From cube-lovers-errors@mc.lcs.mit.edu Sun Mar 8 19:35:17 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA05274; Sun, 8 Mar 1998 19:35:17 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 8 03:41:58 1998 Date: Sun, 8 Mar 1998 09:41:21 +0100 (MET) Message-Id: <1.5.4.16.19980308094102.437739b8@mailsvr.pt.lu> To: rshep@simplex.nl From: Georges Helm Cc: geohelm@pt.lu, schubart@best.com, Cube-Lovers@ai.mit.edu Hi, You once asked a question about early rubik's cube solutions (on Schubart's web page) I have solution from 1979 by ANGEVINE James BEASLEY J.D. CAIRNS Colin / GRIFFITHS Dave CLAXTON Mike DAUPHIN Michel (Mathematique et Pedadogie 24/79) EASTER Bob HOWLETT G.S. JACKSON William Bradley JOHNSON K.C. MADDISON Richard NELSON Roy RODDEWIG Ulrich SWEENEN John TAYLOR Don (1978) TRURAN Trevor (Computer Talk 7.11.1979) Regards Georges Georges Helm geohelm@pt.lu http://ourworld.compuserve.com/homepages/Georges_Helm/ http://www.geocities.com/Athens/2715 From cube-lovers-errors@mc.lcs.mit.edu Mon Mar 9 10:21:54 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA06825; Mon, 9 Mar 1998 10:21:54 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 8 18:58:07 1998 Message-Id: <9803082359.AA00210@jrdmax.jrd.dec.com> Date: Mon, 9 Mar 98 08:59:07 +0900 From: Norman Diamond 09-Mar-1998 0859 To: cube-lovers@ai.mit.edu Subject: Re: Taiwanese Invention of the Cube? Reply-To: diamond@jrdv04.enet.dec-j.co.jp, whuang@ugcs.caltech.edu Wei-Hwa Huang replied to me: >>As for patenting, somehow the mixture of "patent" and "Taiwan" in the >>same sentence strikes me as an oxymoron. >>Somehow the mixture of "trademark" and "Taiwan" strikes me as an >>oxymoron too, even though they're not in the same sentence. >>Want to try "copyright" next? :-) >Is it possible to copyright the Cube? That's why I didn't try it. Some puzzle designers do copyright their designs. When one compares patents with copyrights, copyright makes sense. Patents are intended for inventions that improve the quality of life and will become important in industry after the patents expire, so that the inventors starve. Copyrights are for frivolous entertainment like puzzles and photos, so they bring royalties for the lifetime of the creator plus 50 years to the heirs. One can only wonder why patents were ever granted for puzzles. >In any case, stop sneering -- Taiwan has local copyright, trademark, >and patent laws, and has had them for decades. Sure, they haven't >honored international copyright laws, Guess which part of that I was sneering at. >but then again, most other countries don't think Taiwan exists as an >independent country. The Republic of China also thinks Taiwan doesn't exist as an independent country. >When it became economically viable to honor international >copyright, they did so -- such legislation was passed in 1994. >Perhaps you are getting a biased view from living in Japan? No, my unbiased view was based on observations that I had made for decades. ===== Mr. Huang and I had this discussion in private e-mail already. I didn't know that he was going public with it too. Anyway if I understand correctly, Mr. Huang agreed with my point after that, so there's no need to repeat the rest of the discussion unless I misunderstood. [Moderator's note: In any event, further discussion on this topic should be sent to Wei-Hwa Huang and Norman Diamond, rather than to cube-lovers. I somewhat regret passing _any_ of it on. The topic of intellectual property and its legal status is vast, and has eaten bigger lists than this. ] ===== -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital.] From cube-lovers-errors@mc.lcs.mit.edu Mon Mar 9 11:40:13 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA07026; Mon, 9 Mar 1998 11:40:13 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 8 20:10:06 1998 Message-Id: Date: Sun, 8 Mar 1998 20:09:26 -0500 To: tomkeane@mail.del.net, cube-lovers From: Charlie Dickman Subject: Rubik's Tesseract Solution Tom and other cube-lovers, I have completed a solution to the Rubik Tesseract and have included it in the program and it's associated documentation but neither is ready for prime time just yet. I was wondering if there was anyone who would be kind enough to review the documentation and see if the write-up of the solution is reasonably intelligible and provide me some feedback before I make it and the program generally available. It is an HTML document (332K self-extracting-archive) that you can read with your browser. Thanks, Charlie Dickman charlied@erols.com From cube-lovers-errors@mc.lcs.mit.edu Wed Mar 11 13:07:59 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA14544; Wed, 11 Mar 1998 13:07:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 11 07:40:13 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Blindfold Cube-solving Date: 11 Mar 1998 12:39:04 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6e60l8$2bc@gap.cco.caltech.edu> Is there anyone who knows some good techniques for blindfold cube-solving? I can solve the cube in about 7 "peeks" or so, but that's still quite a ways from looking at the cube once and solving it behind one's back. -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you. From cube-lovers-errors@mc.lcs.mit.edu Wed Mar 11 14:44:29 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA14952; Wed, 11 Mar 1998 14:44:29 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 11 13:58:59 1998 Date: Wed, 11 Mar 1998 13:58:48 -0500 (EST) From: Jiri Fridrich To: Wei-Hwa Huang Cc: cube-lovers@ai.mit.edu Subject: Re: Blindfold Cube-solving In-Reply-To: <6e60l8$2bc@gap.cco.caltech.edu> Message-Id: I believe that solving the cube blindfolded in one shot is very difficult if not impossible. One could memorize the orientation of all cubies and their permutation. Then use algorithms for turning the cubes without moving them, and then algorithms for permuting them. One would need to define orintation of cubies on the cube and then the permutation algorithms would have to preserve that orientation. This system would presume one really long "peek" and excellent memory, of course :) Using my system (http://ssie.binghamton.edu/~jirif), I could probably bring down the number of peeks to four with some practice ... Of course, seven is no sweat. Jiri ********************************************* Jiri FRIDRICH, Research Scientist Center for Intelligent Systems SUNY Binghamton Binghamton, NY 13902-6000 Ph/Fax: (607) 777-2577 E-mail: fridrich@binghamton.edu http://ssie.binghamton.edu/~jirif/jiri.html ********************************************* From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 13 12:20:32 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA24252; Fri, 13 Mar 1998 12:20:31 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Mar 12 13:47:19 1998 Sender: mark@ampersand.com To: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Cc: cube-lovers@ai.mit.edu Subject: Re: Blindfold Cube-solving References: <6e60l8$2bc@gap.cco.caltech.edu> From: Mark Atwood Date: 12 Mar 1998 13:47:11 -0500 In-Reply-To: whuang@ugcs.caltech.edu's message of 11 Mar 1998 12:39:04 GMT Message-Id: whuang@ugcs.caltech.edu (Wei-Hwa Huang) writes: > > Is there anyone who knows some good techniques for blindfold cube-solving? > > I can solve the cube in about 7 "peeks" or so, but that's still quite > a ways from looking at the cube once and solving it behind one's back. I have heard of something like "cubes for the blind". Probably either have a different textured material attached to each cubie face, or a Braille glyph embossed into each cubie face. (Never tried to solve one blind, but I could probably solve on in about a dozen or so glances. But for a while I worked on solving them with my feet, after seeing someone do it on TV.) -- Mark Atwood | Thank you gentlemen, you are everything we have come to zot@ampersand.com | expect from years of government training. -- MIB Zed [ Moderator's note: You'll notice this is a different topic. Perhaps Wei-Hwa Huang should consider his problem "memory solving" rather than "blindfold solving". I've heard that John Conway has a good memory method, I think requiring five peeks (cf Roger Frye, 20 Oct 1981). There are also several mentions of tactile cubes in the archives. ] From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 13 15:11:08 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA24858; Fri, 13 Mar 1998 15:11:07 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Mar 12 17:13:30 1998 Message-Id: In-Reply-To: <6e60l8$2bc@gap.cco.caltech.edu> Date: Wed, 11 Mar 1998 17:33:58 -0500 To: cube-lovers@ai.mit.edu From: Kristin Looney Subject: Re: Blindfold Cube-solving Wei-Hwa Huang wrote: > Is there anyone who knows some good techniques for blindfold > cube-solving? > > I can solve the cube in about 7 "peeks" or so, but that's still quite > a ways from looking at the cube once and solving it behind one's back. This brings back fond memories of the trip to CA for the first National Cube contest back in '81... us nine finalists were taken on a day trip to Disney Land and we had a race to see who could solve the cube the fastest in the line to space mountain. As the line winds inside the building, it is really quite dark, and we were on our hands and knees trying to get whatever light we could from the running lights on the floor. I don't remember who won... but it was a huge amount of fun. -K. kristin@wunderland.com http://www.wunderland.com/wts/kristin To all the fishies in the deep blue sea, Joy. From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 13 16:02:47 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA24990; Fri, 13 Mar 1998 16:02:46 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Mar 12 16:09:47 1998 Date: Thu, 12 Mar 1998 22:09:22 +0100 Message-Id: <199803122109.WAA06383@dataway.ch> To: Cube-Lovers@ai.mit.edu From: Geir Ugelstad Subject: Rules for speed-cubing Hello, What are the exact rules for speed cubeing? I have seen that in the World-campionship it was legal to look at the cube 15 seconds and then put it back on the table. How long time did it take from puting it back on the table (after looking) and the real start??? Ys Geir Ugelstad From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 13 17:08:16 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA25241; Fri, 13 Mar 1998 17:08:16 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Mar 12 23:37:59 1998 Date: Thu, 12 Mar 1998 22:34:54 -0600 (CST) From: "J. David Blackstone" Subject: Oddz On website In-Reply-To: <009C2062.FA899020.3@ice.sbu.ac.uk> To: David Singmaster Cc: skouknudsen@email.dk, cube-lovers@ai.mit.edu Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Thu, 19 Feb 1998, David Singmaster wrote: > common knowledge that it was not Rubik's mechanism. One may be able > to get details from the web site that Oddz On (sp??) has set up. Tom I may have missed it, but could someone provide the URL of this website? ----------------------------------------- J. David Blackstone jxb9451@utarlg.uta.edu http://www.geocities.com/Athens/Acropolis/1341 ----------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 17 10:14:50 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA04592; Tue, 17 Mar 1998 10:14:50 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 13 14:48:44 1998 From: Phil Servita Sender: meister@khitomer.epilogue.com To: cube-lovers@ai.mit.edu Subject: not quite blind cubing Date: Fri, 13 Mar 98 14:48:43 -0500 Message-Id: <9803131448.aa12167@khitomer.epilogue.com> whuang@ugcs.caltech.edu (Wei-Hwa Huang) writes: > > Is there anyone who knows some good techniques for blindfold cube-solving? > > I can solve the cube in about 7 "peeks" or so, but that's still quite > a ways from looking at the cube once and solving it behind one's back. Back when i was still in college, myself and a friend would occasionally perform our "geek party trick", which was that we would sit on the floor, back-to-back, and someone would toss one of us a scrambled cube. Whoever caught it would look at it, make a single quarter-turn on it, and pass it over their shoulder to the other person, who would look at it and make another quarter turn, pass it back, and so on. We could solve it in this fashion in just under 2 minutes. -phil From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 17 10:46:50 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA04737; Tue, 17 Mar 1998 10:46:50 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Mar 14 03:49:32 1998 Message-Id: <3.0.3.32.19980313231431.00835810@netcom13.netcom.com> Date: Fri, 13 Mar 1998 23:14:31 -0800 To: Mark Atwood From: Ray Tayek Subject: Re: Blindfold Cube-solving Cc: cube-lovers@ai.mit.edu In-Reply-To: At 01:47 PM 3/12/98 -0500, Mark Atwood wrote: >... >I have heard of something like "cubes for the blind". Probably either >have a different textured material attached to each cubie face, or a >Braille glyph embossed into each cubie face. >... my wife teaches blind kids. do you know where i could get some braile cubes? thanks Ray (will hack java for food) http://home.pacbell.net/rtayek/ hate Spam? http://www.compulink.co.uk/~net-services/spam/ [ Moderator's note: There are quite a few notes in the archives about adding tactile labels to cubes. Adding characters in Braille should be about the easiest thing to do--I'm sure she has a DYMO embosser. ] From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 17 11:02:50 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA04827; Tue, 17 Mar 1998 11:02:49 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 15 15:27:20 1998 From: roger.broadie@iclweb.com (Roger Broadie) To: "Cube Lovers Submissions" Subject: Ideal's patent for 4^3 Date: Sun, 15 Mar 1998 20:29:20 -0000 Message-Id: <19980315202713.AAA21006@home> On 19 Feb 1998 David Singmaster wrote: In my Cubic Circular 1 (Autumn 1981), I recorded that Wim Osterholt, of the Netherlands, had made and patented a 4^3 which he showed me. I don't remember it and I'm not sure when he brought it to London - perhaps Summer 1981? I also recorded that Rainier Seitz (product manager of Arxon which was Ideal's German agent) showed me some German patents and applications for the 4^3 and 5^3. In Cubic Circular 2 (Spring 1982), I record talking with another person who had devised a 4^3 mechanism. In Cubic Circular 3/4 (Spring/Summer 1982), I describe playing with examples. However, I don't recall ever knowing who devised the mechanism that was produced for Ideal. It was common knowledge that it was not Rubik's mechanism I have just come across Ideal's patent for its 4^3. It is US Patent No 4,421,311. The inventor was Peter Sebesteny, and the original application was made in Germany on 8 Feb 1981, so it may have been one of the patents David Singmaster was shown. It can be viewed at the IBM patent site from http://www.patents.ibm.com/details?patent_number=4421311 One of the references cited by the US Patent Examiner was to page 29 of David Singmaster's "Notes on Rubik's Magic Cube" - undoubtedly the remark "One can imagine the 4x4x4 cube or the 3x3x3x3 hypercube. The first might be makeable but its group seems to be much more complicated. The second is unmakeable, but its group structure may be determinable." The corresponding European patent application was taken through to the point where it was ready for grant, but then allowed to lapse. The next stage would have been quite expensive and have required Ideal to translate the specification into the languages of the European countries in which it was to be in force. And the US was not renewed when the first renewal fees became due in 1986. Presumably by then Ideal had lost interest in the patent - they may have calculated there was zero chance of anyone launching an imitation, given the number of 4^3s that had been left unsold. I don't have ready access to information about the German application, but I suspect it was applied for by Sebesteny on his own behalf, and he then interested Ideal in it. Roger Broadie From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 17 11:43:37 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA05013; Tue, 17 Mar 1998 11:43:37 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 15 06:19:03 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Blindfold Cube-solving Date: 15 Mar 1998 11:17:40 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6egdck$cvj@gap.cco.caltech.edu> References: The Moderator wrote: >[ Moderator's note: You'll notice this is a different topic. Perhaps > Wei-Hwa Huang should consider his problem "memory solving" rather > than "blindfold solving". I've heard that John Conway has a good > memory method, I think requiring five peeks (cf Roger Frye, 20 Oct > 1981). There are also several mentions of tactile cubes in the > archives. ] I used the term "blindfold solving" patterned after "blindfold chess", where two players merely recite moves to each other, using no actual pieces or board. As far as "solving in the dark" goes, it reminds me that I have a cube in which under certain lamps, the yellow and white colors are indistinguishable. Solving such a cube can occasionally give a few trip-ups! -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you. From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 17 14:28:15 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA05547; Tue, 17 Mar 1998 14:28:14 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Mar 17 13:30:17 1998 Date: Tue, 17 Mar 1998 13:30:10 -0500 (Eastern Standard Time) From: Jerry Bryan Subject: Re: Blindfold Cube-solving In-Reply-To: <6egdck$cvj@gap.cco.caltech.edu> To: Wei-Hwa Huang Cc: cube-lovers@ai.mit.edu Message-Id: On Sun, 15 Mar 1998, Wei-Hwa Huang wrote: > As far as "solving in the dark" goes, it reminds me that I have a cube > in which under certain lamps, the yellow and white colors are > indistinguishable. Solving such a cube can occasionally give a few > trip-ups! I have had the same problem with orange and red, especially on my 2x2x2. I have a "latter day" 2x2x2 (my kids lost my first one), and the colors in general do not seem quite true to the colors on my 3x3x3 and 4x4x4 cubes. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 24 12:51:28 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA24618; Tue, 24 Mar 1998 12:51:28 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Mar 24 11:52:07 1998 Message-Id: <3517E49F.DF5B21BF@mail.retina.ar> Date: Tue, 24 Mar 1998 13:51:44 -0300 From: Isidro Reply-To: isidroc@usa.net Organization: Frank Zappa's Fan Club To: Cube Lovers Submissions Subject: 5^3 quiz I need to know the answers for these questions: Who invented 5^3? What is the commercial name? How many cubies it has? -- Isidro: isidroc@usa.net [ Moderator's note: There was a note last July mentioning "Rubik's Wahn (5x5x5) (maybe also called Professor's cube, Ultimate or Master Revenge)"--any other names? The number of cubies is obviously 98--why didn't you just count them? ] From cube-lovers-errors@mc.lcs.mit.edu Wed Mar 25 10:09:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA27206; Wed, 25 Mar 1998 10:09:36 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Mar 24 16:14:21 1998 Message-Id: <3.0.5.16.19980324220550.0bd76334@vip.cybercity.dk> Date: Tue, 24 Mar 1998 22:05:50 To: cube-lovers@ai.mit.edu From: Philip Knudsen Subject: RE: 5^3 quiz To my knowledge, the 5x5x5 was invented by Udo Krell. It was produced by Uwe Meffert in 1983. I read somewhere that Dr. Chr. Bandelow had the Hong Kong factory finish extra puzzles from previously manufactured parts around 1990, don't know if this is true. Bandelow is still selling this puzzle, under the name "Giant Magic Cube". It also seems Meffert reissued the 5x5x5 one or two years ago, under the name "Professor's Cube". This new version might have other colors than the original. I have seen the puzzle under the name "Ultimate Cube" several times, the name "Master Revenge" however is new to me. Since Meffert is the manufacturer, the "most" official name for the 5x5x5 is probably "Professor's Cube". Philip K recording and performing artist Vendersgade 15, 3th DK - 1363 Copenhagen K Phone: +45 33932787 Mobile: +45 21706731 E-mail: skouknudsen@email.dk E-mail: philipknudsen@hotmail.com Sms: 4521706731@sms.tdk.dk (short message, no subject) From cube-lovers-errors@mc.lcs.mit.edu Wed Mar 25 12:56:17 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA27652; Wed, 25 Mar 1998 12:56:16 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Mar 24 16:46:37 1998 Date: Tue, 24 Mar 1998 22:46:49 +0100 (MET) Message-Id: <199803242146.WAA06298@relay.euronet.nl> To: Cube-Lovers@ai.mit.edu From: Sytse <4xs2fs@euronet.nl> Subject: Re: 5^3 quiz Isidro, Who invented 5^3? At least I did. In 1982 I designed and built a 5^3 cube, all in plywood. Although I did not aplly for a patent or other registration, as I was only a schoolboy by then, the local newspaper recorded this event. As the wooden prototype was not as speedy as necessary, I later designed a simulator for the Sinclair ZX Spectrum (a then so called 'personal computer' with an amazing 48K RAM memory). This simulator also included a 6^3 cube. 7^3 was not possible as this did not fit in the screen, which was my parents television set. Oh, those were the days! Nowadays I am an architect. Kind regards, Sytse de Maat P.S. If you happen to know other designers of 5^3, please mail me. [ Moderator's note: Can you describe the design that held the plywood model together while allowing it to turn? ] From cube-lovers-errors@mc.lcs.mit.edu Wed Mar 25 15:19:48 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA28021; Wed, 25 Mar 1998 15:19:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 25 02:37:53 1998 Date: Wed, 25 Mar 1998 08:37:42 +0100 Message-Id: <199803250737.IAA30286@dataway.ch> To: Cube-Lovers@ai.mit.edu From: Geir Ugelstad Subject: Jiri's system for solving Rubiks's cube hello cube-lovers For all of you that haven't been into Jiri's home page at http://ssie.binghamton.edu/~jirif, you should realy look into it! Bouth the method and presentation is of very high standard! I bought myself a system in 1982 but I was so dissapointed that I trow it In the garbage just after. With the system I bought in 1982 it was not possible to make it faster than 2-3 minutes. With Jiri's system it should be possible in about 17 sec.! Ys Geir Ugelstad PS: Question to Jiri. How far are you able to do the foreplanning the 15 sec. before the time start to run? Hopefully longer than "Place the four edges from the first layer"? From cube-lovers-errors@mc.lcs.mit.edu Thu Mar 26 11:46:40 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA00652; Thu, 26 Mar 1998 11:46:40 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 25 16:14:40 1998 Date: Wed, 25 Mar 1998 16:10:54 -0500 (EST) From: Jiri Fridrich To: Geir Ugelstad Cc: Cube-Lovers@ai.mit.edu Subject: Re: Jiri's system for solving Rubiks's cube In-Reply-To: <199803250737.IAA30286@dataway.ch> Message-Id: On Wed, 25 Mar 1998, Geir Ugelstad wrote: > it was not possible to make it faster than 2-3 minutes. With Jiri's > system it should be possible in about 17 sec.! Yes, you are right - with my system AND a lot of time on your hands :) I am pretty sure that the systems of other top speed cubists are at least as as good as mine. The system is only half of the secret. > PS: Question to Jiri. How far are you able to do the foreplanning > the 15 sec. before the time start to run? Hopefully longer than > "Place the four edges from the first layer"? Nope. 15 seconds is not a long time to plan more than the four edges. Of course, as you proceed, you will usually be able to spot the corners with their appropriate cubies from the second layer in some nice position and continue without delays ... Jiri ********************************************* Jiri FRIDRICH, Research Scientist Center for Intelligent Systems SUNY Binghamton Binghamton, NY 13902-6000 Ph/Fax: (607) 777-2577 E-mail: fridrich@binghamton.edu http://ssie.binghamton.edu/~jirif/jiri.html ********************************************* From cube-lovers-errors@mc.lcs.mit.edu Thu Mar 26 12:46:25 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA00805; Thu, 26 Mar 1998 12:46:24 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 25 18:23:23 1998 Message-Id: <9803252324.AA16745@jrdmax.jrd.dec.com> Date: Thu, 26 Mar 98 08:24:24 +0900 From: Norman Diamond 26-Mar-1998 0817 To: cube-lovers@ai.mit.edu Subject: RE: 5^3 quiz I bought my first 5^3 from a department store in Japan in 1985, while it was alongside the 3^3 and 4^3 on the mass market. Bought my second one from Dr. Bandelow some time later. In Japan it was called "Professor Cube" which could be taken as "Professor's Cube" because it would be a bit too awkward to pedantically insert the syllable for possessive form (in Japanese grammar) between two polysyllabic foreign words. (Tangential details: pu-ro-fue-so-ru kyu-u-bu is 5 + 3 syllables, while pu-ro-fue-so-ru no kyu-u-bu would be 5 + 1 + 3 syllables.) The magic dodecahedron reached the mass market around 1989 or so. Those were the days. Some time around 1993, the mass market shifted to computer games. -- Norman Diamond diamond@jrdv04.enet.dec-j.co.jp [Speaking for Norman Diamond not for Digital] From cube-lovers-errors@mc.lcs.mit.edu Thu Mar 26 15:10:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA01187; Thu, 26 Mar 1998 15:10:33 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Mar 26 10:47:00 1998 Date: Thu, 26 Mar 1998 15:36:25 +0000 From: David Singmaster To: skouknudsen@email.dk Cc: cube-lovers@ai.mit.edu Message-Id: <009C3C55.665587E6.39@ice.sbu.ac.uk> Subject: RE: 5^3 quiz Bandelow's leaflet, which he encloses with the 5^3, states that the mechanism was invented by Udo Krell, of Hamburg(?). I haven't seen the patent but perhaps Bandelow has details. