, where V is the unique element of {Xc} such that the urf cubie is properly positioned in the urf cubicle, and W is the unique element of {Yc} such that the urf cubie is properly positioned in the urf cubicle. Any other corner could have been used instead of urf, but once you choose a corner the problem loses its symmetric nature. Well, I guess it still has symmetry, but it is not the uniform symmetry of the cube any more, because there is a preferred orientation. I have found only limited discussion in the archives, but previous investigators have modeled a corners only, centerless cube by leaving one corner fixed. Such a model is clearly a group. For example, if we leave the urf corner fixed, we can generate the group JC as JC=, where we omit all twists of the U, R, and F faces from the set of generators. It is easy to find an isomorphism between GC/C and JC. I would express it as something like {Xc} = {Wc} <--> W, where W is defined as before. W is an element of JC, and as well is an element of {Xc} = {Wc}. {Xc} = {Wc} is an element of GC/C. But W is a particular element of {Xc} = {Wc}, whereas X is an arbitrary element. Also, X is in GC, but X is not in JC unless X = W. The mapping {Wc} <--> W is clearly one-to-one and onto in both directions. For the edges GE, we need to keep one edge cubie fixed, so the centerless cube could be generated by something like JE=, where we keep the uf cubie fixed by omitting all twists of the U and F faces from the set of generators. The isomorphism between GE/C and JE is expressed as {Xc} = {Wc} <--> W, where X is an arbitrary element of GE, and W is the unique element of {Xc} such that the uf cube is properly placed in the uf cubicle. As before, any edge cube would do as well, but once chosen, it is no longer arbitrary. For the whole cube G, at first blush it appears we could model centerless cubes either by keeping a corner cubie fixed, or by keeping an edge cubie fixed. But if we keep a corner cubie fixed, the three immediately adjacent edge cubies are never moved by any Q-turns. We could solve the difficulty by admitting slice turns. But slice quarter-turns are odd on edges and even on corners, so we have to restrict ourselves to slice half-turns. I find this ugly, plus I would prefer to generate G with Q-turns only. Hence, I would prefer to model a centerless full cube as J=, where it is an edge cubie which is held fixed rather than a corner cubie. I said at the beginning that I thought it was impossible for a model of centerless cubes both to be a group and also to maintain cubic symmetry. The reason is as follows: it seems to me that for any model which is a group, it should be possible to find an isomorphism between the model and J (or JC or JE, as appropriate). But J and JC and JE do not have cubic symmetry because there is a preferred orientation. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From @wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU Tue Jul 19 19:09:07 1994 Return-Path: <@wvnvm.wvnet.edu:BRYAN@WVNVM.WVNET.EDU> Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00316; Tue, 19 Jul 94 19:09:07 EDT Message-Id: <9407192309.AA00316@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0827; Tue, 19 Jul 94 11:43:12 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 3312; Tue, 19 Jul 1994 11:43:12 -0400 X-Acknowledge-To: Date: Tue, 19 Jul 1994 11:43:11 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: God's Algorithm, Minor Progress, Q+H Surprisingly, there seems not to be anything in the archives for God's Algorithm for Q+H moves for the whole cube past level 3. Here are some updated results: Distance Number Branching from of Factor Start Cubes 0 1 1 18 18.000 2 243 13.500 3 3,240 13.333 4 43,239 13.345 (new) 5 574,908 13.296 (new) 6 7,618,438 13.252 (new) 7 100,803,036 13.231 (new) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow? From jkato@tmastb.eec.toshiba.co.jp Sun Aug 7 22:18:28 