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu Mar 26 15:58:05 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA01386; Thu, 26 Mar 1998 15:58:04 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Mar 25 19:10:32 1998 Message-Id: <01BD5821.7C9449E0@jburkhardt.ne.mediaone.net> From: John Burkhardt To: Subject: new to list Date: Wed, 25 Mar 1998 19:09:07 -0500 Hi, I just found and joined this list. So I am looking for any and all oddball cube variations I can find. Does anyone have anything to sell or trade. I can trade for a "Magic Dodecahedron" which is the start shaped Hungarian version of the Megaminx and I might be willing to part with a 5x5x5 cube for anything really interesting. I'm looking for an original Tomy Megaminx. Also the octahedron puzzle which is like two Pyraminx's glued together (there might be an official name). I am also searching for a 4x4x4 but I know they are really hard to find these days (mostly because they tend to break). The Dodecahedron puzzle is really amazing. It was actually harder than the 5x5x5 cube. IT took me about 3 hours to work it out! I think once you know the 3x3x3 then all the same moves do similar things and you can easily solve 4x4x4 or 5x5x5 with variations. Of course there are some cool things you can do with these. I must say that I was disappointed with one web page that listed a bunch of moves for the 3x3x3 cube. I was trying some of them out and thinking, my god, how did anyone figure this out, only to then discover that a computer had figured them out. OK, that's certainly an interesting problem, but I have much more fun discovering them on my own. Interstingly enough, solving the dodecahedron led me to some neat new moves for the original cube! So where can we go from here? Have we made all the regular polyhedra into puzzles? Is there hope of actually building 6x6x6 and beyond cubes? Is there really any point to doing it? I suppose they would allow for some nice patterns. Does anyone know of any puzzles that are not in George Helm's collection? I just bought a Magic Cube puzzle at Walgreens for $3. It's a 3x3x3 with psychedelic stickers on it... From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 27 09:48:24 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA03279; Fri, 27 Mar 1998 09:48:24 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 27 06:50:30 1998 Message-Id: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net> From: John Burkhardt To: Subject: Stickers Date: Fri, 27 Mar 1998 06:45:47 -0500 Does anyone know where to find cube stickers? They must come from somewhere! I found some vinyl lettering once and the periods were exactly the right size for a 5x5x5 cube. But they don't come in orange. There must be a way to buy sheets of the stuff. Any ideas? From cube-lovers-errors@mc.lcs.mit.edu Fri Mar 27 13:44:27 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA03643; Fri, 27 Mar 1998 13:44:27 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 27 11:01:41 1998 Date: Fri, 27 Mar 1998 11:02:04 -0500 (EST) From: Nichael Cramer To: John Burkhardt Cc: cube-lovers@ai.mit.edu Subject: Re: Stickers In-Reply-To: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net> Message-Id: John Burkhardt wrote: > Does anyone know where to find cube stickers? They must come from > somewhere! I found some vinyl lettering once and the periods were > exactly the right size for a 5x5x5 cube. But they don't come in > orange. There must be a way to buy sheets of the stuff. Any ideas? Ah, yes, the orange stickers on the 5X .... ;-) Anyway, don't they have sticker sets in any colors other than in the standard cube-pallette? Black or grey come to mind. Not quite the optimal solution, of course, but it would still give you a useable cube. Nichael -- Nichael Cramer work: ncramer@bbn.com home: nichael@sover.net http://www.sover.net/~nichael/ (The cool bit about letters, of course, is that on the 5X5 face in question, you could, say, put almost all the letters of the alphabet --or some other personalized message(s) of your choice-- and give yourself a little something extra to shoot for as you solve the cube.) From cube-lovers-errors@mc.lcs.mit.edu Mon Mar 30 14:54:27 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA04824; Mon, 30 Mar 1998 14:54:26 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 27 20:28:38 1998 Date: Fri, 27 Mar 1998 20:28:57 -0400 (EDT) From: Jerry Bryan Subject: All the Isoglyphs [long] To: Cube-Lovers Message-Id: Dan Hoey introduced glyphs and isoglyphs on 5 August 1997. A glyph is a cube face containing no more than two colors, and an isoglyph is a cube position where every face contains the same glyph. Isoglyphs tend to be very striking and pretty patterns. Each corner and edge facelet of a glyph can be the same or a different color than the center facelet, so there are 2^8 or 256 possible glyphs. Dan reported that there are 51 glyphs unique up to symmetry (70 if chiral pairs are distinguished). On 8 August 1997, Herbert Kociemba reported that there are 35 continuous isoglyphs unique up to symmetry (including Start). A continuous isoglyph is one for which each glyph matches the neighboring glyph along the edge. Herbert did not include the non-continuous glyphs because there are so many, and because non-continuous glyphs are sometimes not so striking and pretty as the continuous glyphs. On 9 August, Dan Hoey classified Herbert's isoglyphs according the their respective glyphs, and provided the usual name for the isoglyphs where a usual name existed. Where a usual name did not exist, Dan provided a reasonable name based on the names of closely related isoglyphs. On 27 August, Mike Reid gave minimal maneuvers for all the continuous isoglyphs in both the quarter-turn and face-turn metrics. I have now calculated all the isoglyphs, using Herbert's Cube Explorer 1.5 program. All I really did was to put each of the 51 glyphs into the program in turn. I can only guess, but this has to be more or less what Herbert did to obtain his results. The only difference is that I asked the program to calculate both continuous and non-continuous isoglyphs, so the task was a bit bigger. My report is much in the spirit of Herbert's original report. I have made no effort to calculate minimal maneuvers, nor have I made any attempt to associate names with the maneuvers. However, my report does include all the glyphs along with their associated isoglyphs. In fact, for each glyph I have included the entire equivalence class of glyphs under the rotations and reflections of the square (either 1, 2, 4, or 8 glyphs in each equivalence class). There is, of course, no necessary relationship between the number of glyphs in the equivalence class and the number of isoglyphs. You only need to put one glyph from the equivalence class into Cube Explorer 1.5 to create the isoglyph, and any one glyph from the equivalence class will do as well as any other. I can report that of the 51 glyphs unique up to symmetry, 8 of them produce only continuous isoglyphs, 17 of them produce only non-continuous isoglyphs, 14 of them produce both continuous and non-continuous isoglyphs, and 12 of them produce no isoglyphs. In addition to confirming Herbert's figure of 35 continuous isoglyphs, I can report that there are 249 non-continuous isoglyphs. In the category of "most isoglyphs", one glyph has 2 continuous and 49 non-continuous isoglyphs, and another has 4 continuous and 46 non-continuous isoglyphs. The only other thing that probably requires explanation about the chart that follows is that there is a two character code below each glyph. This is a hexadecimal representation of a binary number based on the following pattern, 765 4X3 210 where the number includes 2^k if facelet k is the same color as the center facelet. This is not intended as a new classification to replace Dan's. It is just a bookkeeping technique I used (a 16x16 matrix) to keep track of the 256 glyphs. 000 0X0 000 00 D' U L' R B' F D' U (8) * continuous 000 000 00X X00 0X0 0X0 0X0 0X0 00X X00 000 000 01 04 20 80 R2 D L2 U' B2 D' U2 R' F' U R B' L' D' F L2 B2 R U' (19) continuous B2 D F2 U' L2 B' D2 B U B' D2 F L R' D U F' (17) continuous 000 000 000 0X0 0X0 0XX XX0 0X0 0X0 000 000 000 02 08 10 40 D' U B D' L' R F D' B' D' U L (12) * continuous L2 D' B' F L' D U' F L' R U B' F' (13) not F2 D' L2 B' D' U' R B L F L F U' F' (14) not F2 D L2 R2 U' B' U L D L D2 R' F' D' B' D (16) not U R2 D B2 D F D' B' L' B' D2 F' L' F U F' R' (17) not R2 U2 B' F D B2 L' R D2 F' R2 F2 D' U (14) not B2 D2 R2 B' F D' F2 L' R U2 F' L2 D U (14) not D' U' L2 F' D2 L R' B2 D' B F' R2 D2 B2 (14) not B2 D U' L2 F' D2 L R' B2 D' B F' R2 (13) not R2 D2 R' B' L' B D' R2 B' R B' D' R (13) not 000 000 000 000 00X 0XX X00 XX0 0X0 0X0 0XX XX0 0XX 0X0 XX0 0X0 0XX XX0 00X X00 000 000 000 000 03 06 09 14 28 60 90 C0 F2 D F2 D B2 L2 U L2 D' L D L' B' L U' F' U R' U' (19) not F2 U L2 U L2 U' F U' F' D2 B L R U' B' D' R F' (18) not D' R2 D2 B2 U' F2 U' L2 B D R D' U F U' B' U2 B' R' U' (20) not 000 00X X00 X0X 0X0 0X0 0X0 0X0 X0X 00X X00 000 05 21 84 A0 F2 U2 L' R D2 F2 L' R (8) not F2 U2 B2 L2 U' B2 U' B2 L2 D2 L2 U R2 U' (14) not U' L2 D' L2 D B2 F2 L2 R2 D F2 U' F2 U' (14) not U2 L2 F2 D U' B2 L2 D' U' (9) not 000 00X X00 XXX 0X0 0XX XX0 0X0 XXX 00X X00 000 07 29 94 E0 (none) 000 000 0X0 0X0 0XX XX0 0XX XX0 0X0 0X0 000 000 0A 12 48 50 F2 D' R2 D' L' U' L' R B D' U B L F2 L U2 (16) continuous U B2 L D B' F L' D U' L' R F' D2 R' (14) continuous U' B2 R2 U2 F' D2 L' F2 U' F2 D2 F U2 R' U2 (15) continuous D2 U B2 D U' R' D2 B' R2 D2 L' R' D' B2 L B (16) not B2 U2 F2 D2 F2 U R' F' L2 U2 L R U' L2 F2 L' F (17) not U F2 L2 U2 B' U2 L F2 U B2 D2 B D2 R' U2 (15) not F2 D2 B2 D' B2 L2 U2 B D2 R F2 D F2 D2 B' U2 L F2 (18) not U2 B2 F2 D' F2 R2 D2 B U2 L' U2 B' D2 F2 D F2 R' U2 (18) not D2 U' B2 U2 F D2 L D2 F' D2 L2 F2 U B2 R' U2 (16) not F2 U R2 D U B' D' B' D' F L' F D' U2 L U2 (16) not D2 U' B2 U2 F D2 L D2 F U2 R2 B2 U' B2 R' U2 (16) not U2 F2 D F2 L2 U2 F' U2 L F2 D B2 D2 F D2 R' F2 (17) not F2 D L2 D2 B2 R2 B2 L2 F2 U2 R B U L U B U' L' U (19) not 000 000 0XX XX0 0XX XX0 0XX XX0 0XX XX0 000 000 0B 16 68 D0 U2 F2 R2 U' L2 D B R' B R' B R' D' L2 U' (15) continuous 000 000 00X 00X 0X0 0X0 X00 X00 0XX XX0 0X0 XX0 0X0 0X0 0X0 0XX X00 00X 0X0 000 00X X00 0X0 000 0C 11 22 30 41 44 82 88 D U2 L2 U R2 U' L2 U R' B2 L2 F' L2 B' R' F' L D U' (19) continuous U' F2 L2 D2 U F2 U2 F' L' D2 B2 R' D' B R' U L2 B2 F' (19) not 000 000 00X 0XX X00 X0X X0X XX0 0XX XX0 0X0 0X0 0X0 0XX XX0 0X0 X0X X0X 0XX 00X XX0 000 000 X00 0D 15 23 61 86 A8 B0 C4 D2 L2 F2 R2 U2 B2 D2 F2 R2 U2 R2 U2 (12) not U2 L2 B2 L2 U2 F2 U2 F2 L2 U2 R2 U2 (12) not U2 R2 B2 L2 U2 F2 U2 F2 R2 U2 R2 U2 (12) not D2 R2 F2 R2 U2 B2 D2 F2 L2 U2 R2 U2 (12) not 000 000 00X 0X0 0X0 0XX X00 XX0 0XX XX0 0XX 0XX XX0 XX0 XX0 0XX XX0 0XX 0X0 00X X00 000 0X0 000 0E 13 2A 49 54 70 92 C8 (none) 000 000 00X 0XX X00 XX0 XXX XXX 0XX XX0 0XX 0XX XX0 XX0 0XX XX0 XXX XXX 0XX 00X XX0 X00 000 000 0F 17 2B 69 96 D4 E8 F0 D2 R2 F2 U2 F2 U2 F2 U2 R2 B2 (10) not F2 L2 D2 B F R2 B F' R2 (9) not F2 U2 L2 F2 D U R2 F2 D U' B2 (11) not U2 L2 B2 D2 F2 U2 F2 U2 R2 B2 (10) not U2 L2 B2 U2 B2 D2 F2 U2 R2 B2 (10) not U2 F2 L2 B2 U2 B2 D2 F2 U2 R2 (10) not L2 D2 F2 L2 U' B2 L2 R2 F2 D' R2 (11) not D2 R2 F2 D2 B2 D2 F2 U2 R2 B2 (10) not U2 L2 R2 D F2 U' R2 F2 U2 F2 D' U2 F2 U' (14) not D' R2 D' B2 U2 B2 F2 R2 B2 F2 U' F2 U' (13) not U' B2 U' F2 D2 B2 F2 R2 B2 F2 U' F2 U' (13) not B2 U B2 U' L2 D2 F2 U' R2 U F2 (11) not D' R2 D' F2 D2 L2 R2 U' B2 F2 D R2 U' F2 U' (15) not L2 D F2 U' R2 F2 U2 F2 D B2 U' B2 U2 (13) not F2 D F2 U' R2 U2 F2 U' R2 D B2 (11) not D2 B2 U' L2 U B2 U B2 D' R2 D' R2 U' (13) not D' B2 D L2 D2 B2 U B2 U' (9) not D' L2 B2 D L2 D2 B2 U B2 R2 U' (11) not D' U' L2 D2 U2 B2 D' U' (8) not L2 U2 B2 L2 D B2 L2 R2 F2 U' (10) not L2 U2 R2 D' U' B2 R2 B2 D' U' (10) not L2 D2 L2 B2 U2 F2 D2 F2 R2 F2 R2 U2 (12) not L2 D2 B2 D2 F2 R2 B2 R2 F2 U2 R2 U2 (12) not B F D2 L2 B F (6) * not 000 0X0 XXX 0X0 000 0X0 18 42 L2 U2 L2 R2 U2 L' R' (7) * not 000 000 00X 0X0 0X0 0XX X00 XX0 XXX XXX XXX 0X0 0X0 0X0 XXX 0X0 00X X00 000 0XX XX0 0X0 000 0X0 19 1C 38 43 46 62 98 C2 D2 L2 D R2 U B2 U2 B R' B' D B2 R' F R2 F' U R' (18) not 000 0X0 0X0 0X0 XXX 0XX XX0 XXX 0X0 0X0 0X0 000 1A 4A 52 58 D F2 R2 F2 R2 U F2 R F2 R D2 U' F L' F' L D (17) continuous B2 L2 U' B2 F2 D2 B2 R B' F2 U' B' D2 L' B' U L2 D' U' (19) not D' B2 U' B2 F2 D F2 D2 F L2 U L F' D' F2 L' U' (17) not L2 F2 U B2 U2 F2 R2 B2 R2 F R2 D F2 R' D' B' D' B' R' U (20) not F2 D2 R2 B2 D2 F2 D' F2 L' B2 U' L B' L D L B' R2 U' F2 (20) not L2 D R2 U' L2 F2 L2 D' B2 F' L R B D2 R' B F L F2 U' (20) not L2 D2 U2 L' U2 L' R2 D2 U2 R' U2 R' (12) not U2 R2 B2 F2 D2 U2 L' B2 F2 R' D2 L' R (13) not 000 000 0X0 0X0 0XX 0XX XX0 XX0 XXX XXX 0XX XX0 0XX XXX XX0 XXX 0XX XX0 0XX XX0 0X0 000 0X0 000 1B 1E 4B 56 6A 78 D2 D8 U R2 U' F' U2 F2 U2 F R F2 R' U R2 U' (14) continuous B2 D U2 R2 D F2 B' L2 D2 F L F2 L' R2 F' U' F' U (18) not 000 0XX X0X XX0 XXX 0X0 XXX 0X0 X0X 0XX 000 XX0 1D 63 B8 C6 F2 L' R B2 U2 L R' D2 (8) not D' U' B2 L2 D' U R2 F2 U2 (9) not 000 0XX XX0 XXX XXX 0XX XX0 XXX XXX 0XX XX0 000 1F 6B D6 F8 (none) 00X X00 0X0 0X0 X00 00X 24 81 (none) 00X X00 X0X X0X 0X0 0X0 0X0 0X0 X0X X0X 00X X00 25 85 A1 A4 D' B2 L2 F2 R2 F2 U R2 U2 F' R B L D B U' F R' U2 R (20) continuous B2 L2 R2 U R2 B2 U L2 U' B F D2 L' B2 R2 D' U B' L' R' (20) continuous F2 R2 U2 B2 D' R2 D L2 D2 R2 F2 U' R2 U' (14) not 00X 00X 00X 0XX X00 X00 X00 XX0 0X0 0XX XX0 0X0 0X0 0XX XX0 0X0 XX0 X00 X00 X00 0XX 00X 00X 00X 26 2C 34 64 83 89 91 C1 (none) 00X 00X X00 X00 X0X X0X XXX XXX 0X0 0XX 0X0 XX0 0XX XX0 0X0 0X0 XXX X0X XXX X0X 00X X00 00X X00 27 2D 87 95 A9 B4 E1 E4 L2 U' R2 D U2 L' B2 F' D' R' B2 D L2 R2 U2 F' L U (18) continuous B R2 B' F2 L2 B' L' D2 R D' L2 U F' R2 B' L B U (18) not 00X 0XX X00 XX0 0XX XX0 XX0 0XX XX0 X00 0XX 00X 2E 74 93 C9 (none) 00X X00 XXX XXX 0XX XX0 0XX XX0 XXX XXX 00X X00 2F 97 E9 F4 U' L2 R2 F2 U L2 U' F2 R2 L' U B' R D' B2 D2 B' R' (18) continuous R2 B2 D B2 D U R2 D' B' D' R F2 R' D B U' (16) continuous D' L2 U' F2 U F2 U2 F2 D' L2 U B2 U' (13) not 00X 0X0 X00 X0X XX0 0X0 0XX 0X0 00X X0X X00 0X0 31 45 8C A2 (none) 00X 0X0 0X0 X00 XX0 0XX XX0 0XX 0X0 X00 00X 0X0 32 4C 51 8A D U L2 B2 D U' F' U F' R F2 R' F D' B2 L2 D' U' (18) continuous R2 B2 D2 L2 U L2 D B L2 U2 B2 L' R2 F2 D' U R U2 R' (19) continuous 00X 0X0 0X0 0XX X00 X0X X0X XX0 XX0 0XX XX0 XX0 0XX 0XX XX0 0XX 0XX X0X X0X 00X XX0 0X0 0X0 X00 33 4D 55 71 8E AA B2 CC R2 B2 D U L' R' D2 L' R D U (11) not B2 L2 D2 L2 R2 B' F' R2 B F' R2 (11) not B2 R2 B2 R2 F2 U2 B2 R2 U2 R2 (10) not R2 F2 D U L' R' U2 L' R D' U' (11) not R2 F2 D' U' L' R' U2 L R' D U (11) not L2 D' U F2 L R B2 L R D U (11) not F2 L2 B2 R2 F2 U2 B2 R2 D2 R2 (10) not R2 B2 D' U' L' R' D2 L R' D' U' (11) not B2 L2 B2 U R2 U' B2 R2 U2 R2 U B2 U' (13) not R2 F2 U' B2 D' R2 U2 F2 U' F2 D' B2 L2 (13) not B2 F2 R2 D F2 D' L2 U2 B2 U' B2 U R2 (13) not D' R2 D2 B2 R2 B2 U B2 D U B2 U' F2 U' (14) not B2 R2 U2 L2 D' F2 R2 U' L2 D2 B2 U' R2 F2 U' (15) not B2 R2 F2 R2 F2 D F2 D' L2 U2 B2 U' B2 U R2 (15) not F2 U R2 U F2 D2 L2 U B2 U L2 R2 F2 (13) not B2 F2 L2 R2 D' B2 D B2 D2 R2 U F2 U' (13) not B2 R2 D' R2 U F2 D2 L2 U B2 D' L2 F2 (13) not U F2 R2 U2 F2 D' R2 U2 B2 D F2 R2 U' (13) not B2 F2 L2 R2 D U R2 F2 D' U' (10) not F2 L2 F2 R2 F2 U2 F2 D2 U2 R2 (10) not L2 D2 R2 U2 B2 F2 R2 F2 R2 U2 (10) not L2 D2 R2 F2 U2 B2 D2 F2 R2 F2 R2 U2 (12) not L2 D2 B2 U2 F2 L2 B2 R2 F2 U2 R2 U2 (12) not B2 L2 F2 L2 F2 D2 F2 R2 D2 U2 (10) not 00X 0XX X00 X0X X0X X0X X0X XX0 XX0 0X0 0XX 0X0 0X0 0XX XX0 0X0 X0X X0X X0X 0XX XX0 X00 00X X0X 35 65 8D A3 A6 AC B1 C5 B2 L' D2 L' B2 L U2 F2 U2 R' F' L2 D' L F2 U F' L B R (20) not D2 L2 U2 F' L2 F R2 F U B' D2 F2 R U' F2 L' U2 B2 D F (20) not L2 U' L2 B2 U' R2 F R B' D L U B' U' R2 D R2 F L F (20) not 00X 0XX X00 XX0 XX0 0XX 0XX XX0 XX0 X00 0XX 00X 36 6C 8B D1 (none) 00X 0XX X00 X0X X0X XX0 XXX XXX XX0 0XX 0XX 0XX XX0 XX0 0XX XX0 XXX X0X XXX 0XX XX0 X0X X00 00X 37 6D 8F AB B6 D5 EC F1 D2 R2 F2 L2 F2 D R2 D' R2 U2 F2 U' R2 U' (14) not D2 B2 D' L2 D F2 U2 R2 U R2 U F2 (12) not U' L2 U' L2 D2 F2 U' F2 U' F2 R2 B2 L2 (13) not R2 D F2 U R2 D2 L2 B2 D' B2 U L2 F2 U2 (14) not F2 U R2 D' F2 R2 U2 L2 D B2 U' (11) not F2 R2 D' B2 U F2 D2 F2 R2 D' F2 U F2 (13) not R2 B2 L2 U B2 U R2 D2 F2 U L2 U' B2 U2 (14) not U2 L2 F2 L2 F2 U F2 U' F2 U2 L2 U' L2 U' (14) not 00X 0X0 X00 XXX XXX 0X0 XXX 0X0 00X XXX X00 0X0 39 47 9C E2 B2 D2 L R' D2 B2 L R' (8) not U2 R2 F2 D' U B2 L2 D' U' (9) not 00X 0X0 0X0 0X0 0X0 0XX X00 XX0 XXX 0XX XX0 XXX XXX XX0 XXX 0XX 0X0 XX0 0XX 00X X00 0X0 0X0 0X0 3A 4E 53 59 5C 72 9A CA D' U2 B2 U2 L2 U B U' L2 B2 R' B2 R F2 D2 F2 D F' (18) not F2 D' B2 U2 F2 U R2 D B L2 B' R B2 U2 F D2 L' U' F (19) not D F2 D U2 F2 L2 D' B2 F D L' B2 L2 F D F D2 U2 F2 R (20) not 00X 0X0 0X0 0XX X00 XX0 XXX XXX XXX 0XX XX0 XXX XXX XXX 0XX XX0 0XX XXX XXX 00X XX0 X00 0X0 0X0 3B 4F 57 79 9E DC EA F2 D2 R2 B2 L2 U2 F2 U2 B2 R2 U2 R2 U2 (12) not U2 R2 F2 R2 U2 B2 D2 B2 L2 U2 R2 U2 (12) not U2 L2 F2 R2 U2 B2 D2 B2 R2 U2 R2 U2 (12) not D2 L2 B2 L2 U2 F2 U2 B2 L2 U2 R2 U2 (12) not 00X 0XX X00 XX0 XXX 0X0 XXX 0X0 X00 XX0 00X 0XX 3C 66 99 C3 (none) 00X 0XX X00 X0X X0X XX0 XXX XXX XXX 0X0 XXX XXX XXX 0X0 0X0 0X0 X0X XXX X0X 00X X00 XXX 0XX XX0 3D 67 9D B9 BC C7 E3 E6 L2 F2 L2 U R2 D' F2 U' R2 D R2 U R2 U' (14) not D2 R2 B2 D B2 U R2 B2 D2 F2 D' B2 U B2 (14) not F2 L2 U2 F2 D' R2 D L2 D2 R2 F2 U' R2 U' (14) not D2 B2 R2 U B2 U F2 R2 D2 F2 R2 U L2 U' (14) not B2 L2 R2 D' F2 L2 U' B' F' L D2 F2 R D' U' F' L' R' (18) not 00X 0XX 0XX 0XX X00 XX0 XX0 XX0 XXX 0XX XX0 XXX XXX 0XX XX0 XXX XX0 XX0 XX0 X00 0XX 0XX 0XX 00X 3E 6E 76 7C 9B CB D3 D9 (none) 00X 0XX X00 XX0 XXX XXX XXX XXX XXX 0XX XXX XX0 0XX XX0 XXX XXX XXX XXX XXX XXX 0XX XX0 00X X00 3F 6F 9F D7 EB F6 F9 FC L2 D' L B2 U' B' L B U B2 L' B D (13) not U' F2 D' L' U' F U2 L U2 F D R2 F' R' U2 (15) not U R2 D2 B2 U' F L2 B R D R B R' D' F2 U2 (16) not 0X0 XXX 0X0 5A U B2 U2 L2 U F2 R2 B2 U' L2 D2 F2 U' B L2 R2 D2 U2 F' (19) continuous L2 R' B2 F2 D2 B2 F2 L2 R2 U2 R' (11) continuous 0X0 0X0 0XX XX0 XXX XXX XXX XXX 0XX XX0 0X0 0X0 5B 5E 7A DA D U2 R2 D' U' R D B2 R2 B2 R2 D B2 D2 R U' (16) continuous 0X0 0XX X0X XX0 XXX XX0 XXX 0XX X0X 0XX 0X0 XX0 5D 73 BA CE (none) 0X0 0XX XX0 XXX XXX XXX XXX XXX XXX 0XX XX0 0X0 5F 7B DE FA B2 D2 B2 R2 F2 L2 U2 L2 F2 R2 (10) not B2 D2 B2 L2 U' F2 U' F2 R2 U2 L2 U R2 U' (14) not B2 L2 D2 L2 U' F2 U B2 U2 F2 R2 U' R2 U' (14) not D2 L2 D U' L2 F2 D U' F2 U2 (10) not 0XX X0X X0X XX0 XX0 0XX XX0 0XX X0X XX0 0XX X0X 75 AE B3 CD D2 U R2 D' F2 U F2 R2 B R2 F2 U2 L' F2 D2 B2 D B' U' (19) continuous U2 L2 F2 R2 F2 U B2 U' B2 D2 L2 U' L2 U' (14) not 0XX 0XX X0X X0X XX0 XX0 XXX XXX XX0 XXX XXX XXX 0XX XXX 0XX XX0 XXX X0X 0XX XX0 XXX X0X XX0 0XX 77 7D BB BE CF DD EE F3 L2 U' L2 D' L2 D F' L2 R' U B D2 B' D' U' R U (17) continuous L2 B2 F2 D2 L2 B2 U R2 U B2 F D' B2 U2 L F2 L D' B' (19) not 0XX XX0 XXX XXX XX0 0XX 7E DB (none) 0XX XX0 XXX XXX XXX XXX XXX XXX XXX XXX 0XX XX0 7F DF FB FE U' L2 U F' R2 F U' L2 U F' R2 F (12) continuous R2 D B2 D' B2 U' B2 U B2 U R2 U' (12) not X0X 0X0 X0X A5 D2 F2 U' B2 F2 L2 R2 D R' B F D' U L R D2 U2 F' U2 (19) continuous R' D2 U2 L2 B2 F2 L' F D' U R2 B2 F2 R' L' B F' U' (18) continuous B2 F2 L2 R2 D2 U2 (6) * continuous X0X X0X X0X XXX 0X0 0XX XX0 0X0 XXX X0X X0X X0X A7 AD B5 E5 F2 U2 B2 F2 L2 U' B2 L D2 F' R2 B L2 R U' R' D' F' R (19) continuous L2 R2 D2 L2 D' U F L' R U2 B2 U B F' R' D2 R' U2 (18) not B2 U2 F2 U2 L2 D U' B' U2 L' R B2 D' R2 B' F' L' R (18) not B2 R2 D U' F2 U2 R2 B D U B' F' L' R' F' D' U R2 (18) not B2 L2 D' U2 L2 B2 R' B U2 R B L' D' F2 R (15) not R2 D L2 B2 F2 R2 U' R2 D2 U2 (10) not R2 F2 D B2 L2 B F L B2 L2 D U' F' L' R D' (16) not R2 U2 L2 R2 U' R2 D B F' L' B2 R2 D' U B L' R' (17) not U L2 U2 F2 U F2 R2 F' L F U B' D' R' D R2 D2 F' D (19) not U B2 F2 R2 D2 U2 R D2 U2 R2 B D U' L R2 B' D' (17) not D2 U B2 U2 B2 R2 U2 B' R2 D U F U' L R' B L F U' (19) not U2 L2 B2 R2 U2 B2 R2 D U B U B' L' F2 U' L' U R' D U' (20) not U F2 L2 R2 F2 L2 R' B D B2 F2 U' B' L' U' (15) not D' B2 D' L2 F2 D' L' R F D2 L2 F D' U' R B F' (17) not R2 U' B2 L2 F2 U' L2 D F2 L2 F D F U L U B' U' L' (19) not D' U' B2 F2 L' R F R2 D' U F2 L' B' F U2 (15) not R2 D L2 F2 U F2 D R2 B R' D R D2 R2 B' F2 D F R2 (19) not F2 D' B2 D F2 L2 D2 U L2 U R D R2 D2 L' F' U2 B2 U F (20) not U2 B U2 R2 D2 B L2 U' B' U L F2 U R F' D' R2 B' R' (19) not U2 L2 R2 D' F2 U' B2 R F' D R' B' D F R' U L' D2 F U (20) not D2 B2 U2 R2 B2 R2 U2 F2 R2 U2 (10) not X0X X0X XXX XXX 0XX XX0 0XX XX0 XXX XXX X0X X0X AF B7 ED F5 F2 L2 D' R2 B2 L2 R2 F U2 L2 D' L' D' R2 F' D' L' F2 (18) continuous D2 L2 D' F2 D U F' R2 D' L' R F' L' R' B' U' R2 U2 (18) continuous D U F2 R' B D2 U2 F' D2 U2 R F2 D' U' (14) continuous U B2 L B F' L2 R' B' F D U2 L' B2 U' (14) continuous B2 L2 U' L2 U2 B2 R2 U' B U R' D' L' D2 L D B D U' (19) not B2 F2 L2 D2 R2 U L F D' L2 R2 D2 U' F' R' D L2 U2 (18) not R2 U' L2 R2 D B2 D R F' R' B L' R U L' U' F R' (18) not F2 D2 B2 U L2 B2 L2 D' R' B R' D' L2 B' D' B L2 R2 U' (19) not B2 D' R2 D R2 D' B R' F R' D L2 F' U L B' L U' (18) not R2 F2 L2 D' B2 U' B2 L2 F2 L B' L' U' B' D' B D B' D' (19) not B2 L2 F2 D' L2 B2 D' L B L D F2 D B' D U2 F' U2 (18) not D L2 F2 D U2 B2 R2 F' D R U2 L2 F' L2 U' R' U B (18) not U2 R2 F2 D U' B2 R2 B2 R2 D' U' (11) not D2 L2 B2 D' U B2 R2 B2 R2 D' U' (11) not D2 B2 D2 L2 D2 L2 U2 F2 R2 F2 R2 U2 (12) not D2 B2 U2 R2 U2 L2 U2 F2 R2 F2 R2 U2 (12) not B2 F2 D2 U L2 D2 R D' L2 R D' B2 D F D F' U' (17) not B2 F2 D' F2 D U R D' L2 R U' L2 U F D F' U' (17) not R2 U2 B2 L2 F2 R2 D2 F2 U2 F2 R2 U2 (12) not L2 F2 U2 B2 U2 R2 B2 R2 F2 U2 R2 U2 (12) not L2 U2 F2 R2 F2 R2 U2 B2 U2 F2 R2 U2 (12) not R2 F2 D2 F2 D2 R2 B2 L2 F2 U2 R2 U2 (12) not D2 B2 L2 U2 R2 U2 B2 L2 U2 F2 R2 U2 (12) not D2 B2 L2 D2 L2 D2 B2 L2 U2 F2 R2 U2 (12) not U2 R2 F2 D R2 F2 R2 F L D2 L' D' F' L' U2 B2 R' (17) not B2 L2 U R2 D U' L2 B L R2 D' L' B D L' R2 B L' (18) not F2 L2 B2 R2 D2 F2 D2 F2 R2 U2 R2 U2 (12) not F2 R2 B2 R2 U2 B2 U2 F2 L2 U2 R2 U2 (12) not L2 U2 R2 F2 U2 B2 U2 R2 F2 R2 F2 U2 (12) not L2 D2 L2 B2 D2 B2 U2 R2 F2 R2 F2 U2 (12) not F2 R2 B2 R2 D2 F2 D2 F2 L2 U2 R2 U2 (12) not D' R2 B2 R2 D' R2 B' D' F' L' U' B' U L F D R2 (17) not L2 U2 R2 D2 R2 U2 B D' U2 R F D2 U2 B' L' D' B' (17) not L2 B2 D' B2 L2 B2 F R B R2 U F2 R U B U F' (17) not R2 D F2 D U' R2 L B D L2 R2 D' L' B F2 D' R2 (17) not B2 D U' L2 D2 F' D U' R F D U' R' D' U' (15) not L2 D2 L2 D' U' F2 L' D' U B L D' U B' D' U' (16) not U' L2 F2 L2 D F2 L2 D' U L B' L' D' L2 D B D L (18) not R2 B2 R2 D' F2 L2 U L' D' L2 R F' R' D R F R' U' (18) not U' F2 D' F2 L2 D2 U B' L' B D L' U' L' F2 L' U (17) not D' L2 D' L2 B2 D' B' L' B D L' U' L' F2 L' U (16) not R2 U F2 L2 B2 U L2 D' R2 F2 R2 U' (12) not D L2 D' F2 D' R2 U2 R2 U2 B D2 F' R' U R' D2 U B (18) not R2 B2 D2 F2 D2 R2 F2 L2 F2 U2 R2 U2 (12) not U F2 R2 F2 D' L2 D U2 B2 L2 F D' R' F R' D R F R' (19) not L2 B2 L2 B2 D2 R2 U2 R2 B2 U2 F2 U2 (12) not B2 R2 U2 R2 D2 R2 B F' R2 B' F' (11) not B2 R2 D2 L2 U2 R2 B F' R2 B' F' (11) not U' B2 D' L2 U' L2 R2 D L D U' F D F D' U L D (18) not B2 D' B2 F2 D' L2 U2 B2 R' B R U2 L' F' L' U' F2 L2 U (19) not X0X XXX XXX 0X0 X0X XXX BD E7 L2 B F' L2 R2 B F' R2 (8) not F2 L2 U2 L2 R2 B' D2 U2 F U2 R2 F2 (12) not R2 B2 F2 R2 U' B2 F2 D2 L2 R2 U' (11) not L2 R2 D B2 F2 R2 B2 F2 R2 U' (10) not U2 B2 R2 D2 U2 R2 F2 U2 (8) not X0X XXX XXX XXX XXX 0XX XX0 XXX XXX XXX XXX X0X BF EF F7 FD U R' D' U F2 D U' R' U' (9) * continuous L2 U' F2 B D' R' D2 R B' F2 L U (12) * continuous D' B' R2 B' D U' L B2 D2 U2 R D2 U' (13) not D2 U R' D2 U2 B2 L' D' U B R2 B D (13) not F2 D' L2 R2 B' L' R D B2 D L' R F' D' F2 (15) not U F2 U2 F L' U' B' U2 B L U F' U (13) not U' F2 U F2 R B' U' R' U R B U2 R' F2 (14) not U' R U L' R B' R' B U' L R' F (12) not U L2 B2 D' F2 R2 B2 U' F2 U' (10) not L2 R2 U' L2 R2 D' L' R F2 L R' (11) not L2 R2 U L2 B2 R2 D' R2 B' L2 U2 R2 F L2 (14) not F2 D U2 R2 B R2 U2 R2 B R2 D' F2 (12) not D2 R2 B2 R2 D' R2 B2 R2 D (9) not F2 R2 U L2 U' R2 F2 L R F' U2 F L' R' (14) not R2 D B R' D' R' B' R' D B R' (11) * not R2 D2 B D2 R2 B2 L B2 U2 F2 R F' U2 (13) not D2 B2 U2 L' U2 B' D2 R2 U2 F U2 R (12) not L2 D2 F2 D2 R D2 F' R2 D2 L2 B U2 R (13) not F2 U' F2 D2 B2 D2 F2 U' F2 (9) not R2 D R2 B R2 D R2 B' R2 D' R2 B' (12) not D2 B2 R2 D2 R2 B R2 D2 R2 B' D2 (11) not U2 F2 U2 L2 U2 F U2 L2 U2 F' U2 (11) not R2 D2 B2 D2 U2 F D2 L2 D2 F' U2 (11) not D2 R2 B2 L2 U' R2 F2 R2 U L2 R2 (11) not U F' U2 L D2 B2 U2 R D2 F' U' (11) * not D' U' B D B U R2 D' L R2 B' L' (12) not R2 D F D' L' B F L' B' U L F2 R2 (13) not F2 D' L2 R2 U L R' U2 L R' (10) not D' R2 B2 R2 D R2 B2 R2 D2 (9) not D2 B2 R2 D2 R2 D2 R2 U2 F2 U2 (10) not U R' F L R' D' R D' L' R F U' (12) not F2 L2 F' L2 R2 B L2 B L2 R2 F (11) not D2 R2 U2 F2 D2 U2 B' U2 L2 U2 B (11) not R2 D' L2 B2 L2 D R2 U2 R' F2 D2 F2 R U2 (14) not D2 R2 U' R2 F2 L2 D R2 B L2 U2 R2 F' D2 (14) not D U B' R' D2 B' D' R B D' U' B D' (13) not R2 F2 L2 D' B2 L B2 D2 F2 R' B2 U (12) not U' R B U' R' U' B' U' R B U2 (11) * not L2 U L2 F2 R2 D' L2 B' L2 D2 R2 F (12) not U2 R2 B' L' R' B2 L' R' F' L2 (10) * not D2 B D2 U2 F D U' R2 D' U' (10) * not R2 U' L' U' B' L' B U L U B R2 (12) * not D2 L D2 F' R2 D2 L2 B D2 R' U2 F2 (12) * not U' L D' U F L R' U' L' R F' D (12) * not R2 F2 L' F' L' R U L U L R' F R2 (13) not U' L2 D' L' D' U B D B D U' L U (13) not D U R' D' U F' D U' R D' U F U2 (13) not U' L2 B L' R D' L D' L R' B L U (13) not D U F2 U2 B2 D' U' F D2 U2 B' R2 (12) not D U2 B2 U B U' B' U' B2 F L' D L F' D' U2 (16) not B2 F2 R2 D' F2 R2 F2 D' L' U2 F' L2 U2 L2 F' D2 R F2 (18) not XXX XXX XXX FF (0) continuous (this is Start) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Mar 30 15:45:45 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA05004; Mon, 30 Mar 1998 15:45:45 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Mar 27 21:26:20 1998 Message-Id: In-Reply-To: <01BD594B.F9EFBF20@jburkhardt.ne.mediaone.net> Date: Fri, 27 Mar 1998 21:26:52 -0500 To: cube-lovers@ai.mit.edu From: Charlie Dickman Subject: Re: Stickers >Does anyone know where to find cube stickers? They must come from >somewhere! I found some vinyl lettering once and the periods were >exactly the right size for a 5x5x5 cube. But they don't come in >orange. There must be a way to buy sheets of the stuff. Any ideas? I have found some adhesive backed vinyl sheets at a local Art Emporium but they are mostly irridescent shades and you have to cut the pieces to size yourself. I seem to recall that there was an orange color but I'm not sure. Charlie Dickman charlied@erols.com From cube-lovers-errors@mc.lcs.mit.edu Mon Mar 30 16:23:46 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA05149; Mon, 30 Mar 1998 16:23:46 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 04:29:11 1998 Message-Id: <3.0.5.16.19980329094205.097f34b6@vip.cybercity.dk> Date: Sun, 29 Mar 1998 09:42:05 To: cube-lovers@ai.mit.edu From: Philip Knudsen Subject: Eclipse and Pyramorphix There are two new puzzles out, by the two most prominent veterans respectively: 1) Rubik's Eclipse, which is some sort of two-player game and, according to the people who have it, a real gem. 2) Pyramorphix, by Meffert. David Byrden's Twisty Puzzles page shows a picture of a 2x2x2 Pyraminx together with the text "A solid version of this amazing puzzle is now available from Uwe Meffert, called the Pyramorphix". Now the 2x2x2 pyraminx looks like an old east german puzzle, which was a 2x2x2 cube in tetrahedral shape. The shape changed when the puzzle was scrambled, so the name Pyramorphix would apply. However the east german puzzle was not by Meffert. Now if anyone knows more about these new puzzles, or where to get them, please reply. Philip K recording and performing artist Vendersgade 15, 3th DK - 1363 Copenhagen K Phone: +45 33932787 Mobile: +45 21706731 E-mail: skouknudsen@email.dk E-mail: philipknudsen@hotmail.com Sms: 4521706731@sms.tdk.dk (short message, no subject) From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 31 10:02:23 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA07294; Tue, 31 Mar 1998 10:02:22 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Mar 30 21:08:16 1998 Message-Id: <19980331020806.13788.qmail@hotmail.com> X-Originating-Ip: [206.114.5.101] From: "HADER MESA" To: zot@ampersand.com, rtayek@netcom.com Cc: cube-lovers@ai.mit.edu Subject: i need information!!! Date: Mon, 30 Mar 1998 18:08:05 PST Hello, I am a fond of the cube of Rubik, but in my country it is very difficult to get it. She/he would want to know if you can give me information about where I can get the cube and their different variants. For the information that you can to give, I thank him a lot. Cordially: Hader Mesa From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 1 10:55:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA10617; Wed, 1 Apr 1998 10:55:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 18:20:22 1998 Date: Sun, 29 Mar 1998 18:20:42 -0400 (EDT) From: Jerry Bryan Subject: All the Partial Isoglyphs To: Cube-Lovers Message-Id: I have been able to calculate all the partial isoglyphs a little more quickly than expected. I can report that there are 10 continuous partial isoglyphs and 130 non-continuous partial isoglyphs, unique up to symmetry. Here is a breakdown of how the solid faces can be arranged. 97 - two solid faces, opposite to each other 11 - two solid faces, adjacent to each other 25 - one solid face 1 - three solid faces, mutually adjacent to each other 2 - three solid faces, not mutually adjacent to each other 3 - four solid faces, other two opposite to each other 1 - four solid faces, other two adjacent to each other --- 140 The partial isoglyphs are all included in the chart which follows. If nothing is listed with respect to the manner in which the solid faces are arranged, then there are two solid faces opposite to each other. Otherwise, the arrangement of the solid faces is listed explicitly. This chart follows the same format as the previous one I posted for all the isoglyphs, except that this time I included only a single representative glyph for each partial isoglyph, rather than the complete equivalence class of glyphs. 000 0X0 000 00 D B2 F2 D U' L2 R2 U' (8) continuous 000 0X0 00X 01 (none) 000 0X0 0X0 02 D B2 L2 R2 F2 U' L2 B2 F2 R2 (10) not D' B2 F2 D U L2 R2 U' (8) not F2 D2 B2 D U L2 D' U' (8) not B2 D U' L2 D U' (6) * not D B2 D L2 . B F' D2 R' B2 R2 D' U F L R' (15) not 000 0X0 0XX 03 L2 U2 B2 D' B2 R2 D2 B2 R2 U' B2 L2 U2 R2 U' (15) not 000 0X0 X0X 05 D U F2 D' L2 . B' F U2 L' D U' F2 R2 F' L R' (16) not D B2 L2 R2 F2 U' L2 B2 F2 R2 U2 (11) not L2 F2 U B2 F2 U2 B2 F2 U' F2 R2 (11) not 000 0X0 XXX 07 D' B2 F2 D' U L2 R2 U' (8) continuous 000 0XX 0X0 0A L2 D2 R2 F2 U2 R2 F2 U2 F2 U2 (10) not D B2 D' U' L2 R2 U2 R2 U' (9) not 000 0XX 0XX 0B B2 L2 U2 L2 U' L2 B2 D2 F2 U F2 R2 U2 R2 U' (15) not 000 0XX X00 0C F2 L2 U2 L2 U L2 F2 D2 B2 U' B2 R2 U2 R2 U' (15) not 000 0XX X0X 0D L2 D2 R2 F2 U2 R2 F2 U2 F2 (9) not B2 R2 F2 D U B2 L2 F2 L2 D U (11) not F2 R2 B2 D' U' B2 R2 F2 R2 D' U' (11) not 000 0XX XX0 0E L2 U2 F2 D F2 R2 D2 F2 R2 U F2 L2 U2 R2 U' (15) not R2 U2 F2 D' F2 L2 U2 B2 R2 U' B2 R2 U2 R2 U' (15) not F2 U2 R2 U F2 D2 F2 R2 D' F2 R2 D2 F2 R2 U' (15) not 000 0XX XXX 0F B2 D U' L2 D U (6) * not 000 XXX 000 18 D2 U2 (2) * continuous D' U (2) * continuous L2 F2 L2 R2 F2 R2 (6) * not F2 U2 B2 F2 U2 F2 (6) * not 000 XXX 00X 19 (none) 000 XXX 0X0 1A D F2 R2 B2 F2 R2 D' U R2 U' (10) not D' B2 U2 B2 L2 R2 D2 F2 L2 R2 D' (11) not L2 B2 F2 R2 D' U2 L2 B2 F2 R2 U' (11) not U' L2 R2 U2 L2 R2 U' (7) * not 000 XXX 0XX 1B B2 D2 L2 U' F2 D2 F2 L2 D F2 L2 U2 B2 R2 U' (15) not 000 XXX X0X 1D L2 B2 F2 R2 D U2 L2 B2 F2 R2 U' (11) not F2 U L2 R2 D2 L2 R2 U' F2 (9) not 000 XXX XXX 1F D (1) * continuous D2 (1) * continuous 00X 0X0 X00 24 D' L2 F2 U2 B2 R2 U2 L2 D U' R2 U' (12) not 00X 0X0 X0X 25 (none) 00X 0X0 XX0 26 B2 L2 D U F2 L2 D U F2 R2 (10) not L2 D' U' F2 D U . L R' U2 L R (11) not D' U' F2 D' U . L R' U2 L R (10) not U' B2 L2 D2 R2 F2 U2 F2 R2 U' (10) not 00X 0X0 XXX 27 L2 D2 R2 B2 U R2 B2 D2 L2 F2 U' F2 D2 R2 U' (15) not 00X 0XX XX0 2E (none) 00X 0XX XXX 2F L2 D2 B2 D B2 L2 U2 B2 L2 D' B2 R2 U2 R2 U' (15) not 00X XX0 00X 31 L2 F2 L2 R2 F2 R2 U2 (7) * not 00X XX0 0X0 32 F2 D2 L2 U B2 D2 B2 L2 D' B2 L2 U2 F2 R2 U' (15) not 00X XX0 0XX 33 D F2 R2 B2 F2 R2 D' U R2 U (10) not 00X XX0 X0X 35 F2 L2 D2 R2 D' R2 F2 D2 B2 D' B2 R2 U2 R2 U' (15) not 00X XX0 XX0 36 (none) 00X XX0 XXX 37 L2 U2 F2 D F2 R2 U2 F2 R2 D' F2 L2 D2 R2 U' (15) not B2 U2 F2 L2 D L2 D2 R2 B2 D B2 D2 F2 R2 U' (15) not F2 R2 D2 L2 D L2 F2 U2 F2 D F2 R2 U2 R2 U' (15) not 00X XXX 00X 39 U2 F2 U L2 . B' F U2 R' F2 R2 D U' B L R' (15) not B2 L2 R2 F2 D' L2 B2 F2 R2 U' (10) not B2 U' B2 L2 R2 F2 D' F2 (8) not 00X XXX 0X0 3A B2 U2 R2 U' B2 D2 B2 R2 D B2 R2 D2 B2 R2 U' (15) not 00X XXX 0XX 3B L2 D2 L2 F2 U2 R2 B2 U2 F2 (9) not U' R2 B2 R2 D F2 D' R2 B2 R2 U (11) not B2 R2 F2 D' U' B2 L2 F2 R2 D' U' (11) not 00X XXX X00 3C D' L2 B2 U2 F2 R2 U2 L2 D' U R2 U' (12) not F2 R2 D U L2 B2 R2 B2 D' U' F2 R2 (12) not D' R2 B2 U2 B2 R2 U2 R2 D' U R2 U' (12) not 00X XXX X0X 3D (none) 00X XXX XX0 3E B2 L2 D2 B2 R2 F2 L2 U2 F2 R2 (10) not D' U' L2 D' U . B F' D2 B F (10) not U2 L2 . B' L2 D2 U2 R2 F' R2 (9) not D2 L2 . B' D2 L2 R2 U2 F' R2 (9) not 00X XXX XXX 3F R2 U2 F2 D' F2 L2 D2 B2 R2 D B2 L2 D2 R2 U' (15) not 0X0 XXX 0X0 5A D F2 R2 F2 D' U R2 F2 R2 U' (10) not B2 F2 D2 L2 R2 D B2 F2 U2 L2 R2 U' (12) not 0X0 XXX 0XX 5B (none) 0X0 XXX X0X 5D B2 F2 L2 R2 D B2 F2 L2 R2 (9) not B2 F2 L2 R2 D2 B2 F2 L2 R2 (9) not 0X0 XXX XXX 5F D' L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not L2 B2 D' B2 L2 R2 F2 U' F2 R2 (10) not 0XX XX0 X0X 75 F2 D2 B2 L2 D L2 U2 L2 F2 D F2 U2 F2 R2 U' (15) not 0XX XX0 XXX 77 R2 D2 B2 D' B2 R2 D2 F2 L2 D F2 R2 U2 R2 U' (15) not 0XX XXX XX0 7E D' R2 F2 U2 F2 R2 U2 R2 D U' R2 U' (12) not 0XX XXX XXX 7F (none) X0X 0X0 X0X A5 D2 R2 U2 L2 R2 U2 R2 U2 (8) not (four solid, other two opposite) D2 L2 F2 L2 R2 F2 R2 U2 (8) not X0X 0X0 XXX A7 D' B2 U' L2 . B' F U2 R' D U' B2 L2 B L R' (15) not B2 L2 R2 F2 D' U2 L2 B2 F2 R2 U' (11) not U2 F2 D' U' R2 U2 . R B2 F L' R D L' B2 F2 R B (17) not *1 D2 L2 B2 R2 U' F2 L2 D U' . R B2 U' F' D2 U' R F2 D L' R2 (20) not *2 B2 R2 F2 U' L2 U . R B D2 B' R' D' R' F2 L R2 B2 U' (18) not *2 D F2 L2 F2 D' U' R2 D' R2 . B' D2 B' D' L' U L2 R' U' R' (19) not *2 D' L2 R2 D' U' B2 F2 U' (8) not F2 L2 D2 B2 U2 B2 F2 R2 F2 U2 (10) not F2 D' F2 D B2 U B2 F2 U2 L2 U F2 . R B U' B2 U B' R' (19) not D' F2 R2 B2 F2 R2 D' U R2 U (10) not *1 two solid faces, adjacent *2 one solid face X0X 0XX XXX AF D . F' D2 U2 B R B' D2 U2 F L' D' (12) * continuous *1 R2 U2 . L B L U R' U R' D' F' D' (12) * not *2 L2 U2 R2 D2 R2 U2 (6) * not *4 U F2 D U2 L2 U' F2 . L' U' F D2 U L' F2 U2 B' R (17) not *3 D B2 D' U2 . F D F U' R' U F' U' R' U' B2 (15) not *2 F2 U' L2 U L2 U' B2 . L' U' B U2 B' U L' F2 D' R2 (17) not R2 D' R2 U' R2 U . R U L F2 D2 L' U' B' D B D (17) not *2 L2 U2 R2 D2 R2 . B' L' B' U' F U' F D R D F2 (16) not *2 B2 D2 B2 U2 F2 L2 D' . F' D' L' U L' U R B R (16) not *2 L2 U2 R2 F2 D2 R2 F2 D2 F2 U2 (10) not F2 L2 U2 B2 D R2 . B' L' D' L' D' B' U' B' F U' B U2 (18) not R2 D2 B2 D2 F2 U . F' D' L' D' B D L D F' R2 (16) not *3 U F2 L2 U2 L2 F2 R2 D' U' . B' D' L' B' R' B' R B L (18) not *2 B2 L2 . B' D' L' U B' R2 U' L F' D' F (13) not *2 B2 R2 F2 L2 U2 F2 R2 U2 F2 U2 (10) not *1 three solid faces mutually adjacent *2 one solid face *3 two solid faces adjacent *4 four solid, other faces opposite X0X XXX X0X BD D B2 F2 D' U L2 R2 U' (8) continuous D2 B2 D2 U2 F2 U2 (6) * not *4 R2 . F' U2 L2 D2 B2 L2 U2 R2 F' R2 (11) * not *2 F2 D2 L2 . F D2 U2 B' R2 U2 F2 (10) * not *3 F2 R2 U2 . B' D2 U2 F D2 R2 F2 (10) * not *1 D2 U2 B2 U2 R2 . F' D2 U2 B L2 U2 F2 (12) not *5 L2 D' U' B2 F2 D' U' R2 (8) not L2 D2 R2 . B' U2 F U2 L2 U2 F R2 F' (12) not *5 B2 F2 L2 R2 U' B2 F2 D2 L2 R2 U' (11) not L2 R2 U2 B2 F2 U B2 F2 U2 L2 R2 U' (12) not L2 R2 D2 L2 B2 U B2 F2 D' F2 R2 U2 (12) not D B2 L2 B2 D U' R2 F2 R2 U' (10) not *1 - three solid faces, not mutually adjacent *2 - four solid faces, other two faces adjacent *3 - two solid faces, adjacent *4 - four solid faces, other two faces opposite *5 - one solid face X0X XXX XXX BF D' U R2 F2 D U' . R' D U' B' L2 B D' U R' (15) continuous L2 D U . B D' B' U' L2 D L D L' D2 (13) continuous *1 B2 D U' L2 D' U (6) * not D B2 U2 . L' U2 B2 D2 R' D (9) * not *2 R2 D U' . B D' B' D' U R D R (11) * not *2 F L R' D2 L' R F (7) * not *2 L2 U2 . B U2 L2 D2 F D2 (8) * not *1 L2 . F L R' D2 L' R F L2 (9) * not *2 U2 L2 D2 . B' L2 U2 R2 F' (8) * not *1 L2 D U' . F' L F D' U L' B' L' (11) * not *2 F2 U2 L2 D2 . B' L2 U2 R2 F (9) * not *2 D2 . B' L' R D2 L R' B' D2 (9) * not *3 R2 U2 . B D2 L2 U2 F D2 (8) * not *1 U2 B2 U2 L2 U2 . B D2 R2 U2 F' D2 (11) not *2 D . R B2 F2 L' U' L B2 F2 R' (10) * not *2 D . R' B F' U R' U' B' F R (10) * not *2 D . F' R' B' L' D' L B R F (10) * not *1 B2 D L2 U . R U R' F U2 L D' L B' (13) not *2 R2 D . F D' F' R2 D' B' D B (10) * not *1 D F2 D R2 . F R2 D2 R2 F R2 D F2 D' (13) not *2 D' F2 U2 B2 U2 F2 D' (7) * not F2 R2 U2 . B' U2 R2 U2 B' U2 F2 (10) * not *2 F2 D2 . F D2 R2 D2 F D2 R2 F2 (10) * not *2 B2 R2 U' L2 U R2 B2 R2 U F2 U' R2 (12) not D L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not F' L2 R2 B2 L2 R2 F' (7) * not L2 . B L' B' D2 R' B' R B D2 L' (11) * not *1 D U' . B F' U' B' F R2 D' U F' (11) * not *1 - 2 solid, adjacent *2 - 1 solid *3 - 3 solid, not mutually adjacent XXX XXX XXX FF (none) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 3 17:18:01 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA16814; Fri, 3 Apr 1998 17:18:01 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 1 06:05:52 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: new to list Date: 1 Apr 1998 09:19:33 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6ft0r5$6kj@gap.cco.caltech.edu> References: John Burkhardt writes: >The Dodecahedron puzzle is really amazing. It was actually harder >than the 5x5x5 cube. IT took me about 3 hours to work it out! I >think once you know the 3x3x3 then all the same moves do similar >things and you can easily solve 4x4x4 or 5x5x5 with variations. Of >course there are some cool things you can do with these. Really?? I found the Dodecahedron significantly easier than the 4x4x4. The Dodecahedron gives more "space" for moves... -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you. From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 3 18:45:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA16932; Fri, 3 Apr 1998 18:45:48 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 18:56:43 1998 Date: Sun, 29 Mar 1998 18:57:08 -0400 (EDT) From: Jerry Bryan Subject: Re: All the Partial Isoglyphs In-Reply-To: To: Cube-Lovers Message-Id: On Sun, 29 Mar 1998, Jerry Bryan wrote: > Here is a breakdown of how the solid faces can be arranged. > > 97 - two solid faces, opposite to each other > 11 - two solid faces, adjacent to each other > 25 - one solid face > 1 - three solid faces, mutually adjacent to each other > 2 - three solid faces, not mutually adjacent to each other > 3 - four solid faces, other two opposite to each other > 1 - four solid faces, other two adjacent to each other > --- > 140 > As this table shows, the vast majority of partial isoglyphs involve two solid faces opposite to each other. The basic reason for this is the corners. If the corners are not fixed, then the only partial isoglyphs which are possible have two solid faces opposite to each other. Conversely, the 43 partial isoglyphs which do not have two solid faces opposite to each other do fix the corners. In fact, 67 of the partial isoglyphs derive from just 5 of the glyphs, namely those which fix the corners. If the corners of the partial isoglyph are fixed, you can think of the edges as consisting of a set of strongly constrained edge flips and swaps. (Be careful -- if the corners are fixed, then *any* resultant position can be thought of as just a bunch of edge flips and swaps. But for partial isoglyphs, the possible edge flips and swaps are strongly constrained.) The glyph which yields the most partial isoglyphs is the one my charts call BF, whick looks like the following. X0X XXX XXX With this glyph, each face of a partial isoglyph can have at most one edge cubie which is swapped or flipped, but on a cube-wide basis there are quite a few different ways to arrange for this to happen. Another interesting glyph which fixes the corners is called BD on my charts, and which appears as follows. X0X XXX X0X As an isoglyph, this glyph yields five different patterns on the 6-H theme. As a partial isoglyph, this glyph yields a number of pretty 2-H, 3-H, 4-H, and 5-H patterns. You may also think of the H patterns as complicated edge swappers/flippers, with exactly zero or two edges swapped/flipped on each face, and with the coloring requirements for partial isoglyphs being maintained. The following two glyphs (A7 and AF in my charts) are in the same spirit as the H, except that the configuration of the edges on each face which are swapped/flipped is slightly different than for the H. X0X X0X 0X0 0XX XXX XXX Finally, for completeness in the list of glyphs which fix the corners, the glyph called A5 on my charts appears as follows. X0X 0X0 X0X However, this glyph only yields two partial isoglyphs. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 3 19:32:23 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA16989; Fri, 3 Apr 1998 19:32:23 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 19:35:16 1998 Date: Sun, 29 Mar 1998 19:35:43 -0400 (EDT) From: Jerry Bryan Subject: Re: partial isoglyphs In-Reply-To: <199708210441.AAA22489@life.ai.mit.edu> To: Cube-Lovers Message-Id: On Thu, 21 Aug 1997, michael reid wrote: > dan recently introduced the concept of "partial isoglyphs", in which > some faces are solid, and the others are glyphs of the same pattern. > i looked into this a little and didn't find much. only the case > of two opposite solid faces seems to have many possible glyph types, > although some of these possible types may have many solutions. > > here's what i found Note that all the glyph types which Mike lists (01, 02, 0D, 04, and 03 in Dan's taxonomy) fix the corners. Thus, his note below points out that in order to have anything other than two solid faces opposite to each other, you must fix the corners. The correspondence between Dan's taxonomy and my charts is 01=BF, 02=AF, 03=A7, 04=A5, and 0D=BD. As I said earlier, the identfication numbers on my charts are not a taxonomy. Rather, they provide a unique identification for each of the 2^8 glyphs. > > 6 solid faces: start > 5 solid faces: no possibilities > 4 solid faces: > other two faces opposite: types 02, 0D and 04 are possible All three possibilities do occur in my chart. > other two faces adjacent: type 0D is possible This possibility does occur in my chart. > 3 solid faces: > mutually adjacent: type 02 is possible This possibility does occur in my chart. > not mutually adjacent: types 01 and 0D are possible Both possibilities do occur in my chart. > 2 solid faces: > adjacent: types 01, 02, 0D and 03 are possible All four possibilities do occur in my chart. > opposite: many possible types Indeed! > 1 solid face: types 01, 02 and 0D are possible > All three possibilities do occur in my chart. In addition, I found three partial isoglyphs of type 03 with one solid face. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Sun Apr 5 16:13:15 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA20772; Sun, 5 Apr 1998 16:13:15 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Mar 30 20:40:51 1998 Date: Mon, 30 Mar 1998 20:41:13 -0400 (EDT) From: Jerry Bryan Subject: Pretty vs. Not-So-Pretty Isoglyphs To: Cube-Lovers Message-Id: After looking at a lot of isoglyphs and partial isoglyphs in the last little while, I wonder if it's not the case that some of the non-continuous isoglyphs are prettier than some of the continuous ones, and that some of the partial isoglyphs are prettier than some of the isoglyphs? Continuous isoglyphs do *in general* seem prettier than non-continuous ones, and isoglyphs do *in general* seem prettier than partial isoglyphs. But consider the following two (counter?) examples. The glyph 000 XXX 000 yields (among other things) L2 F2 L2 R2 F2 R2, which is a non-continuous partial isoglyph. It looks about as follows (quite pretty and striking, in my opinion): XXX XXX XXX 0X0 0X0 0X0 0X0 0X0 0X0 0X0 0X0 0X0 0X0 0X0 0X0 XXX XXX XXX On the other hand, U B2 R2 F2 L2 U L2 F2 U2 R' B' R F' L' U2 B2 R2 B' D' U' is a real mess in my opinion, even though it is a continuous isoglyph. It looks something like the following. X00 0X0 XXX XOX XXX X00 00X XX0 0X0 0X0 0XX X00 00X XXX X0X 00X 0XX X0X Notice that the partial isoglyph which was my first example "looks" fairly continuous, even though it really isn't. The reason it looks that way is that it is continuous along all the edges where the non-solid glyphs come together. Call such a non-continuous partial isoglyph quasi-continuous. I think your eye tends to ignore the solid faces anyway, so that a quasi-continuous partial isoglyph tends to be very striking and very pretty. For example, there are a number of 4-H and 4-T patterns among the partial isoglyphs which are quasi-continuous and which are very pretty. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Sun Apr 5 23:28:33 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id XAA21469; Sun, 5 Apr 1998 23:28:32 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Apr 5 18:06:04 1998 Date: Sun, 5 Apr 1998 18:05:59 -0400 (EDT) From: der Mouse Message-Id: <199804052205.SAA03822@Twig.Rodents.Montreal.QC.CA> To: cube-lovers@ai.mit.edu Subject: Re: Pretty vs. Not-So-Pretty Isoglyphs > On the other hand, > U B2 R2 F2 L2 U L2 F2 U2 R' B' R F' L' U2 B2 R2 B' D' U' > is a real mess in my opinion, even though it is a continuous > isoglyph. I think this (the pattern, not the operator to produce it) is actually rather striking and pretty - provided you look at the cube along the URB-LDF corner-to-corner axis. der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 8 12:17:06 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA28530; Wed, 8 Apr 1998 12:17:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 8 11:04:08 1998 To: Cube-Lovers@ai.mit.edu Date: Wed, 8 Apr 1998 07:55:06 -0700 Subject: A workable 6x6x6 cube design (probably) Message-Id: <19980408.075506.7150.0.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) I have found that the 6x6x6 cube can only be made practical if the outer rows of cubies are slightly larger (about 3mm or 1/8 inch). If the rows are all the same size then some cross-sections of pieces (e.g. the corner pieces) are less than 3 sq-mm, and other pieces are extremely thin (0.6mm in some places). If the plastic is black (or white) and the stickers are all the same size then the inequality in the size of the cubies will be effectively masked. The stickers would have to be spaced evenly. The cube will look as if it has a small 'border' but the perception will be that the cubies are the same size. This design is actually almost as strong as the 4x4x4 cube. It contains an internal frame plus 256 movable pieces of ten different types. No cross section of a piece is smaller than 7 sq-mm (the 4x4x4 has center pieces with 9 sq-mm cross section). Two of the types of piece (FACE EDGE PIECE, SPACER PIECE 2) come in two mirror image forms, so the number of molds that would be needed to produce this is 14 (counting two for the internal frame). The internal mechanism would need to be greased to allow it to turn smoothly, but it should be no worse than the 5x5x5. The following is an exact geometric description of each piece. To be able to understand this you need to know how to use Cartesian and Polar coordinates. All pieces are intersections of planes, spheres, and hyperboloids (which can probably be approximated as cones). The SPACER PIECE 2 could probably be replaced by some sort of rectangular but rounded blob-like thing, it does not need to be an exact shape and the cube might turn more smoothly if it is rounded. It also might then be possible to make it symmetrical so they could be produced with a single mold, which would slightly reduce production cost. Comments, suggestions and quibbles are welcome. LEGEND - x,y,z are Cartesian coordinates, r is distance from origin Dx, Dy, Dz is distance from x, y, z axis respectively NO TOLERANCES - pieces must be shrunk away from all sides a little bit DIMENSIONS assume that the size of an inner CUBIE is 100 and the size of an outer CUBIE is 125, this allows the pieces to be much stronger than if the cubies were all the same size. The puzzle occupies the space such that -325175, y>175, z>175, 280320 AND all points such that 0175, z>175, 280360 AND all points such that 100175, z>175, 320175, z>175, 280360 AND all points such that 0175, 320175, 280360 all points such that 100175, 320175, 280360 AND all points such that 0175, 2800, 24060, y>60, z>0, 20030, y>30, z>0, 1000, 2000, 100sqrt(x^2+60^2), Dy>sqrt(y^2+60^2), Dz>sqrt(z^2+60^2), x>0, y>0, z>0, 200sqrt(x^2+30^2), Dy>sqrt(y^2+30^2), Dz>sqrt(z^2+30^2), x>0, y>0, z>0, 100120, y>120, z>120, 240120, y>120, 0175, 320 Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA01718; Thu, 9 Apr 1998 16:30:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 8 18:05:08 1998 To: Cube-Lovers@ai.mit.edu Date: Wed, 8 Apr 1998 13:45:07 -0700 Subject: A workable 6x6x6 cube design (probably) - correction Message-Id: <19980408.144131.8926.2.tenie1@juno.com> From: tenie1@juno.com (Tenie Remmel) Yikes, there were errors in my geometric description. Here is a (hopefully) correct version: CORNER PIECE consists of: all points such that 200320 AND all points such that 0360 AND all points such that 100175, z>175, 280360 AND all points such that 0175, 280360 all points such that 100175, 280360 AND all points such that 00, 24060, y>60, z>0, 20030, y>30, z>0, 1000, 2000, 100sqrt(x^2+60^2), Dy>sqrt(y^2+60^2), Dz>sqrt(z^2+60^2), x>0, y>0, z>0, 200sqrt(x^2+30^2), Dy>sqrt(y^2+30^2), Dz>sqrt(z^2+30^2), x>0, y>0, z>0, 100120, y>120, z>120, 240120, y>120, 0 Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA11586; Mon, 13 Apr 1998 12:07:53 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Apr 11 21:10:52 1998 Message-Id: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net> From: John Burkhardt To: "Cube-Lovers@ai.mit.edu" Subject: RE: A workable 6x6x6 cube design (probably) - correction Date: Sat, 11 Apr 1998 21:05:25 -0400 So who gets to try and make one? I understood that the dies for the 5x5x5 cube are too expensive to build now due to "lack of interest". On the other hand, we should try to build one because we can. If we can that is :) -JRB From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 15 15:02:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA17097; Wed, 15 Apr 1998 15:02:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 15 13:07:56 1998 Date: Wed, 15 Apr 1998 18:07:57 +0100 From: David Singmaster To: cube-lovers@ai.mit.edu Message-Id: <009C4C21.E208C3B3.8@ice.sbu.ac.uk> Subject: Hamiltonian circuits on the cube The discussion of isoglyphs, etc., has reminded me of a problem which I worked on in the early 1980s but never resolved. I took an all white cube and traced a Hamitonian circuit through all the 54 facelets. If you jumble this up, it is essentially impossible to restore. Indeed there are probably many solutions to the problem. This led me to ask some questions about such Hamiltonian circuits through the 54 facelets. A. How many are there? B. Are there any such circuits where the pattern is the same on each face? I thought I could prove that such did not exist, but I think I assumed that the circuit entered and left each face once, but this need not be the case. I was able to find a circuit with two types of face pattern and the two types were mirror images. If you index the facelets on a face by 11, 12, ..., 33, then the path on the face is: 11, 12, 22, 21, 31, 32, 33, 23, 13. If the circuit enters and leaves each face just once, then the sequence of faces visited forms a Hamiltonian circuit on the faces of the cube, which is better viewed as the vertices of an octahedron. It is easy to see that there are just two such circuits on the octahedron (up to isomorphism). One of these circuits has two kinds of vertex behavior and hence is not suitable. Does this question interest anyone? The reason for the second question was that if just one type of face pattern could be used, then it would be easy to print up stickers for sale - one would just do the same pattern six times! DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 15 16:23:24 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA17354; Wed, 15 Apr 1998 16:23:24 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 15 15:35:06 1998 Date: Wed, 15 Apr 1998 15:38:00 -0400 (Eastern Daylight Time) From: Dale Newfield Reply-To: DNewfield@cs.virginia.edu To: cube-lovers@ai.mit.edu Subject: Re: Hamiltonian circuits on the cube In-Reply-To: <009C4C21.E208C3B3.8@ice.sbu.ac.uk> Message-Id: On Wed, 15 Apr 1998, David Singmaster wrote: > The discussion of isoglyphs, etc., has reminded me of a problem which I > worked on in the early 1980s but never resolved. I took an all white cube and > traced a Hamitonian circuit through all the 54 facelets. If you jumble this > up, it is essentially impossible to restore. Indeed there are probably many > solutions to the problem. This led me to ask some questions about such > Hamiltonian circuits through the 54 facelets. This is quite reminiscent of "Oddmaze," (http://www.edoc.com/zarf/custom-cubes.html) which is a creation by Andrew Plotkin realized using Kristin Looney's "Custom Cube Technology" (http://www.wunderland.com/WTS/Kristin/Technology.html). On its surface is a labyrinth with no branches or dead ends. Each facelet has exactly two paths through it. In the "start" position, at least, the path obeys the Celtic knotwork property (over/under alternations). It is really quite interesting, and well described on the above mentioned page. (This doesn't help answer your questions, but might put you in contact with another that has given them some thought.) -Dale Newfield Dale@Newfield.org From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 15 17:12:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA17446; Wed, 15 Apr 1998 17:12:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 15 16:25:48 1998 Date: Wed, 15 Apr 1998 16:29:26 -0400 (EDT) From: Nicholas Bodley To: John Burkhardt Cc: "Cube-Lovers@ai.mit.edu" Subject: RE: A workable 6x6x6 cube design (probably) In-Reply-To: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net> Message-Id: Am I missing something? The geometrical description seemed plausible and fine, but unless I'm far off base, it seems that some quite-clever mechanical design is essential. Fairly sure that Douglas Hofstadter noted in passing (I think in Go"del (G"odel ? :), Escher, Bach...) that a physical prototype of the 6^3 has been built. I have pulled apart and studied all "sizes" from the 2^3 to the 5^3, and the innards of each are rather different; the 5 is based on the 3, but the 4 (Rubik's Revenge) has a ball inside, as probably most List readers know. The innards of the 2 are quite distinctive, again; (also, borderline impossible to assemble/disassemble!). It's remarkable how a simple increment of one, so to speak, has such a profound effect on the basic internal design. My awareness of most abstruse corners of math. is quite comparable with that of, let's say, a turtle. However, I do know modest bits about formal kinematics, four-bar linkages, and some underlying principles of the linkage variety of mechanical analog computers, for instance, so my ignorance is somewhat better that that of a rock. I also know the innards of mechanical calculators rather well. However, with such non-qualifications, I suspect that there is no theory of such mechanisms as we find inside our cubes and related puzzles. Mathematicians seem to be able to handle braids (Emil Artin?) rather well, and knots seem to be doing well, but I really doubt that there's any significant theory that can be used to develop a design such as the innards of a 5^3. Ordinary geometry, I feel fairly confident, is of relatively little help. One can at least define the geometry of the requisite constraints and "freedoms" of motion, but to create the requisite shapes, seems to me, requires a special and clever kind of mind. Honestly, I'd welcome having big holes figuratively shot through my contentions! I'm sure I'd learn something. For limited (and probably very costly) prototype runs, the technology that goes by various names such as 3-D printing, rapid prototyping, and (ugh!) stereolithography should do well to create the shapes. (Seems to me it's a fairly formidable challenge to a CAD program to create some of the weird shapes, but I plead ignorance! (The "stereo" part of that long word is fine, but it's really stretching a point to think of it as writing on stone.) My best to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When will the non-word "alot" first be listed |* Amateur musician *|* in a dictionary? Maybe 2030? -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 15 18:36:13 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA17618; Wed, 15 Apr 1998 18:36:12 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 15 16:32:14 1998 Date: Wed, 15 Apr 1998 16:35:57 -0400 (EDT) From: Nicholas Bodley To: John Burkhardt Cc: "Cube-Lovers@ai.mit.edu" Subject: RE: A workable 6x6x6 cube design (probably) - another comment In-Reply-To: <01BD658D.8DD543C0@jburkhardt.ne.mediaone.net> Message-Id: On Sat, 11 Apr 1998, John Burkhardt wrote: }So who gets to try and make one? I understood that the dies for the }5x5x5 cube are too expensive to build now due to "lack of interest". On Does anyone know if the dies still exist? I wouldn't be a bit surprised if the whole set weighs several tons, even if they are single-cavity types. Tooling for injection molding is fiercely expensive! (Tooling for a decent ("serious") plastic soprano recorder runs probably a third to a half $US million, for instance. (Mostly bigger parts, a few very critical tolerances, and far fewer parts, also.)) Best, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* I might need to switch to shore.net, but will |* Amateur musician *|* do my best to minimize the nuisance if so. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon Apr 20 15:57:40 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA00263; Mon, 20 Apr 1998 15:57:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Apr 20 11:51:47 1998 To: cube-lovers@ai.mit.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Hamiltonian circuits on the cube Date: 20 Apr 1998 15:55:44 GMT Organization: California Institute of Technology, Pasadena Message-Id: <6hfr60$lfq@gap.cco.caltech.edu> References: David Singmaster writes: > The discussion of isoglyphs, etc., has reminded me of a problem which I >worked on in the early 1980s but never resolved. I took an all white cube and >traced a Hamitonian circuit through all the 54 facelets. If you jumble this >up, it is essentially impossible to restore. Indeed there are probably many >solutions to the problem. This led me to ask some questions about such >Hamiltonian circuits through the 54 facelets. > A. How many are there? > B. Are there any such circuits where the pattern is the same on each >face? I thought I could prove that such did not exist, but I think I assumed >that the circuit entered and left each face once, but this need not be the >case. The answer to B is "Yes"!! I was pretty surprised to come up with this within ten minutes of reading the question: +--+--+--+ |42|43|44| +--+--+--+ |47|46|45| +--+--+--+ |54| 3| 4| +--+--+--+--+--+--+--+--+--+--+--+--+ | 1| 2| 5| 6| 7| 8|26|27|40|41|48|53| +--+--+--+--+--+--+--+--+--+--+--+--+ |14|13|12|11|10| 9|25|28|39|38|49|52| +--+--+--+--+--+--+--+--+--+--+--+--+ |15|16|17|18|21|22|24|29|36|37|50|51| +--+--+--+--+--+--+--+--+--+--+--+--+ |33|32|19| +--+--+--+ |34|31|20| +--+--+--+ |35|30|23| +--+--+--+ X=====X=====X=====X H H H H ---------------+ H H H H | H X=====X=====X==|==X H H H | H ---------------+ H H H H H X=====X=====X=====X H H H H ---+ H +-----+ H H | H | H | H X==|==X==|==X==|==X=====X=====X=====X=====X=====X=====X==|==X==|==X==|==X H | H | H | H H H H H H H | H | H | H H +-----+ H +-----------------+ H +-----+ H +-----+ H | H | H H H H H H H | H | H | H | H H | H | H X=====X=====X=====X=====X=====X==|==X==|==X==|==X==|==X=====X==|==X==|==X H H H H H H | H | H | H | H H | H | H H +-----------------------------+ H | H | H +-----+ H | H | H H | H H H H H H | H | H H | H | H | H X==|==X=====X=====X=====X=====X=====X==|==X==|==X=====X==|==X==|==X==|==X H | H H H H H H | H | H H | H | H | H H +-----------------+ H +-----+ H | H | H +-----+ H +-----+ H H H H H | H | H | H | H | H | H H H H X=====X=====X=====X==|==X==|==X==|==X==|==X==|==X==|==X=====X=====X=====X H H H H H +-----+ H +--- H | H | H | H X==|==X==|==X==|==X H | H | H | H H | H | H +--- H | H | H H X==|==X==|==X=====X H | H | H H H | H | H +--- H | H | H | H X==|==X==|==X==|==X -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you. From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 22 11:53:02 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA00421; Wed, 22 Apr 1998 11:53:02 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 22 11:43:05 1998 Date: Wed, 22 Apr 98 11:42:49 EDT Message-Id: <9804221542.AA10123@sun28.aic.nrl.navy.mil> From: Dan Hoey To: whuang@ugcs.caltech.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <6hfr60$lfq@gap.cco.caltech.edu> Subject: Re: Hamiltonian circuits on the cube whuang@ugcs.caltech.edu (Wei-Hwa Huang) writes: > I was pretty surprised to come up with this within ten minutes of reading > the question: Wow, I'm impressed. I thought I'd have to write a program to find them, and here's a nice symmetric solution. The symmetry is more visible in a different unfolding: +-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+ It shouldn't be that hard to solve a cube with these markings--there are only two different kinds of corner cubies, three kinds of edge cubies, and the face centers need only be oriented mod 180 degrees. Working from one of the symmetric corners, it's not hard to see that this is the only continuous solution. I've noticed a minor modification to your pattern that also admits an isoglyphic Hamiltonian path: +-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+ Anyone who's working on an exhaustive search to see if there are any others, send me e-mail before I hack again! Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 22 12:36:05 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA00597; Wed, 22 Apr 1998 12:36:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 22 12:19:02 1998 Message-Id: <353E1961.6231@sgi.com> Date: Wed, 22 Apr 1998 09:22:57 -0700 From: Derek Bosch To: Dan Hoey Cc: cube-lovers@ai.mit.edu Subject: Re: Hamiltonian circuits on the cube - kind of References: <9804221542.AA10123@sun28.aic.nrl.navy.mil> On a similar note, has anyone stickers with: | / - - / | or | | ----- | | (or any of those rotations?) Kind of a cross between a rubik's Tangle and a rubik's cube? Especially if each of the lines has a different color? D -- Derek Bosch "A little nonsense now and then (650) 933-2115 is relished by the wisest men"... W.Wonka bosch@sgi.com From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 22 14:41:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA01093; Wed, 22 Apr 1998 14:41:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 22 14:20:19 1998 Date: Wed, 22 Apr 1998 14:24:21 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Hamiltonian circuits on the cube In-Reply-To: <9804221542.AA10123@sun28.aic.nrl.navy.mil> To: Dan Hoey Cc: whuang@ugcs.caltech.edu, cube-lovers@ai.mit.edu Message-Id: On Wed, 22 Apr 1998, Dan Hoey wrote: > whuang@ugcs.caltech.edu (Wei-Hwa Huang) writes: > > > I was pretty surprised to come up with this within ten minutes of reading > > the question: > > Wow, I'm impressed. I thought I'd have to write a program to find > them, and here's a nice symmetric solution. The symmetry is more > visible in a different unfolding: > Not to minimize the difficulty of the problem or the beauty of the solution (quite the contrary), but the solution seems almost trivial when viewed in the light of Dan's particular unfolding of the surface of the cube. The same comment is true of Dan's isoglyphic solution. It makes me wonder of you actually saw Dan's unfolding in your mind's eye, as it were, as you worked out your solution. Or another way to put it, did you work out your solution in 2-D or in 3-D? It also makes me wonder if there is any other unfolding that would lead as naturally to a Hamiltonian circuit. I tend to think not, but I could well be wrong. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 23 11:51:20 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA04842; Thu, 23 Apr 1998 11:51:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Apr 23 11:42:59 1998 From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Message-Id: <199804231547.IAA09346@gluttony.ugcs.caltech.edu> Subject: Re: Hamiltonian circuits on the cube To: jbryan@pstcc.cc.tn.us (Jerry Bryan) Date: Wed, 22 Apr 1998 16:58:41 -0700 (PDT) Cc: cube-lovers@ai.mit.edu In-Reply-To: <9804231425.AA10935@sun28.aic.nrl.navy.mil> from "Dan Hoey" at Apr 23, 98 10:25:30 am Reply-To: whuang@ugcs.caltech.edu Jerry Bryan typed something like this in a previous message: > It makes me wonder of you actually saw Dan's unfolding in your mind's > eye, as it were, as you worked out your solution. Or another way to put > it, did you work out your solution in 2-D or in 3-D? It also makes me > wonder if there is any other unfolding that would lead as naturally to a > Hamiltonian circuit. I tend to think not, but I could well be wrong. > Actually, I didn't visualize any unfolding at all, so I guess I did it in 3-D. Here's approximately the line of reasoning that led to my solution. As Dr. Singmaster notes, there is only one way to draw a Hamiltonian on a 1x1x1 cube where all the faces are identical, and that is with a right angle on each face. Naturally one's first impulse is to find a path that enters each 3x3 face in one place and exits in another -- and these two ends must be on edges 90-degree apart. One quickly sees that the two exits must be on edge cubies, since if any were on corner cubies there would be a parity problem between "inner corners" and "outer corners." But if they were edge cubies, then no Hamiltonian path exists (as the inner corner must join to the ends already). However, another extension is the "three parallel paths" pattern: put this on each face: A B C | | | | | +-D | +----E +-------F This leads to three paths on the cube, where the center one is the traditional 1x1x1 Hamiltonian. If this can be rearranged to a solution, we must try to reconnect the ends so that there is some "interaction" between the three paths. C must connect to D, but we can connect A to B instead -- and this leads to a solution, which surprised me when I visualized it on a 3-d cube. (I most definitely find visualizing in 3-D easier than visualizing the links in an unfolded cube.) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you. From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 23 20:24:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA05974; Thu, 23 Apr 1998 20:24:58 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Apr 23 20:22:45 1998 Date: Thu, 23 Apr 98 20:21:11 EDT Message-Id: <9804240021.AA11374@sun28.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Cc: whuang@ugcs.caltech.edu Subject: Re: Hamiltonian circuits on the cube I wrote: "...send me e-mail before I hack again!" Too late. The only chiral Hamiltonian isopaths are the two we've already seen, and: +---+---+-@-+---+-@-+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+---+-@-+---+-@-+---+-@-+---+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+---+-@-+---+-@-+---+-@-+---+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+-@-+---+-@-+---+---+ I actually generated all the continuous chiral isopaths, and the following is the other extreme--the only one with nine disjoint paths. Yet one of the paths goes through one third of the facelets. +-@-+-@-+---+-@-+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+---+-@-+-@-+-@-+---+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+---+-@-+-@-+-@-+---+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+-@-+---+-@-+-@-+ Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 24 09:41:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA07001; Fri, 24 Apr 1998 09:41:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Apr 24 09:38:22 1998 Date: Fri, 24 Apr 98 09:38:06 EDT Message-Id: <9804241338.AA11821@sun28.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Cc: whuang@ugcs.caltech.edu Subject: Re: Hamiltonian circuits on the cube I wrote: > I actually generated all the continuous chiral isopaths, and the > following is the other extreme--the only one with nine disjoint paths. Which was bogus. I actually generated only the continuous chiral isopaths in which no circuit lies entirely on one face. That's fine for the Hamiltonian circuit problem, but for the maximum number of disjoint circuits we probably want the 14-circuit pattern +-@-+-@-+-@-+---+---+---+ |@@ @@@@@ | @@@@@ @@| + + + + @ + @ + @ + |@@ @@@@@ | @@@@@ @@| + @ + @ + @ + + + + |@@ @@@@@ | @@@@@ @@| +-@-+-@-+-@-+---+---+---+-@-+-@-+-@-+ |@@ @@@@@ | @@@@@ @@| + + + + @ + @ + @ + |@@ @@@@@ | @@@@@ @@| + @ + @ + @ + + + + |@@ @@@@@ | @@@@@ @@| +-@-+-@-+-@-+---+---+---+-@-+-@-+-@-+ |@@ @@@@@ | @@@@@ @@| + + + + @ + @ + @ + |@@ @@@@@ | @@@@@ @@| + @ + @ + @ + + + + |@@ @@@@@ | @@@@@ @@| +---+---+---+-@-+-@-+-@-+ which should be familiar to Tartan fans. Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Sat Apr 25 20:15:48 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA10540; Sat, 25 Apr 1998 20:15:47 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Apr 24 14:24:49 1998 Date: Fri, 24 Apr 1998 14:21:43 -0400 (Eastern Daylight Time) From: Dale Newfield Reply-To: DNewfield@cs.virginia.edu To: cube-lovers@ai.mit.edu Subject: 4x4x4 pieces, and in quantity Message-Id: [ Moderators note: Dale Newfield passes on this notice. Contact Mike Green for details. ] Date: Fri, 24 Apr 1998 01:17:15 -0700 From: Mike Green To: Dale Newfield Cc: Dale Newfield , Dale Newfield Subject: "Rubik's Revenge" - 4x4x4 Dale, Thank you for your inquiry. We do have a limited number of "Rubik's Revenge" parts for those of you who have a broken cube: ITC-030a 4x4x4 Center Cubie - Ideal Toy Co. $ 2.50 each ITC-030b 4x4x4 Ball Center - Ideal Toy Co. $10.00 each ITC-030c 4x4x4 Corner Cubie - Ideal Toy Co. $ 2.00 each ITC-030d 4x4x4 Edge Cubie - Ideal Toy Co. $ 2.00 each ITC-030e 4x4x4 Sticker - Ideal Toy Co. $ .50 each You want 1 corner and 2 centers? You will reuse your stickers? How will you pay? Postage will probably be $2.00. Recently the price of a "Rubik's Revenge" has hit as high as $200.00 each on the "Web". Can you believe that! The last five we sold, fortunately for our customers, went for $65.00 each. How would you like to see it back in the market for less than $30.00? Possibly even less than $25.00. Would you buy more than one? For us to bring it back we have to place a minimum order of between 10,000 to 30,000 pieces and pay for new tooling - all up front. Tell your friends and have them tell their friends, and their friend's friends to get on our wish list. Have your local puzzle retailer contact us as well. By using the power of the "Internet", e-mail, and word of mouth I'm sure we can get the numbers up there and make this happen in less than a year. I'm ready and willing are you? In the meantime, we also carry as standard stock the Rubik's 2x2x2 for $5.99, Rubik's 3x3x3 for $10.99, 3x3x3 Magic Cube for $6.99, 5x5x5 for $38.99, Square 1 for $14.99, and Skewb for $32. We also pull in on a fairly regular basis Megaminx, Impossiball, Pyraminx, Mickey's Challenge, Masterballs, and various other sequential movement puzzles when we can. Prices and quantities vary, but we're always on the hunt. We'd very much like to bring the 4x4x4 back to market. You can help greatly by spreading the word. Thank you. Sincerely, Mike D. Green President From cube-lovers-errors@mc.lcs.mit.edu Sat Apr 25 21:20:14 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA10646; Sat, 25 Apr 1998 21:20:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Apr 25 20:50:39 1998 Date: Sat, 25 Apr 98 20:50:27 EDT Message-Id: <25Apr1998.202137.Hoey@AIC.NRL.Navy.Mil> From: Dan Hoey To: bosch@sgi.com Cc: cube-lovers@ai.mit.edu In-Reply-To: <353E1961.6231@sgi.com> (message from Derek Bosch on Wed, 22 Apr 1998 09:22:57 -0700) Subject: Re: Hamiltonian circuits on the cube - kind of Derek Bosch asks for a cross between a Rubik's tangle and a Rubik's cube. Here's a Hamiltonian chiral isotangle. .__._____._____.__.__._____._____.__. | \ : / : \ | \ : | : \ | +-. `-+-' .-+-. `-+-. `-+--+--+-. `-+ |..\..:../..:..\..|..\..:..|..:..\..| | | : / : / | / : / : | | +--+--+-' .-+-' .-+-' .-+-' .-+--+--+ |..|..:../..:../..|../..:../..:..|..| | \ : | : \ | \ : / : \ | +-. `-+--+--+-. `-+-. `-+-' .-+-. `-+ .__._____._____.__|__\__:__|__:__\__|__\__:__/__:__\__| | \ : / : \ | \ : | : \ | +-. `-+-' .-+-. `-+-. `-+--+--+-. `-+ |..\..:../..:..\..|..\..:..|..:..\..| | | : / : / | / : / : | | +--+--+-' .-+-' .-+-' .-+-' .-+--+--+ |..|..:../..:../..|../..:../..:..|..| | \ : | : \ | \ : / : \ | +-. `-+--+--+-. `-+-. `-+-' .-+-. `-+ .__._____._____.__|__\__:__|__:__\__|__\__:__/__:__\__| | \ : / : \ | \ : | : \ | +-. `-+-' .-+-. `-+-. `-+--+--+-. `-+ |..\..:../..:..\..|..\..:..|..:..\..| | | : / : / | / : / : | | +--+--+-' .-+-' .-+-' .-+-' .-+--+--+ |..|..:../..:../..|../..:../..:..|..| | \ : | : \ | \ : / : \ | +-. `-+--+--+-. `-+-. `-+-' .-+-. `-+ |__\__:__|__:__\__|__\__:__/__:__\__| There's only one path, so it's all one color. Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 29 10:54:31 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA00312; Wed, 29 Apr 1998 10:48:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 29 00:53:36 1998 Message-Id: <3546B17A.3419@idirect.com> Date: Wed, 29 Apr 1998 00:50:02 -0400 From: Mark Longridge To: cube-lovers@ai.mit.edu Cc: cubeman@idirect.com Subject: Various Cube Thoughts Ok, I'm back into cubing again... a few interesting, if somewhat disjoint observations: Summary of the 3 different types of optimal superflip sequences: 1) Superflip with minimal q turns & symmetric moves Process has central reflection symmetry R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 D3 U1 (24q, 22f) 2) Superflip with minimal q turns & asymmetric moves U1 R2 F3 R1 D3 L1 B3 R1 U3 R1 U3 D1 F3 U1 F3 U3 D3 B1 L3 F3 B3 D3 L3 (24q, 23f) 3) Superflip with minimal f turns & asymmetric moves U1 R2 F1 B1 R1 B2 R1 U2 L1 B2 R1 U3 D3 R2 F1 D2 B2 U2 R3 L1 (28q, 20f) ------------------------------------------------------------------ No matter which cube you start searching from, e.g. pons asinorum, 12 flip, or any random cube, the dispersion of cubes is the same: 1, 12, 114, 1068, 10011... etc So much for trying to search backwards from the 12-flip to number the positions from (perhaps) antipode to start! ------------------------------------------------------------------ I have got Mike Reid's optimal solver to work under the dos shell in windows 95. I finally managed to compile it using WATCOM 11.0 thusly: wcl386 /k10000000 search.c I had to give it a 10 megabyte stack for it to work! It found the sequence ( F R B L )^5 to require 20 q turns, so there is nothing better. Next I tried ( F R B L )^6 to see if that would be 24 q but a 20 q solution was found. Mike Reid confirmed the result on another computer running Linux. ------------------------------------------------------------------- Lastly, some non-mathematical ideas on how to do optimal searches of rubik's cube patterns. Using my own human solving algorithm I solve the 4 down edge cubes last. One of the patterns I get was solved optimally by Mike's program thusly: D' R' D' F B' D' L' D L D F' B D R If we assign a value of 1 to each face and add them we get: D = 6 U = 0 F = 2 B = 2 L = 2 R = 2 Note that most of the action occurs with the D face, which I find suggestive. After all, nothing is moved except the 4 bottom edge cubes. Also all the other faces have an even number of turns! My idea is perhaps with some pre-processing of a goal state it is possible to prune the number of moves down to such a degree that the number of moves actually tried is quite small. Also note that this particular goal state has only 2 pairs of cubes swapped, and all the other cubes are in place. Now I may be using too much hindsight, but to me it is counter- intuitive that it is possible to have 3 separate turns of the D face! So, sequences with 3 uses of the D face should not be considered. My theory is that ultimately with enough pre-processing only the optimal sequences will be even considered! -> Mark <- From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 30 10:09:06 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA02688; Thu, 30 Apr 1998 10:09:06 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Apr 30 09:59:19 1998 Date: Thu, 30 Apr 1998 09:57:20 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Various Cube Thoughts In-Reply-To: <3546B17A.3419@idirect.com> To: Mark Longridge Cc: cube-lovers@ai.mit.edu Message-Id: On Wed, 29 Apr 1998, Mark Longridge wrote: > ------------------------------------------------------------------ > > No matter which cube you start searching from, e.g. pons asinorum, > 12 flip, or any random cube, the dispersion of cubes is the same: > > 1, 12, 114, 1068, 10011... etc > > So much for trying to search backwards from the 12-flip to number > the positions from (perhaps) antipode to start! > > ------------------------------------------------------------------ This has been discussed before on Cube-Lovers. There are several ways to look at why it is true. I think at the most basic level that it depends on the inverse property of groups. Let A be any non-empty subset (not necessarily a subgroup) of G, and let x be any element of G. Then xA contains the same number of elements as A. Hence, if A is (for example) the set of all positions which are n moves from Start, then xA is the set of all positions which are n moves from x, and xA is the same size as A (remember that the distance from Start to a is the same as the distance from x to xa for any a in A). Notice that if A is a subgroup of G rather than just being a subset, then xA is a coset. The fact that cosets are either equal or disjoint, combined with the fact that A is the same size as xA, constitute the basis for the proof that the size of a subgroup must divide evenly the size of the group. The inverse property is involved in showing that A and xA are the same size as follows. Suppose we have A={a,b,c} which contains three elements. Then we have xA={xa,xb,xc} which also appears to contain three elements. The only way that xA would not have three elements would be if some of the apparently distinct elements were really the same, for example if xa and xc were really two different names for the same element. But if xa=xc, then we have x'(xa)=x'(xc) so that (x'x)a=(x'x)c so that ia=ic so that a=c. We know by definition that a and c are distinct. Hence, xa and xc must be distinct. Just to give one more illustration of the importance of the inverse property in showing that A and xA are the same size, here is a false counterexample. Consider the multiplicative group of the real numbers or of the rational numbers. Suppose A={ 2/3, 3/4, 7} and x=4. Then, xA={ 8/3, 3, 28}. So far, so good because both A and xA have three elements. But suppose x=0. Then xA={0, 0, 0}={0} which has only one element. Here we have A with three elements and xA with only one element. So what is wrong. The problem is that any multiplicative group of what we might call "normal" numbers (e.g., real or rational or complex) must omit zero because 0 does not have a multiplicative inverse. That is, there is no solution to the equation 0*x=1. So when I let x=0, I was cheating by multiplying by a number which is not in the multiplicative group and which does not have a multiplicative inverse. The reason I know that this has been discussed before was that I was involved in the discussion. At one point I incorrectly asserted that what you are calling "the dispersion of the cubes" did depend on which position was at the root of the search. Cube-Lovers was quick to correct me, of course. However egregious was my error, it was still an honest error. The reason for the honest error is that I accomplish nearly all my searches by counting patterns (M-conjugacy classes) rather than by counting positions. And when you count by patterns, "the dispersion of the cubes" does depend upon which pattern is at the root of the search. So my mistake was to make a statement about positions which should have been applied only to patterns. Your note reminded me of a question I have thought about off and on ever since that previous discussion. Suppose you are searching by patterns. Under what circumstances can you start the search with two different patterns and still have the "dispersion of the cubes" be the same? I suspect that there is a very simple answer, but I am having trouble ascertaining what it is. I suspect that the only possibility is if the two positions differ by superflip, that is if one of them is x then the other one must be xf=fx, where f is the superflip. But I am simply not sure if there are any more possibilities. Note that having the two different patterns be M-conjugate is not an answer to the question because if two patterns are M-conjugate then they are really just one pattern. As a last comment, readers of Cube-Lovers should be familiar with the sequence 1, 12, 114... for positions in quarter turn searches. A search for patterns in quarter turns begins 1, 1, 5... The first 1 is Start. The second 1 (1q from Start) is Q, the set of twelve quarter turns. The 5 (2q from Start) represents the following five patterns: 1) any face twisted twice in the same direction, 2) any two opposite faces twisted once each in the same direction (an antislice), 3) any two opposite faces twisted once each in the opposite direction (a slice), 4) any two adjacent faces twisted once each in the same direction (e.g., UF or U'F'), and 5) any two adjacent faces twisted once each in the opposite direction (e.g., UF' or U'F). Beyond 2q from Start, it becomes too complicated to calculate the patterns in my head. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 30 14:16:03 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA03343; Thu, 30 Apr 1998 14:16:03 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Apr 30 13:01:01 1998 Message-Id: Date: Thu, 30 Apr 1998 12:59:19 -0500 To: cube-lovers@ai.mit.edu From: kristin@wunderland.com (Kristin Looney) Subject: Garden Variety Rubik's Cube Cube Lovers - a new cube pic on the image wall... for your viewing pleasure... http://wunderland.com/EBooks/ImageWall/Pages/GardenVarietyCube.html Peace - -K. kristin@wunderland.com http://www.wunderland.com/wts/kristin To all the fishies in the deep blue sea, Joy. From cube-lovers-errors@mc.lcs.mit.edu Fri May 1 10:45:38 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA06031; Fri, 1 May 1998 10:45:37 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 1 01:54:06 1998 From: Andrew John Walker Message-Id: <199805010552.PAA00579@wumpus.its.uow.edu.au> Subject: Square like groups To: cube-lovers@ai.mit.edu Date: Fri, 1 May 1998 15:52:34 +1000 (EST) Does anyone have any information on patterns where each face only contains opposite colours, but are not in the square group? L' R U2 L R' may be an example. If square moves are applied to such patterns to form new groups, how many such groups exist? Andrew Walker From cube-lovers-errors@mc.lcs.mit.edu Fri May 1 19:58:06 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA07381; Fri, 1 May 1998 19:58:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 1 19:57:13 1998 Date: Fri, 1 May 98 19:56:56 EDT Message-Id: <9805012356.AA16835@sun28.aic.nrl.navy.mil> From: Dan Hoey To: ajw01@uow.edu.au Cc: cube-lovers@ai.mit.edu In-Reply-To: <199805010552.PAA00579@wumpus.its.uow.edu.au> (message from Andrew John Walker on Fri, 1 May 1998 15:52:34 +1000 (EST)) Subject: Re: Square like groups Andrew Walker asks: > Does anyone have any information on patterns where each > face only contains opposite colours, but are not in the square > group? We may call this the "pseudosquare" group P. It consists of orientation-preserving permutations that operate separately on the three equatorial quadruples of edge cubies and the two tetrahedra of corner cubies, and for which the total permutation parity is even. So Size(P) = 4!^5 / 2 = 3981312. > L' R U2 L R' may be an example. No, that's in the square group, says GAP. Also, Mark Longridge noticed (8 Aug 1993) that the square group is mapped to itself under conjugation by an antislice (though I don't recall a proof--is there an easy one?). Your position is (L R)' R2 T2 R2 (L R), so this result would apply. Does anyone have a square process for it? > If square moves are applied to such patterns to form new groups, how > many such groups exist? Consider the subgroup of P consisting of positions in which the parity of the corner permutation is even. (The edge permutation will then also be even, and the parity of the permutations of the two edge tetrahedrons will be equal). Call it AP, for "alternating P". Size(AP) = Size(P)/2 = 1990656. The square group S is a subgroup of index 3 in AP, so Size(S)=Size(AP)/3=663552. I don't have a very criterion for choosing elements of AP to be in S, except that it has to do with a correlation between the permutations of the two tetrahedrons of corners, provided those permutations are of the same parity (as they must be for the position to be in AP). According to GAP, these are the only three possibilities. To be explicit, let us label the cube's corners 1 D B 3 C 2 4 A Then we can partition S4 into six cosets: C1 = { (), (3,4)(1,2), (1,4)(2,3), (2,4)(1,3) } C3 = { (1,2,3), (1,4,2), (1,3,4), (2,4,3) } C2 = { (1,3,2), (1,4,3), (2,3,4), (1,2,4) } C4 = { (1,2), (1,4,2,3), (1,3,2,4), (3,4) } C5 = { (2,3), (1,4), (1,3,4,2), (1,2,4,3) } C6 = { (1,3), (2,4), (1,4,3,2), (1,2,3,4) } and similarly D1,D2,...,D4 for S4 acting on {A,B,C,D}. Now let c be an arbitrary permutation in P that fixes {A,B,C,D} elementwise, and let Coset(c) be the coset to which c's operation on {1,2,3,4} belongs. Let d be an arbitrary permutation in P that fixes {1,2,3,4} elementwise, and let Coset(d) be the coset to which d's operation on {A,B,C,D} belongs. Then the group generated by depends only on Coset(c) and Coset(d): Coset(d) D1 D2 D3 D4 D5 D6 Coset(c) C1 S AP AP P P P C2 AP S AP P P P C3 AP AP S P P P C4 P P P S AP AP C5 P P P AP S AP C6 P P P AP AP S There may be some wisdom to be gained in seeing that C1 is normal in S4, so S4/C1 is isomorphic to S3. We can represent the Ci and Di by their action on {1,2,3,A,B,C}. The above table shows whether the group , has order 6, 18, or 24. I'd love to hear a more explanatory description of this phenomenon, especially if it explains the absence of a subgroup of index 3 in P. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Sat May 2 17:23:32 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA09204; Sat, 2 May 1998 17:23:31 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 1 22:41:24 1998 Message-Id: <354A8671.730D@idirect.com> Date: Fri, 01 May 1998 22:35:29 -0400 From: Mark Longridge Reply-To: cubeman@idirect.com To: Dan Hoey Cc: cube-lovers@ai.mit.edu Subject: Re: Square like groups References: <9805012356.AA16835@sun28.aic.nrl.navy.mil> Dan Hoey wrote: > > Andrew Walker asks: > > > Does anyone have any information on patterns where each > > face only contains opposite colours, but are not in the square > > group? > > We may call this the "pseudosquare" group P. It consists of > orientation-preserving permutations that operate separately on the > three equatorial quadruples of edge cubies and the two tetrahedra of > corner cubies, and for which the total permutation parity is even. So > Size(P) = 4!^5 / 2 = 3981312. > > > L' R U2 L R' may be an example. R2 F2 R2 U2 R2 F2 R2 U2 F2 > > No, that's in the square group, says GAP. Also, Mark Longridge > noticed (8 Aug 1993) that the square group is mapped to itself under > conjugation by an antislice (though I don't recall a proof--is there > an easy one?). Your position is (L R)' R2 T2 R2 (L R), so this result > would apply. Does anyone have a square process for it? I almost forgot about all that info back in 1993! But I hardly think a proof is necessary. After the moves (L' R) all the following moves are in the square's group. Then we are just doing the inverse of (L` R) at the end. Not very rigourous, but... I'll search for a counter-example. -> Mark <- From cube-lovers-errors@mc.lcs.mit.edu Sat May 2 18:35:38 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA09349; Sat, 2 May 1998 18:35:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat May 2 18:31:11 1998 Date: Sat, 2 May 98 18:30:59 EDT Message-Id: <9805022230.AA17631@sun28.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Subject: Re: Square like groups With respect to the square group, I wrote: > I'd love to hear a more explanatory description of this phenomenon, > especially if it explains the absence of a subgroup of index 3 in P. I should really have waited until I got back home to Singmaster's book, which has a marvelous explanation of why the squares group has index 6 in the pseudosquare group. First, the edges are permuted in in all possible ways consistent with 1. remaining in their "equators" of four edges, 2. not being flipped, and 3. having a permutation parity equal to that of the corners. so we need only consider the 2x2x2 cube, and then we fix the BLD corner in place. Corners don't get twisted, so we consider only the permutation. We express the generators as permutations of the seven movable corners, expressed as follows: 2-------A / / \ / T / \ F^2 = B^2 = (1,4)(B,C), / / \ R^2 = L^2 = (1,3)(A,C), B-------1 R 3 T^2 = D^2 = (1,2)(A,B). \ \ / \ F \ / \ \ / 4-------C The neat part is to notice that the permutation on {A,B,C} is determined by the permutation on {1,2,3,4}. We do this by representing these generators as symmetries on a tetrahedron, labelled as follows. 1-----------C-----------2 \`-. .-'/ \ `A. .B' / \ `-. .-' / \ `4' / \ : / B : A \ : / \ C / \ : / \ : / \:/ 3 Notice that the symmetry that permutes the tetrahedron's vertex labels as (1,4) also permutes the edge labels as (B,C), corresponding to F^2 in the cube's action. Similarly (1,3) implies (A,C) and (1,2) implies (A,B). With respect to Mark Longridge's having noticed that the square group is mapped to itself under conjugation by an antislice (L R), the proof turns out to be pretty easy. First, we notice that we may consider conjugation by a slice (L R') since that differs by a square (R^2) from the antislice. Now we work in the group that includes whole-cube orientations, and perform the slice in the mechanically easy way, as a 4-cycle of face centers and an equatorial 4-cycle of edges. Note that all the edges of the equator are flipped (with respect to the orientation that is preserved by the psueudo-square and square groups) by the slice. So if S is a square-group process that rotates the edges in an equator E, the process Slice' S Slice S' has the following actions: 1. Identity on the corners and the two equators other than E, because they are not moved by the slice, 2. Identity on the face centers, because they are not moved by S, 3. Flips each edge of E twice (once in Slice' and once in Slice), so restores the orientation, and 4. Is an even permutation of the edges in E (odd in Slice, odd in Slice', and equal in S and S'). The even permutation (4) of the edges in E is a slice group process, as Mark noted, as for instance the 3-cycle (R^2 F^2 R^2 T^2)^2 F^2. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Mon May 4 10:31:18 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA12732; Mon, 4 May 1998 10:31:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun May 3 17:19:27 1998 Message-Id: <199805032117.RAA07495@life.ai.mit.edu> Date: Sun, 3 May 1998 17:18:53 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: Square like groups andrew walker asks > Does anyone have any information on patterns where each > face only contains opposite colours, but are not in the square > group? L' R U2 L R' may be an example. the set of such patterns is what i called the "target subgroup" for my optimal solver. it is the intersection of the three subgroups , and (or the intersection of any two of them). the position he mentions is in the square group (mark longridge gives a minimal maneuver for it). dan hoey remarks that the square group has index 6 in this "pseudo-square" group. christoph bandelow's book "inside rubik's cube and beyond" gives a nice criterion for a pseudo- square pattern to be in the square group. bandelow's criterion (slightly paraphrased) is the four U corners must be coplanar, the four F corners must be coplanar, and the four R corners must be coplanar. (equivalently, all twelve sets of four coplanar corners remain coplanar.) in fact, this forces the parity of the corner permutation to be even (and thus the same for the edge permutation). this reminds me of an interesting idea i had for a puzzle: a 3x4x5 box, whose faces and slices are restricted to 180 degree turns. this sort of thing could also be done with any dimensions. mike From cube-lovers-errors@mc.lcs.mit.edu Mon May 4 11:24:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA12818; Mon, 4 May 1998 11:24:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon May 4 10:36:17 1998 From: "Noel Dillabough" To: Subject: Revenge and the 5x5x5 Date: Mon, 4 May 1998 10:35:52 -0400 Message-Id: <000001bd7769$f0ced480$02c0c0c0@nat> Since we all know that Rubik's Revenge (4x4x4) puzzles are nearly impossible to find (all of mine have long ago broken) and the 5x5x5 cubes fall apart so easily that they are basically unusable. Well, as a solution to this, I took a Virtual Cube simulation and added sizing buttons (the cube program supports 2x2x2 to 5x5x5 sized cubes), a keyboard interface, and allowed it to receive sequences in standard UDFBLR notation. I also added locking of the center pieces to make using a paired up Revenge easier. The cube is located at http://www.mud.ca/cube/cube.html. Any thoughts, comments, suggestions about the program should be sent to: mailto://noel@mud.ca. Enjoy, -Noel Dillabough From cube-lovers-errors@mc.lcs.mit.edu Mon May 4 14:18:15 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA13472; Mon, 4 May 1998 14:18:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon May 4 12:28:19 1998 Message-Id: <19980504162440.4037.qmail@hotmail.com> From: "Philip Knudsen" To: Cube-Lovers@ai.mit.edu Subject: Re: Revenge and the 5x5x5 Date: Mon, 04 May 1998 09:24:39 PDT Noel writes: > Since we all know that Rubik's Revenge (4x4x4) puzzles are > nearly impossible to find (all of mine have long ago broken) > and the 5x5x5 cubes fall apart so easily that they are basically > unusable. You are right about the 4x4x4 availability. I have, however, never had any problems with my 5x5x5 cube. Actually the 5x5x5 mechanism is quite ingenious. I never heard of any broken one. The only problem i can think of is the orange sticker tendency fall off. Philip K From cube-lovers-errors@mc.lcs.mit.edu Mon May 4 15:28:57 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA13664; Mon, 4 May 1998 15:28:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon May 4 12:40:08 1998 Message-Id: <199805041642.MAA16954@nineCo.com> To: Cube-Lovers@ai.mit.edu Subject: 4x4x4 (Rubik's Revenge) puzzles for sale Reply-To: yanowitz@gamesville.com Date: Mon, 04 May 1998 12:42:51 -0400 From: Jason Yanowitz Hi, I have 6 Rubik's Revenge puzzles (in the original packaging) that I'm considering selling. If people are interested in purchasing one, send me an offer (yanowitz@gamesville.com). I apologize for the commercial nature of this post, but I've seen a few other commercial posts. thanks, -- Jason [ Moderator's note: Announcements of on-topic stuff for sale is generally okay, up until it starts clogging the list. I usually snip any detailed descriptions of the auction process, catalogues of other products, corporate history, etc.--you can get that from Jason (though he thoughtfully omitted excess in his message). ] From cube-lovers-errors@mc.lcs.mit.edu Mon May 4 15:58:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA13709; Mon, 4 May 1998 15:58:52 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon May 4 15:11:31 1998 Date: Mon, 4 May 1998 15:09:33 -0400 (EDT) From: Nichael Cramer To: Cube-Lovers@ai.mit.edu Subject: Re: Revenge and the 5x5x5 In-Reply-To: <19980504162440.4037.qmail@hotmail.com> Message-Id: Philip Knudsen wrote: > You are right about the 4x4x4 availability. I have, however, > never had any problems with my 5x5x5 cube. BTW, for interested (and near-by) folks, Games People Play in Harvard Sq had several 5Xs on the shelf when I dropped through the store last Thurs. Nichael From cube-lovers-errors@mc.lcs.mit.edu Wed May 6 09:18:29 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA18760; Wed, 6 May 1998 09:18:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue May 5 13:57:15 1998 Message-Id: <19980505174802.8836.qmail@hotmail.com> From: "Philip Knudsen" To: Cube-Lovers@ai.mit.edu Subject: Re: Revenge and the 5x5x5 Date: Tue, 05 May 1998 10:48:00 PDT I suggest people with 5x5x5 that tend to fall apart try and fasten the small screw underneath the center caps. This might help, at least it did on mine. Mine didn't fall apart though, it just got loose, and sometimes the pieces between the corners and the centres would sort of make a wrong twist. After i tightened the screws that problem disappeared. Philip K From cube-lovers-errors@mc.lcs.mit.edu Tue May 12 15:55:03 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA06770; Tue, 12 May 1998 15:55:02 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue May 12 14:24:43 1998 Message-Id: <35576405.5EC25A05@frontiernet.net> Date: Mon, 11 May 1998 16:48:05 -0400 From: John Bailey To: Cube-Lovers Subject: Solving a 4 Dimensional Rubik's type Cube Announcing a web page at http://www.frontiernet.net/~jmb184/solution.html which gives the explicit steps to solve a challenge configuration for a 2x2x2x2 (that's four dimensions) Rubik type cube. The challenge configuration is available at http://www.frontiernet.net/~jmb184/Nteract4.html. These pages do NOT require a Java enabled browser however, they do require Netscape 4.0 or Microsoft Explorer 4.0. This note is to solicit your judgments regarding the difficulty of the 4 Dimensional Rubik with no edge cubes (2x2x2x2.) I believe it is relatively easy, provided only that the simulation provides for the cyclic permutation move, (NE-->SW, SW-->NW, NW--->NE) Background: Posted on rec.puzzles dl April 21, 1998: A four dimensional articulated cube is on the web at http://www.frontiernet.net/~jmb184/4cube.html The result of marrying a Rubik's cube with a tesseract, this cube is 2x2x2x2. It has 16 corners and 24 faces. It does not have edge cubes and the corners have no orientation requirement. Only 4 colors are used. The solution space is thus roughly equivalent to that of a 3x3x3 Rubik if not smaller. It is rendered in Javascript and will run on Netscape 3.0 and 4.0 This posting caused about 25 hits to the page, but got no follow-up dialog on the rec.puzzles news dl. Note that in this first version, the corners are only identified by color, not by correct position. I wrote the page without having a clue as to how to solve it. In the process of just testing code I discovered that it is remarkably unchallenging, once you get a sense of which corners the various buttons rotate. (Flipping a glove from left-handed to right-handed can be done in 4-space, but is impossible in 3-space.) I may not be an unprejudiced solver, but I would rate the challenge only slightly harder than a 15 square slider puzzle. To increase the level of difficulty, a second version of the puzzle was developed. In this version, the solution requires that the corners are returned to their correct location. They still do not requires 4-space orientation. This version was announced in the following posting. Posted on the rec.puzzles dl May 2, 1998: A Four dimensional Rubik's Cube with solution. At http://www.frontiernet.net/~jmb184/Nteract4.html Re-designed to allow importing of 3D Rubik methods, this version uses (a slightly extended version of) standard Rubik cube naming of moves and positions, has a shortcut button for one of the common permutation moves and a scramble button to provide a challenge position. I rate the challenge as equivalent to solving two faces of a 3D Rubik cube. I am looking forward to your comments, opinions, and suggestions. I am especially interested in positions which cannot be solved or cannot be solved without extensive permutation moves other than the one included. This page has received about 50 hits. But again, there was no responding dialog on rec.puzzles news dl. The difficulty of the second version is higher, but I rated the challenge as equivalent to solving two layers of a 3x3x3 cube. The only obstacle, an ordinary solver might face, is finding the longish sequence required to permutate 3 of 4 corners. That's why I provided the shortcut button (which applies the actions: L'URU'R'LRUR'U' with one click.) Discussion: My concern is that people assume the puzzle is really hard and not worth the effort. It may be seen as somewhat like the sequences from one time pads which would be cryptographers who post and ask if anyone can decrypt them. To make it clear that a solution is not that difficult, I have now made a page which gives an explicit solution, with illustrations of each step and even some animation at http://www.frontiernet.net/~jmb184/solution.html There are obviously shorter sequences to obtain a solution, however this one has the value of providing clear checkpoints along the way, such that a solver can determine if they have missed a twist. I want and would welcome your judgment about how easy or hard the puzzle is. John Bailey jmb184@frontiernet.net http://www.frontiernet.net/~jmb184 May 11, 1998 From cube-lovers-errors@mc.lcs.mit.edu Tue May 12 17:33:46 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA06959; Tue, 12 May 1998 17:33:46 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue May 12 14:24:43 1998 Message-Id: <35576405.5EC25A05@frontiernet.net> Date: Mon, 11 May 1998 16:48:05 -0400 From: John Bailey To: Cube-Lovers Subject: Solving a 4 Dimensional Rubik's type Cube Announcing a web page at http://www.frontiernet.net/~jmb184/solution.html which gives the explicit steps to solve a challenge configuration for a 2x2x2x2 (that's four dimensions) Rubik type cube. The challenge configuration is available at http://www.frontiernet.net/~jmb184/Nteract4.html. These pages do NOT require a Java enabled browser however, they do require Netscape 4.0 or Microsoft Explorer 4.0. This note is to solicit your judgments regarding the difficulty of the 4 Dimensional Rubik with no edge cubes (2x2x2x2.) I believe it is relatively easy, provided only that the simulation provides for the cyclic permutation move, (NE-->SW, SW-->NW, NW--->NE) Background: Posted on rec.puzzles dl April 21, 1998: A four dimensional articulated cube is on the web at http://www.frontiernet.net/~jmb184/4cube.html The result of marrying a Rubik's cube with a tesseract, this cube is 2x2x2x2. It has 16 corners and 24 faces. It does not have edge cubes and the corners have no orientation requirement. Only 4 colors are used. The solution space is thus roughly equivalent to that of a 3x3x3 Rubik if not smaller. It is rendered in Javascript and will run on Netscape 3.0 and 4.0 This posting caused about 25 hits to the page, but got no follow-up dialog on the rec.puzzles news dl. Note that in this first version, the corners are only identified by color, not by correct position. I wrote the page without having a clue as to how to solve it. In the process of just testing code I discovered that it is remarkably unchallenging, once you get a sense of which corners the various buttons rotate. (Flipping a glove from left-handed to right-handed can be done in 4-space, but is impossible in 3-space.) I may not be an unprejudiced solver, but I would rate the challenge only slightly harder than a 15 square slider puzzle. To increase the level of difficulty, a second version of the puzzle was developed. In this version, the solution requires that the corners are returned to their correct location. They still do not requires 4-space orientation. This version was announced in the following posting. Posted on the rec.puzzles dl May 2, 1998: A Four dimensional Rubik's Cube with solution. At http://www.frontiernet.net/~jmb184/Nteract4.html Re-designed to allow importing of 3D Rubik methods, this version uses (a slightly extended version of) standard Rubik cube naming of moves and positions, has a shortcut button for one of the common permutation moves and a scramble button to provide a challenge position. I rate the challenge as equivalent to solving two faces of a 3D Rubik cube. I am looking forward to your comments, opinions, and suggestions. I am especially interested in positions which cannot be solved or cannot be solved without extensive permutation moves other than the one included. This page has received about 50 hits. But again, there was no responding dialog on rec.puzzles news dl. The difficulty of the second version is higher, but I rated the challenge as equivalent to solving two layers of a 3x3x3 cube. The only obstacle, an ordinary solver might face, is finding the longish sequence required to permutate 3 of 4 corners. That's why I provided the shortcut button (which applies the actions: L'URU'R'LRUR'U' with one click.) Discussion: My concern is that people assume the puzzle is really hard and not worth the effort. It may be seen as somewhat like the sequences from one time pads which would be cryptographers who post and ask if anyone can decrypt them. To make it clear that a solution is not that difficult, I have now made a page which gives an explicit solution, with illustrations of each step and even some animation at http://www.frontiernet.net/~jmb184/solution.html There are obviously shorter sequences to obtain a solution, however this one has the value of providing clear checkpoints along the way, such that a solver can determine if they have missed a twist. I want and would welcome your judgment about how easy or hard the puzzle is. John Bailey jmb184@frontiernet.net http://www.frontiernet.net/~jmb184 May 11, 1998 From cube-lovers-errors@mc.lcs.mit.edu Thu May 14 10:52:20 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA10874; Thu, 14 May 1998 10:52:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu May 14 09:18:42 1998 Date: Thu, 14 May 1998 14:03:21 +0100 From: David Singmaster To: Cube-Lovers@AI.MIT.Edu Cc: zingmast@ice.sbu.ac.uk Message-Id: <009C62C9.843B05E9.31@ice.sbu.ac.uk> Subject: New radio programme TO: FRIENDS AND COLLEAGUES I am participating in a new weekly program called 'Puzzle Panel' on BBC Radio 4, beginning on Thursday, 4 June at 1:30. We recorded a pilot in January and the commissioning producers were delighted with it. There will be a group of three to five panelists and we will discuss both mathematical and verbal puzzles. Some will be sent in by listeners and some will be set to the listeners by the panellists. At the pilot, the panel was myself, Chris Maslanka (of the Guardian) as chair, William Hartston (of the Independent, etc.) and Ann Bradford (compiler of a Crossword dictionary), but the membership may vary. I'll let you know of any changes of time/date, etc. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu May 21 13:24:10 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA13269; Thu, 21 May 1998 13:24:09 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu May 21 12:24:47 1998 Message-Id: <35646320.2295@ping.be> Date: Thu, 21 May 1998 18:23:45 +0100 From: Geoffroy Van Lerberghe To: Cube-Lovers Subject: Cristoph's Jewel internal mechanism The Christoph's Magic Jewel is a disguised Rubik's cube (cf. Metamagical Themas by Douglas R. Hofstadter p.339 : Stan Isaacs's coloring scheme) but what about the internal mechanism? Is it simply a Rubik's cube with only edge and centre cubes or is the mechanism different from the classic cube. I haven't managed to disassemble the Magic Jewel yet. Geoffroy.VanLerberghe@ping.be From cube-lovers-errors@mc.lcs.mit.edu Thu May 21 17:26:51 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA13891; Thu, 21 May 1998 17:26:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu May 21 14:06:58 1998 Message-Id: <19980521180258.25636.qmail@hotmail.com> From: "Philip Knudsen" To: Cube-Lovers@ai.mit.edu Subject: Re: Cristoph's Jewel internal mechanism Date: Thu, 21 May 1998 11:02:57 PDT The Jewel is basically an octahedron, but the vertex pieces are absent. This does not make the puzzle easier. Apart from the jewel i also have a taiwanese and a polish made octahedreon (with vertex pieces). A third version exists, made by Uwe Meffert, but quite rare. The turning quality of the jewel is very close to that of the polish made octahedron, so i believe that is where the jewel originates (Correct me if i'm wrong, Christoph!) The disassembled polish octahedron has a mechanism very close to that of the Pyraminx puzzle, also by Uwe Meffert. It is not a cube mechanism. >The Christoph's Magic Jewel is a disguised Rubik's cube (cf. Metamagical >Themas by Douglas R. Hofstadter p.339 : Stan Isaacs's coloring scheme) >but what about the internal mechanism? Is it simply a Rubik's cube with >only edge and centre cubes or is the mechanism different from the >classic cube. >I haven't managed to disassemble the Magic Jewel yet. > >Geoffroy.VanLerberghe@ping.be ____________________________________ Philip K recording and performing artist Vendersgade 15, 3th DK - 1363 Copenhagen K Phone: +45 33932787 Mobile: +45 21706731 E-mail: philipknudsen@hotmail.com E-mail: skouknudsen@get2net.dk E-mail: skouknudsen@email.dk E-mail: 4521706731@sms.tdm.dk (leave subject blank!) From cube-lovers-errors@mc.lcs.mit.edu Fri May 22 12:26:20 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA16202; Fri, 22 May 1998 12:26:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 22 06:24:16 1998 Message-Id: <19980522101332.6763.qmail@hotmail.com> From: "Philip Knudsen" To: cube-lovers@ai.mit.edu Subject: spare piece for domino variant Date: Fri, 22 May 1998 03:13:31 PDT I just received puzzle from a fellow collector: It is like a Magic Domino, but only about 47 mm along the long edges. The pieces are red and white. The 9 red pieces have a drawing of Superman and the 9 white pieces a drawing of Superwoman! Unfortunately the puzzle was broken on arrival. Does anyone on the list have a similar broken puzzle, and maybe could spare a piece (edge)? ____________________________________ Philip K recording and performing artist Vendersgade 15, 3th DK - 1363 Copenhagen K Denmark Phone: +45 33932787 Mobile: +45 21706731 E-mail: skouknudsen@get2net.dk E-mail: philipknudsen@hotmail.com E-mail: skouknudsen@email.dk (soon to expire) E-mail: 4521706731@sms.tdm.dk (leave subject blank!) From cube-lovers-errors@mc.lcs.mit.edu Fri May 22 19:06:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA17343; Fri, 22 May 1998 19:06:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 22 19:02:24 1998 Date: Fri, 22 May 1998 18:59:24 -0400 (EDT) From: Nicholas Bodley To: Cube Mailing List Subject: Magic Jack Message-Id: Sorry if my memory's faulty, but I don't recall any recent mention of the Magic Jack. This is a 3-cubed, 3-D array of 26 small cubes constrained by an outer cage to slide past their neighbors. At first glance, it looks like a Rubik's Cube, but immediately one realizes it's quite different. It's about the same size. Disassembly looks impossible unless the outer "cage" is cut. As you'd expect, it's a 3^3 array, but with one position empty. It's a 3-D analog of the 15 Puzzle. The individual cubes are not connected in any sense to their neighbors. While the moves in a 15 Puzzle are in one plane and easily defined by amateur mathematicians, in the Magic Jack, there are many more possible ways of moving a given cube to another position. Also, not surprisingly, cube moves are strictly translational. The fun begins when one attempts to create patterns. Each cube has specific surface markings. The simplest configuration creates an exterior in which all cubes have a random, fine-grained, glittery diffraction-grating-like surface. More complicated, and difficult, are the colored patterns, which when solved, create (iirc) a continuous path around the whole puzzle. There are three, I'm fairly sure; one creates a message. Solving is made more difficult by the fact that most cube faces are obscured by their neighbors. As to its intrinsic mathematical difficulty, I'm not close to being well informed/educated enough to judge. The practical problem of hidden faces does add to the practical difficulty, and the number of "degrees" of freedom for a given cube (from 3 to 6, depending on position) certainly increases the available choices. I saw this puzzle at Games People Play in Cambridge; it's a German import. Quality of construction was good, although there was no detenting, and it could be easier to move the cubes. It might actually be easier to constrain potential interferers, and let gravity do the work. The difficulty was essentially caused by other cubes' getting out of position, not poor quality. Price in the store is $25. Not sure whether they're interested in mail orders, but it might be worth a try. While I have no connections with G.P.P., perhaps it wouldn't be out of order to give some info.: The Games People Play 1100 Massachusetts Ave. (Abbreviation = Mass. is OK!) Cambridge, Mass. 02138 (617) 492-0711 Afaik, they had possibly as many as a dozen in stock. G.P.P. also periodically imports 5^3s from Germany, perhaps not from Dr. Bandelow. They have a nice collection of movable-piece puzzles. |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Are you designing an icon for a GUI? |* Amateur musician *|* China has been doing it for millennia. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon May 25 15:42:05 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA23077; Mon, 25 May 1998 15:42:04 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 22 19:38:36 1998 Date: Fri, 22 May 1998 19:35:00 -0400 Message-Id: <22May1998.192434.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Alan@lcs.mit.edu To: nbodley@tiac.net Cc: Cube-lovers@ai.mit.edu In-Reply-To: (message from Nicholas Bodley on Fri, 22 May 1998 18:59:24 -0400 (EDT)) Subject: Re: Magic Jack Date: Fri, 22 May 1998 18:59:24 -0400 (EDT) From: Nicholas Bodley ... Not sure whether they're interested in mail orders, but it might be worth a try. While I have no connections with G.P.P., perhaps it wouldn't be out of order to give some info.: The Games People Play 1100 Massachusetts Ave. (Abbreviation = Mass. is OK!) Cambridge, Mass. 02138 (617) 492-0711 Afaik, they had possibly as many as a dozen in stock. Check your local puzzle outlet first -- Magic Jack may be pretty widely available. When I was in the hospital last summer, my father brought one of these with him when he came to vist me from Philadelphia. I don't recall the name of the store there where he purchased it. I still haven't solved it. The first step would clearly be to just catalog the 26 different cubies, but I haven't even done that... - Alan From cube-lovers-errors@mc.lcs.mit.edu Mon May 25 16:14:10 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA23149; Mon, 25 May 1998 16:14:09 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri May 22 20:01:43 1998 Date: Fri, 22 May 1998 19:58:40 -0400 (EDT) From: Nicholas Bodley To: Alan Bawden Cc: Cube-lovers@ai.mit.edu Subject: Magic Jack website (!) In-Reply-To: <22May1998.192434.Alan@LCS.MIT.EDU> Message-Id: Sorry, all; the 'Net still has its surprises. Guess what: The Magic Jack has its own Web site: www.magicjack.com They list the retailers who carry it; there are very roughly a dozen or so. The site looks worth a visit. Gosh, Alan, I guess we all should welcome you back, if my recollection's clear! May you continue to be well! My best to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* Are you designing an icon for a GUI? |* Amateur musician *|* China has been doing it for millennia. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon May 25 16:48:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA23216; Mon, 25 May 1998 16:48:52 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat May 23 03:33:16 1998 From: canttype@earthlink.net Message-Id: In-Reply-To: Date: Sat, 23 May 1998 00:34:41 -0700 To: Subject: Re: Magic Jack vs. Vadasz Cube Nicholas Bodley wrote > Sorry if my memory's faulty, but I don't recall any recent mention of >the Magic Jack.... check out http://members.aol.com/islandcom/ for information about the Vadasz Cube which is a variation of the Magic Jack described above. I have a "3x3x3 Classic Cube Solid" and have been able to solve it. The Vadasz Cube allows you to easily disassemble it, if desired. Also, each of the 26 cubies can be disassembled and reconfigured allowing you to create variations of the puzzle. The cubies are made out of plastic tiles so that you can re-arrange the construction and colors of each of the 26 cubies if you desire. Five different puzzles are available: 2x2x2 3x3x3 4x4x4 and multi 3x3x3 multi 4x4x4 From cube-lovers-errors@mc.lcs.mit.edu Wed May 27 07:01:51 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id HAA27559; Wed, 27 May 1998 07:01:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed May 27 03:39:50 1998 Message-Id: <19980527073534.13389.qmail@hotmail.com> From: "Philip Knudsen" To: cube-lovers@ai.mit.edu Subject: Re: Magic jack Date: Wed, 27 May 1998 00:35:33 PDT Two comments on the Magic Jack: Apart from Magic Jack and Vadasz Cube, there also exists a german produced puzzle called "IQUBE". Like the Magic Jack, this is a 3x3x3 sliding puzzle with 26 smaller cubes. Cubes have colours red, green and yellow, and it is possible to arrange them so the entire surface is either red or green. Yellow is possible with red or green centres. IQUBE comes with a leaflet that suggests a total 12 different solution possibilities. The puzzle is suitable for the blind, since the three different colours also feel differently. I bought mine from Spielkiste (http://www.twfg.de/puzzle/default.htm). A 2x2x2 sliding puzzle is mentioned in "Rubik's Cubic Compendium", in the part that is written by David Singmaster (quote): "The only such (three dimensional moving-piece puzzle) puzzle that i know of is a sliding cube puzzle of Piet Hein which is so rare that both Rubik and I recently re-invented it before learning that it had been done by Hein." There is also an illustration which shows a 2x2x2 sliding cube puzzle similar to the small Vadasz Cube. Philip K From cube-lovers-errors@mc.lcs.mit.edu Wed May 27 12:52:33 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA28328; Wed, 27 May 1998 12:52:33 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed May 27 11:37:46 1998 Message-Id: <01BD8995.A6F13540.Johan.Myrberger@ebc.ericsson.se> From: Johan Myrberger Reply-To: To: Subject: RE: Magic jack Date: Wed, 27 May 1998 17:34:05 +0200 Organization: Ericsson Business Networks AB On 27 May 1998 09:36, Philip Knudsen wrote: > Apart from Magic Jack and Vadasz Cube, there also exists a german > produced puzzle called "IQUBE". Like the Magic Jack, this is a 3x3x3 > sliding puzzle with 26 smaller cubes. Cubes have colours red, green and > yellow, and it is possible to arrange them so the entire surface is > either red or green. Yellow is possible with red or green centres. IQUBE > comes with a leaflet that suggests a total 12 different solution > possibilities.... Some years ago (around 1989?) I made a computer search on this kind of puzzle. The idea was "is there a way of colouring the 27 cubies (and then remove one) so that a 3x3x3 cube can be arranged (with sliding block moves) to show all external sides of either of three colours". Since a 3x3x3 cube shows 9x6=54 cubie sides at one time, and 27 cubies have in all 27x6=54x3 cubie sides all "cubie sides" would be used in one configuration each. So - I hunted for the answers to: 1) Is such a colouring possible? 2) Which cubie would be nicest to remove? My search showed that 1) was indeed possible, and that there is one distinct way for the colouring (not counting reflections etc) and 2) It is possible to choose a cubie to remove so that the space will be positioned in one of the space diagonals for each of the three solutions. If anyone is interested I can dig out the specific colouring. Regards /Johan Myrberger mailto:Johan.Myrberger@ebc.ericsson.se http://home.bip.net/johan.myrberger From cube-lovers-errors@mc.lcs.mit.edu Thu May 28 12:18:52 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA02019; Thu, 28 May 1998 12:18:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed May 27 16:46:38 1998 From: David A Bagley Message-Id: <199805272042.QAA15526@gwyn.tux.org> Subject: Updated 3x3x3x3 (Rubik's Tesseract) To: cube-lovers@ai.mit.edu Date: Wed, 27 May 1998 16:42:53 -0400 (EDT) Cc: charlied@erols.com Hi All A new version of Charlie Dickman's Rubik Tesseract program and its accompanying documentation is now available from http://www.tux.org/~bagleyd/ (under the heading of "Neat 4D stuff I wish I wrote" :) ). This latest version contains a general solution to unscramble an arbitrarily scrambled Rubik Tesseract as well as some improved bells and whistles. The solution is given in the docs and is also implemented in the Macintosh program. All mail about the Tesseract docs and program should be addressed to Charlie Dickman . -- Cheers, /X\ David A. Bagley (( X bagleyd@bigfoot.com http://www.tux.org/~bagleyd/ \X/ xlockmore and more ftp://ftp.tux.org/pub/tux/bagleyd From cube-lovers-errors@mc.lcs.mit.edu Thu May 28 13:26:48 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA02149; Thu, 28 May 1998 13:26:47 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu May 28 07:41:10 1998 Message-Id: <01BD8A0B.2B00FBC0@jburkhardt.ne.mediaone.net> From: John Burkhardt To: Cube Mailing List Subject: RE: Magic Jack Date: Thu, 28 May 1998 07:35:19 -0400 I bought one of these from Games People Play and it is unsolvable on one of the colors. At least, getting the piping to wander all the way around on the red colors was not possible. It looks like one of the stickers is oriented incorrectly. I told the folks at FunTech and they told me to send it back to them and they would look. Also, when I solved the message version I couldn't get the rest of it to line up. Some of the pipes on the other edges didn't line up and it wasn't possible to make the rest silver. This was kind of disappointing. From cube-lovers-errors@mc.lcs.mit.edu Fri Jun 5 19:41:11 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA23314; Fri, 5 Jun 1998 19:41:10 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jun 1 19:17:58 1998 Message-Id: <357329D5.FD5A7E89@t-online.de> Date: Tue, 02 Jun 1998 00:23:17 +0200 Organization: BERA Softwaretechnik GmbH To: Cube Mailing List Subject: New member From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Dear cube lovers, a few days ago I have subscribed to the list. I have downloaded all the archives and also found some software (great stuff). Some of you may know my name from Mr. Bandelow's book - I'm the guy who won a prize with the shortest maneuver fro the super-flip-twist. I always thought the cube is dead. Now I am really surprised to see this completely wrong :) The level of discussion is really amazing. I have are some questions concerning terminology: What does M-conjugacy mean (this doesn't seem to be a standard term from group theory). [ Moderator's note: See "Symmetry and Local Maxima", 14 December 1980. This also introduces the group M and some of its subgroups, which are helpful in a lot of the cube-lovers discussion. Jerry Bryan also tried an explanation of M-conjugacy on 3 October 1996. ] Some mails in the archives mention numbers like p102 for patterns ?!?!? [ M: I believe Mark Longwood uses numbers of that form to catalogue patterns. ] Does anybody know the current upper bound for God's algorithm (in q/f metric). [ M: 29 face turns, 42 quarter turns. The best known lower bounds are 20 face turns, or 24 quarter turns (both achieved by superflip). This was true on 13 February 1996, and I don't think there has been an advance since then. ] Is there any serious research on the 4*4*4 or 5*5*5 cube. Computer search is probably beyond available/affordable hardware :( Are there any maneuver search programs that can handle slice metric ? I think that slice metric makes sense since q and f metric have no natural extension for "higher" cubes. [ M: See 1 June 1983 for "Eccentric Slabism", a genereralization of the q metric that could be adapted to a f metric. ] Does anybody have some nice patterns on the 4^3 or 5^3 cube ? [ M: I reported some 4^3 patterns on 15 June 1982. Have there been others? Any 5^3 patterns? ] I think I have some in my old (and thick) cube folder (paper, not on my PC :)) [ M: Could you type some in? ] Some of these questions probably have been discussed already. Sorry, I haven't read ALL old mails. [ M: Note that I left quite a few of these questions unanswered--other replies are welcome, either pointers to archive messages I forgot or new answers. But this highlights a major failing of the archives: We don't have a FAQ, or even an index to the major articles. Is anyone interested in working on something like this? I have very little time for it just now. ] adS -- -------------------------------------------------------------------------- Rainer aus dem Spring email Rainer.adS.BERA_GmbH@t-online.de (home) Schimmelbuschstr. 10 email TEEADS@TEE.toshiba.de (business only) 40699 Erkrath tel. +49 (0)02104 35157 (private) Germany tel. +49 (0)02104 936150 (business) --------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Fri Jun 5 21:42:40 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA23518; Fri, 5 Jun 1998 21:42:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jun 5 20:53:28 1998 Date: Fri, 5 Jun 1998 20:36:06 -0400 Message-Id: <0020A7D5.001706@scudder.com> From: jdavenport@scudder.com (Jacob Davenport) Subject: Re: New member To: Cube Mailing List >Does anybody have some nice patterns on the 4^3 or 5^3 cube ? I made a chess board out of four 5^3 cubes, which you can check out along with our other cube sculptures on www.wunderland.com/WTS/Jake/CubeArt. If anyone has any good pattern ideas for four 5^3 cubes, I'd like to hear them, particularly before I peel off the stickers on one of them and give it a color pattern similar to Colorspace created by Andy Plotkin (which you can see at www.wunderland.com/WTS/Kristin/CustomCubes.html). -Jacob From cube-lovers-errors@mc.lcs.mit.edu Fri Jun 5 22:31:09 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA23625; Fri, 5 Jun 1998 22:31:09 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jun 5 21:44:38 1998 Message-Id: In-Reply-To: <357329D5.FD5A7E89@t-online.de> Date: Fri, 5 Jun 1998 21:41:23 -0400 To: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring), Cube Mailing List From: Kristin Subject: Re: New member Rainer aus dem Spring wrote: >Some of you may know my name from Mr. Bandelow's book - I'm the guy >who won a prize with the shortest maneuver fro the super-flip-twist. > >I always thought the cube is dead. Now I am really surprised to see >this completely wrong :) The level of discussion is really amazing. Your words perfectly describe a day a couple of years ago when I first found this list. Peace - -Kristin kristin@wunderland.com wunderland.com/home/rubik.html From cube-lovers-errors@mc.lcs.mit.edu Mon Jun 8 15:52:11 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA00875; Mon, 8 Jun 1998 15:52:10 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Jun 6 09:39:26 1998 Message-Id: <3.0.5.16.19980606155910.0f372bfe@ryle.get2net.dk> Date: Sat, 06 Jun 1998 15:59:10 To: cube-lovers@ai.mit.edu From: Philip Knudsen Subject: 4*4*4 patterns [Re: New member] > Does anybody have some nice patterns on the 4^3 or 5^3 cube ? > [ M: I reported some 4^3 patterns on 15 June 1982. Have there been > others? Any 5^3 patterns? ] Maybe you find the following pattern for the 4*4*4 interesting (I'm not sure I am using the proper notation, but by Capital letters I mean side moves, small letters are slice moves): R r U2 u2 R2 r2 U3 u3 F3 L D2 L3 D3 F2 U2 F2 D L D2 L3 F L3 F U2 F3 U3 L2 D2 L2 U F U2 F3 L D F2 B2 D2 F2 B2 D The last three lines alone make a similar pattern on a 3*3*3 cube. I have a shorter sequense for it somewhere that I can't remember by head. One can also make a similar pattern on the 5*5*5, I'll try and dig it out... Philip K From cube-lovers-errors@mc.lcs.mit.edu Fri Jun 19 11:15:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA27099; Fri, 19 Jun 1998 11:15:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jun 19 03:22:16 1998 Message-Id: <3.0.5.16.19980619094150.0c8f65de@ryle.get2net.dk> Date: Fri, 19 Jun 1998 09:41:50 To: cube-lovers@ai.mit.edu From: Philip Knudsen Subject: New Puzzle: "Dogic" I just received a new Puzzle called "Dogic - Test Your Logic". It's in the shape of an Icosahedron, and moves in the following manner: 5 triangles can rotate around their common vertex somewhat like the Impossiball. Each triangle is again subdivided into 4 smaller triangles which move separately, i.e. one can also rotate 5 smaller triangles around the same vertex. Thus there are 60 "vertex" triangles and 20 "middle" triangles, the latter are in fact equivalent to the Impossiball. The "vertex" triangles are unicolored, the "middle" triangles have three colours. The whole Puzzle has twelve colours, one for each vertex. I count the number of distinguishable positions: 20! 3^19 60! ------------ = 2,199110779324 x 10^82 2 5!