1994 Received: from inet-tsb.toshiba.co.jp by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23184; Sun, 7 Aug 94 22:18:28 EDT Received: from tis2.tis.toshiba.co.jp (tis2) by inet-tsb.toshiba.co.jp (5.67+1.6W/2.8Wb) id AA20727; Mon, 8 Aug 94 11:18:23 JST Received: from tis4.tis.toshiba.co.jp (tis4) by tis2.tis.toshiba.co.jp (5.67+1.6W/6.4J.6-R05) id AA23581; Mon, 8 Aug 94 11:18:31 JST Received: from eecisa by tis4.tis.toshiba.co.jp (5.52/6.4J.6-R06) id AA21120; Mon, 8 Aug 94 11:16:30 JST Received: from tmastb.eec.toshiba.co.jp by eecisa.eec.toshiba.co.jp (4.1/6.4J.6-R1) id AA26984; Fri, 5 Aug 94 19:29:39 JST Received: by tmastb.eec.toshiba.co.jp (4.0/6.4J.6-R1) id AA03406; Fri, 5 Aug 94 19:30:13 JST Date: Fri, 5 Aug 94 19:30:13 JST From: jkato@tmastb.eec.toshiba.co.jp (Toshi Kato) Return-Path: Message-Id: <9408051030.AA03406@tmastb.eec.toshiba.co.jp> To: Cube-Lovers@ai.mit.edu Subject: HIKIMI Wooden Puzzle Competition To promote puzzles throughout the world and to convey the warmth of wood to as many people as possible THE 6th HIKIMI WOODEN PUZZLE COMPETITION will be held in October 1994. We invite puzzlers from around the world to enter. APPLICATION GUIDE 1.Conditions 1)Puzzle must be made of wood 2)Puzzle must be original 3)Puzzle has never been sold commercially 4)Puzzle can be easily mass produced 5)Do not submit puzzles that emphasize artistic design 6)Puzzle may have two or more inventors 7)Entry with more than one puzzle permitted 2.Points for consideration 1)Entry into the competition is free of charge, but each contestant must bear the expense for the sending the puzzle to Hikimi. 2)We may request to purchase the puzzle within the constraints of our budget. 3)Copyright of the puzzles entered belongs to the inventor, but Hikimi Chamber of Commerce reserves the right to first negotiation. 4)Deadline: Puzzle and application must be received by October 14,1994. 3.For an application form, write to: Hikimi-cho Shokokai,Hikimi-cho,Mino-gun,Shimane Prefecture 698-12 Japan Applications in Japanese are prefered. However, since this may be a difficult requirement for non-Japanese entrants, you may send your application in English. 4.Judging Judging will be held sometime during October,1994. All applicants will be notified directly of the results. The commendation ceremony will be held on November 12,1994 in Hikimi Town. 5.Judges Chief Judge: Saburo Oguro Judge : Nob Yoshigahara Judge : Shigeo Takagi Judge : A. Yamashita Judge : T. Ohhata 6.Prizes Grand Prize (one person): ¥500,000(about 5,000 US$) 2nd Prize (two persons): ¥300,000(about 3,000 US$)each 3rd Prize(three persons): ¥100,000(about 1,000 US$)each Runner-ups (several) : ¥ 50,000(about 500 US$)each Sponsored by: Hikimicho Shokokai(Hikimi Chamber of Commerce) Tel:+81-856-56-0310 Fax:+81-856-56-0753 APPLICATION FORM FOR THE 6TH HIKIMI WOODEN PUZZLE COMPETITION Date: +--------------------------------------+--------------------------------------+ |Applicant's name: Age( )|Co-inventors of puzzle | | | Name Age Occupation | +--------------------------------------+-------------------+---+--------------+ |Address: | | | | | | | | | |Tel: Fax: | | | | |Occupation: | | | | +--------------------------------------+-------------------+---+--------------+ |Name of company where you're employed:|Rating by Judges *| | +--------------------------------------+ |Address: |Application number *| | +--------------------------------------+ |Tel: Fax: |Remarks *| +--------------------------------------+ | |Name of puzzle: | | | | | |Number of pieces in the puzzle: | | +--------------------------------------+--------------------------------------+ |Object of puzzle: |Solution: | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +--------------------------------------+ | |Applicant's comments on the puzzle: | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +--------------------------------------+----------------------------------