^12 60 I'm not sure these calculations are correct, but if they are, this Puzzle is at the very top of Mark Longridge's "Great Cosmic Ranking List", even above the good old 5x5x5 Cube! The Puzzle is very well "Made In Hungary". A true must for anyone who likes cube-type Puzzles. Available from Spielkiste/Germany, check out: www.twfg.de/puzzle/default.htm Philip Knudsen Recording and Performing Artist Vendersgade 15, 3th DK - 1363 Copenhagen K Denmark Phone: +45 33932787 Mobile: +45 21706731 E-mail: skouknudsen@get2net.dk E-mail: philipknudsen@hotmail.com From cube-lovers-errors@mc.lcs.mit.edu Tue Jun 23 10:18:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA03441; Tue, 23 Jun 1998 10:18:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jun 22 09:08:14 1998 Message-Id: <199806221304.AA08381@world.std.com> To: "Cube-Lovers@ai.mit.edu" Subject: Puzzles newly available in US Date: Mon, 22 Jun 98 09:02:41 -0500 From: "Michael C. Masonjones" I apologize that I don't have more information, since I am away from the stuff that I bought and I can't look at the packaging. I was in Toy Works in W. Springfield, MA, and was very surprised to find a series of puzzles I believe are newly available in the US after a long hiatus. The puzzles are Pyraminx, Skewb, and Meffert's Ball (with the four colored rings arranged on a spherical Skewb device. I think only the Skewb was called by its real name. All were in the same basic packaging, looked pretty authentic, and I think they all had Meffert patent/copyright info on them (at least the ball did). And they were all marked down from $10 to $6. Not bad at all. There must be more out there, but my nearest KayBee didn't have anything. I'm not sure if Toy Works is a big chain or not. It seems to be from the stuff they carry and the size of the store. Happy hunting. Mike Masonjones From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 10 12:57:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA06279; Fri, 10 Jul 1998 12:57:33 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jul 10 08:48:35 1998 Date: Fri, 10 Jul 1998 08:48:05 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Ten Face Moves from Start To: cube-lovers@ai.mit.edu Message-Id: Face Moves Patterns Positions Branching Positions/ from Factor Patterns Start 0 1 1 1.0 1 2 18 18 9.0 2 9 243 13.5 27.0 3 75 3240 13.333 43.2 4 934 43239 13.345 46.294 5 12077 574908 13.296 47.604 6 159131 7618438 13.252 47.875 7 2101575 100803036 13.231 47.965 8 27762103 1332343288 13.217 47.991 9 366611212 17596479795 13.207 47.998 10 4838564147 232248063316 13.199 47.999 This run took about three weeks on a Pentium 300. The next level from Start is going to be difficult. With the current algorithm and hardware, it would take about thirty to forty weeks. In addition, the memory requirements will go up considerably. Currently, I store only the positions up to five moves from Start in memory. To calculate the next level, I will have to store the positions up to six moves from Start. I still suggest (see "How Big is Big?" in the archives) that the problem can be calculated all the way to the bitter end, eventually. The Cube problem simply is not as big as, for example, Chess or Go. As a possible strategy, if we could add one level per decade, we could probably calculate the problem all the way to the end within about 100 years. Moore's Law (the power of computers doubles about every eighteen months) suggests that such a schedule might be possible. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 10 16:23:48 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id QAA06984; Fri, 10 Jul 1998 16:23:48 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jul 10 13:31:51 1998 Date: Fri, 10 Jul 1998 13:31:22 -0400 Message-Id: <199807101731.NAA02872@corwin.ece.cmu.edu> From: "Jonathan R. Ferro" Organization: Electrical and Computer Engineering, CMU To: cube-lovers@ai.mit.edu In-Reply-To: (message from Jerry Bryan on Fri, 10 Jul 1998 08:48:05 -0400 (Eastern Daylight Time)) Subject: Re: Ten Face Moves from Start "Jerry" == Jerry Bryan writes: Jerry> As a possible strategy, if we could add one level per decade, we Jerry> could probably calculate the problem all the way to the end Jerry> within about 100 years. Moore's Law (the power of computers Jerry> doubles about every eighteen months) suggests that such a Jerry> schedule might be possible. This method is called Zarf's Linearization: For any exponential-time problem, just wait the linear amount of time for the current generation of computation to make it possible to solve your instance in one hour, then solve your instance in one hour. From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 14 17:15:12 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA22065; Tue, 14 Jul 1998 17:15:12 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 14 13:46:15 1998 Message-Id: <19980714163444.6844.rocketmail@send1a.yahoomail.com> Date: Tue, 14 Jul 1998 09:34:44 -0700 (PDT) From: Eddy Liao Subject: Cubes for sale To: Cube-Lovers@ai.mit.edu Dear Madam/Sir, I have the following items for sale: Rubic's cube(6-color) - $5.50 Rubic's cube(poker) - $5.50 Magic snake - $5.50 Rubic's cube keychain (1.5 inch) - $4.50 Rubic's cube keychain (3/4 inch) - $3.50 Pyramid key chain (1.5 inch) - $4.50 Magic snake keychain - $4.50 List your orders plus $1 shipping charge of entire order (plus $5 if you prefer COD(Cash on Delivery)) Please send check or money order to: Eddy Liao 694 Yorkhaven Rd. Cincinnati, OH 45246 If you have any questions, please Email me at: liao_1@yahoo.com From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 16 12:05:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA29553; Thu, 16 Jul 1998 12:05:33 -0400 (EDT) Message-Id: <199807161605.MAA29553@mc.lcs.mit.edu> Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 14 13:46:15 1998 Date: Tue, 14 Jul 1998 09:34:44 -0700 (PDT) From: Dan Hoey Subject: Spammer says: Cubes for sale To: Cube-Lovers@AI.MIT.Edu I greatly regret allowing the advertisement from Eddy Liao entitled "cubes for sale". Having just received his ad spammed in my personal mailbox, I must conclude he is an abuser of the network. So if you buy anything from him, you're supporting network abuse, and for all I know he may steal your money as blithely as he steals the network's resources. Dan Hoey, Moderator Cube-Lovers-Request@AI.MIT.Edu From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 24 13:30:22 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA02692; Fri, 24 Jul 1998 13:30:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 23 18:21:31 1998 Date: Thu, 23 Jul 1998 18:21:02 -0400 From: michael reid Message-Id: <199807232221.SAA07643@hilbert.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: patterns on 5x5x5 cube a while ago, rainer asked for patterns on the 5x5x5 cube. here are some i know (the hardest part seems to be finding the scraps of paper on which the maneuvers are written). standard notation uses R and r for 90 degree clockwise twists of the outer layer and second layer, respectively. i've found it convenient to have notation for Rr , so i use _R_ (that is, capital R underlined). i guess this notation is more convenient for handwritten maneuvers, but not so convenient for e-mail. i'll use _R_ to denote R underlined and _( F L U B ... )_ to mean that the whole thing inside the parentheses is underlined. the first pattern is a "double" snake; it meanders onto each face twice. _R'_ b' _U_ F2 _U'_ b _U_ F2 _(U' R2 F')_ u2 _(F U L')_ u2 _(L U' R' L)_ f _D'_ B2 _D_ f' _D'_ B2 _(D L2 B)_ d2 _(B' D' R)_ d2 _(R' D L)_ if i understand the terminology correctly, this is a continuous non-chiral isoglyph with the pattern ...*. **.** .*... .*... .*... i still remember that when i found this pattern some 10-12 years ago, i saw the URF faces together. then i turned the cube around, and was surprised by how it continued on the other three faces. (i shouldn't have been surprised, but you know how that goes ... ) i came across this pattern accidently. then i went snake hunting, and found several others: snake 2 _(R F2 R2 U2)_ r2 _(U2 R2 F2 R' D' F2 B2 D R F2 R2 U2)_ r2 _(U2 R2 F2 R' D')_ r2 _(F2 B2)_ L2 _(R2 U' D F2 B2 U)_ those two have the property that the two snake segments on each face have the same color. if this condition is relaxed, we also have snake 3 _(R L' F U2 R F2 R2 U2)_ r2 _(D2 L2 F B' D' F B' U' D F R L D' B2 L' F B' D')_ f2 _(U2 D2)_ f2 _(U' D2)_ this one can be modified slightly; change the U and D faces .*.*. .*.*. .*.*. .***. .*.*. ..... from .*.*. to .***. .*.*. .*.*. if only one is changed, then we get two separate snakes. there's also snake 4 _(D F2 B2)_ l2 _(F2 B2 R')_ R2 _(F2 R2 U2)_ r2 _(U2 R2 F2 R' D' F2 B2 D R F2 R2 U2)_ r2 _(U' D' F' U2 D2 B U' D L2 B2 L' U2 D F2 B2)_ another interesting pattern is U R' U' F' _U'_ R' _U_ f _U'_ R _(U F')_ F2 U R U' _B_ l' _D2_ l _D_ f' _D2_ f _(D' B')_ D' L D B _D_ L _D'_ b' _D_ L' _(D' B)_ B2 D' L' D _F'_ r _U2_ r' _U'_ b _U2_ b' _(U F)_ which gives a continuous non-chiral isoglyph with the pattern .*... .*... .*... ***** ...*. the same maneuver produces an analogous pattern on the 4x4x4 cube, but there's probably an easier maneuver. another isoglyph (also continuous and non-chiral) with the same pattern is R f' U2 f U l' U2 l U' R' _D'_ L b2 L' _B'_ U b2 U' _(B D)_ L' b D2 b' D' r D2 r' D L _U_ R' f2 R _F_ D' f2 D _(F' U')_ modifying this pattern is how i found the first double snake. mike From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 26 14:10:13 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA09548; Sun, 26 Jul 1998 14:10:13 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Jul 25 17:52:37 1998 Message-Id: <35BA52EF.35D831D6@t-online.de> Date: Sat, 25 Jul 1998 23:49:35 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: michael reid Cc: cube-lovers@ai.mit.edu Subject: Re: patterns on 5x5x5 cube References: <199807232221.SAA07643@hilbert.math.brown.edu> From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Thanks for the patterns, the problem is - my 5*5*5 cube is scrambled and I have to figure out how to unscramble it. I haven't touched it for 15 years. I also have several hundred pages of hand-written stuff. I have several 4*4*4 patterns. I wonder if enough people on the mailing list have a (La)TeX system so that I can post the patterns in LaTeX format. Concerning this I need feedback !! By the way, my last 4*4*4 cube starts to fall apart. Does anybody know if it is still available ? michael reid wrote: > > a while ago, rainer asked for patterns on the 5x5x5 cube. here > are some i know (the hardest part seems to be finding the scraps > of paper on which the maneuvers are written). > snip .... Rainer adS PS Does anybody else have patterns for the 4 or 5 cube ? If so, pls. send them to me. I will create a document in ps and/or pdf format about patterns. [ Moderator's note: My sense of this is that short notes in latex can be made readable enough as text that it might be workable on this list. Postscript and PDF are not acceptable in this medium, though they can be placed in the archive at ftp://ftp.ai.mit.edu/pub/cube-lovers/contrib/ --Dan] From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 27 17:33:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA15152; Mon, 27 Jul 1998 17:33:48 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Jul 26 15:51:06 1998 Message-Id: <35BB827C.10B7C9A7@t-online.de> Date: Sun, 26 Jul 1998 21:24:44 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: Cube Mailing List Subject: 4*4*4 patterns From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Dear cube lovers, as promised the other day here comes my collection of 4*4*4 patterns. My favorites are the single twisted rings. I still find it surprising to see that there is no second ring on the "other" side. The maneuvers use all sorts of slice moves which are probably not accepted as moves by most cubeologists. I am too lazy to rewrite them. Does anybody have any idea which format I should post for people without a TeX system? Plain ASCII is not really what one needs to display cube maneuvers. I can offer Mathematica notebook, postscript, WINWORD (arrrrgh)and (perhaps?) pdfd. Does anybody know of a 4*4*4 emulator or even solver? Anything like a (sub)optimal solver is probably beyond the current PC powers. Rainer PS I am NOT a LaTeX expert :) hints are welcome ! [Moderator's note: The notation is fairly straightforward, but may be new to cube-lovers. Subscripts (encoded with underscores) show which 1x4x4 slabs relative to the given face are turned; omitted subscripts apparently mean the outer face, as "_1" would mean if it were used. Exponents have their usual meanings as repetition. The only advantage to running this through TeX seems to be that you get true superscripts and subscripts and somewhat nicer fonts. I've added a few commands that seem required by LaTeX. I've also replaced a number of narrow spaces (coded as backslash-comma) with ordinary spaces, so that this is more readable as text and so that some of the worst long-line problems are reduced. Perhaps some of these processes would be more readable if printed on multiple lines, or punctuated somehow. I wonder if there could be some simplification with the [X,Y] = X Y X^{-1} Y^{-1} commutator notation or the X^Y = Y^{-1} X Y conjugate notation, or if this would make the processes too hard to follow. --Dan] \documentstyle{article} \begin{document} \section{Patterns} \subsection{Dot Patterns} \subsubsection*{2 Dots (u,d)} $(R^2_2 F_{23}^2)^2$ Since this pattern exists it is obviously possible to create all dot patterns, i.e.\ all $6! = 720$ elements of the dot permutation group. \subsubsection*{2 Dots (f,r)} $D_2 R_{23}^2 D_2^{-1} L^2 B^2 U_2^{-1} R_{23}^2 U_2 R_{23}^2 B^2 L_{123}^2$ \subsubsection*{3 Dots (f,b,r)} $L_{123}^2 B^2 U_2^2 R_{23}^2 U_2 R_{23}^2 U_2 B^2 L^2 D_2 R_{23}^2 D_2^{-1}$ \subsubsection*{3 Dots (f,u,r)} $F_2^{-1} U^2 F_2 D_{23} F_2^{-1} U^2 F_2 D_{23}^{-1} B_2 U^2 B_2^{-1} D_{23} B_2 U^2 B_2^{-1} D_{23}^{-1}$ \subsubsection*{4 Dots (f,r)(l,b)} $D_2 R_{23}^2 D_2^{-1} U_2^{-1} R_{23}^2 U_2$ \subsubsection*{4 Dots (f,b)(r,l)} $R_{23}^2 D_{23} R_{23}^2 D_{23}^{-1}$ \subsubsection*{4 Dots (f,u)(r,l)} $R_2^{-1} U^2 R_2^2 B^2 R_2 R_{23}^2 D_{23} R_{23}^2 D_{23}^{-1} R_2^{-1} B^2 R_2^2 U^2 R_2$ \subsubsection*{6 Dots (f,b)(r,l)(u,d)} $D_{23} F_{23}^2 D_{23}^{-1} R_2^2 F_{23}^2 R_2^2$ \subsection{Brick Patterns} \subsubsection*{Exchanged 1*1*1 Cubes} $B^{-1} U^{-1} B L^2 F^{-1} D R_2^2 B_2^2 R_{12}^2 B_2^2 R^2 B_2^2 F^{-1} D^{-1} F^2 L^2$ \subsubsection*{Exchanged 1*1*2 Bricks} $R^2 U^2 R_{123}^{-1} D_{12}^{-1} R_{123} U^2 R_{123}^{-1} D_{12} R_{123} U^2 F_{12} U^2 F_{12} U^2 F_{12}^{-1} U^2 R_2^2 F_{12}^2 R_2^2 F_{12}^2 R_{12}^2$ \subsubsection*{Exchanged 1*1*3 Bricks} $B D^{-1} U^2 B^{-1} R^{-1} B U^2 F^{-1} L F^{-1} L^{-1} F^2 D B^{-1}$ Of course, this is a 3*3*3 maneuver. \subsubsection*{Exchanged 1*2*2 Bricks} $R_{12} B R_{12}^{-1} F_{12}^2 R_{12} B^{-1} R_{12} D R_{12}^2 F_{12}^2$ \subsubsection*{Exchanged 1*2*3 Bricks} $D^{-1} B_{12}^{-1} L^2 U^2 F_{12}^{-1} R_{123}^{-1} D_{12}^{-1} R_{123} U^2 R_{123}^{-1} D_{12} R_{123} U^2 F_{12} U^2 F_{12} L^2 B_{12} D$ \subsubsection*{Exchanged 1*3*3 Bricks} $F_2^2 R_{23}^2 F_2^2 B^{-1} U^{-1} B L^2 F^{-1} D L_2^2 B_2^2 L_{12}^2 B_2^2 L^2 B_2^2 F^{-1} D^{-1} F^2 L_{123}^2$ \subsubsection*{Exchanged 2*2*2 Cubes} $B_{12}^{-1} U_{12}^{-1} B_{12} L_{12}^2 F_{12}^{-1} D_{12} F_{12}^{-1} D_{12}^{-1} F_{12}^2 L_{12}^2$ Of course, this is a 2*2 maneuver. \subsubsection*{Exchanged 2*2*3 Bricks} $U_2 L_{12}^2 U_2^{-1} D_2^{-1} L_{12}^2 D_2 R_{12} B R_{12}^{-1} F_{12}^2 R_{12} B^{-1} R_{12} D R_{12}^2 F_{12}^2$ \subsubsection*{Exchanged 2*3*3 Bricks} ??? \subsubsection*{Exchanged 3*3*3 Cubes} $F^2 L^2 D F^{-1} B_{12}^2 R^2 B_2^2 R_{12}^2 B_2^2 R_2^2 U^{-1} R B^2 R^{-1} D_{23} R F_{23}^2 R^{-1} D R $ \subsubsection*{4 Chess Boards} $U^2 D_2^2 R^2 L_2^2 F^2 B_2^2 R^2 L_2^2$ \subsubsection*{6 Bars} $F^2 R^2 F_{23}^2 L^2 F_2^2 D_{23}^2 F_{12}^2 D_{23}^2$ \subsubsection*{2 Twisted Rings} $L_{12}^2 F_{12}^{-1} L_{12}^{-1} F_{12} L_{12}^{-1} U_{12} B_{12}^{-1} U_{12} B_{12} U_{12}^2 (B^{-1} U^2 B R^{-1} U^2 R)^2$ Certainly not the shortest maneuver. \subsubsection*{1 Small Twisted Ring} $F^{-1} L_2^2 F R^2 B_2 U^{-1} B_2^{-1} D_{12}^{-1} B_2 U B_2^{-1} D_{12} R^2 F^{-1} L_2^2 F$ \subsubsection*{1 Large Twisted Ring} $F_{12}^{-1} R_{12} D_2^2 R_{12}^{-1} U_{12}^{-1} R_{12} D_2^2 U L_2 D_2^{-1} L_2^{-1} U^{-1} L_2 D_2 L_2^{-1} R_{12}^{-1} U_{12} F_{12}$ \subsubsection*{4 Diagonals} $U(R^2 F R^2 D_{23}^2)^2 U_{12}^{-1} F^2 R_{12}^2 D_{23} R_2^2 D_{23}^{-1} R^2 F^2 U_2$ \subsubsection*{2 Small Twisted Peaks} $B_2^2 D_2^2 L_{12}^2 U F^2 L^2 D^{-1} L^{-1} D L^{-1} F U^{-1} F L_{12}^2 D_2^2 B_2^2$ \subsubsection*{2 Large Twisted Peaks} $D_{12}^2 R_{12}^{-1} D_{12}^{-1} R_{12} D_{12}^{-1} F_{12} L_{12}^{-1} F_{12} L_{12} F^2 U_2^2 R_2^2 F^{-1} R D_{23}^2 R^{-1} U^{-1} R D_{23}^2 R^{-1} U F R_2^2 U_2^2 F_2^2$ \subsection{Snakes} \subsubsection*{Snake 1} $L_{12}^{-1} U_2^2 L_{12} F_{12}^2 L_2 F_{12}^2 R_2^{-1} D_{12}^2 R_2 U_{12}^2 R_2^{-1} D_{12}^2 R_2 U_{12}^2$ \subsubsection*{Snake 2} $F^2 B^2 D^2 L_2 D^2 L_2^{-1} D^2 R_2 D^2 R_2^{-1} B^2 L_2 F^2 R_{12} U_2^2 R_{12}^{-1}$ \subsubsection*{Snake 3} $D_{12} R_{12}^2 F_{12} R_{12}^{-1} B_2 R_{12} F_{12}^{-1} R_{12}^2 D_{12}^{-1} R_{12}^{-1} B_{12}^2 D_{12}^{-1} B_{12} L_2^{-1} B_{12}^{-1} D_{12} B_{12}^2 R_{12}$ \end{document} From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 27 19:57:45 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id TAA15569; Mon, 27 Jul 1998 19:57:44 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jul 27 09:23:29 1998 Date: Mon, 27 Jul 1998 09:23:10 -0400 (EDT) From: Nicholas Bodley To: Rainer aus dem Spring Cc: michael reid , cube-lovers@ai.mit.edu Subject: Restoring a 5^3 to solved state (Was: Re: patterns on 5x5x5 cube) In-Reply-To: <35BA52EF.35D831D6@t-online.de> Message-Id: [Also an endorsement for Acrobat (*.PDF) at the end of this msg.] On Sat, 25 Jul 1998, Rainer aus dem Spring wrote: {Snips} }Thanks for the patterns, } }the problem is - my 5*5*5 cube is scrambled and I have to figure out }how to unscramble it. I haven't touched it for 15 years. Although it's using your mind in a different fashion, you could disassemble it, sort the pieces (takes a while!) and reassemble it in the solved state. With any reasonable degree of care, you won't harm a 5^3, I'm just about sure.* Have a clean work surface. I have done it maybe 5 or 6 times on mine (from Meffert, ca. 1987). If you have a cat, don't even think of letting it in the same apartment or house while it's apart! :) The insides are really quite amazing to see. The internal "foot" that retains a corner cubie is an amazing shape. *A 4^3 requires much more care. The center cubies are fragile! There was a message a while back from someone who's selling parts for 4^3s. To start the disassembly, align all layers (obviously), so it's a cube. Then rotate one face, leaving the other four layers aligned. Rotate it either less or more than 45 degrees, so that a left or right edge cubie of the rotated face is aligned with the edge of the other four layers. Plainly, it doesn't matter which you choose, because of physical symmetry internally. With the rotated face on top, pry up the left (or right) edge cubie, away from the edge you aligned it with. Use your thumb, thumbnail facing down. Once it disengages, the rest won't fall apart uncontrollably; a few pieces will fall out, but most will need to be actually removed, one by one. Study the structure as you pull it apart. Amazement is one reason, and the other is to get a better idea of how to reassemble it. Sort the pieces (it might take longer than you think!). Your color references for rebuilding will obviously be the center and corner cubies. Build one face completely, place that face down onto your work surface, and build progressively up from there. The last cubie will be in the same position you removed to start. My 3-D sense happens to be extremely good (apparently hereditary), so I had very little trouble figuring out what goes where and how. It might be harder for some others. ===== [ Moderator's note: Nicholas Bodley's and Rainer aus dem Spring's discussions of the merits of Acrobat and other graphics languages are not on topic for Cube-Lovers. Send them e-mail for the discussion--the addresses are in the headers of this message. We may eventually get some in the archives, which will be announced. ] My regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* The personal computer industry will have become |* Amateur musician *|* mature when crashes become unacceptable. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 27 21:55:55 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id VAA15829; Mon, 27 Jul 1998 21:55:54 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jul 27 14:45:54 1998 Date: Mon, 27 Jul 1998 14:45:33 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Ten Face Moves from Start In-Reply-To: To: cube-lovers@ai.mit.edu Message-Id: On Fri, 10 Jul 1998 08:48:05 -0400 (Eastern Daylight Time) Jerry Bryan wrote: > This run took about three weeks on a Pentium 300. Here is a bit of a follow-up. I didn't say so explicitly, but only the results 10f from Start were new. The search had been calculated through 9f from Start previously. In some ways, there was nothing new in the program to calculate 10f from Start vs. 9f from Start, because the memory requirements are the same either way (all the positions up through 5f from Start have to be stored either way). Basically, the only difference was to let the program run longer. A faster machine helped a great deal. Also, I did add a simple checkpointing capability which helped a great deal. I received some private E-mails suggesting using the net as a massively parallel computer to calculate the problem further from Start, similar to what has already been done on the net to break certain ciphers. The checkpointing I added to the program is a step in the direction of parallel processing. As has been described in the Cube-Lovers archives, the program uses an algorithm that produces permutations in lexicographic order. Such an algorithm inherently decomposes easily into parallel processing. So by analogy to processing a phone book or a dictionary, it should be possible for one machine to process the A's, another machine to process the B's, a third machine to process the C's, etc., and then to add the results together. (Actually, you would use finer decomposition than that. One machine would process the AB's, another would process the AC's, etc., or perhaps you would use even a finer decomposition. Note that there are no AA's because these are permutations we are talking about -- no letter repeats within a word.) What is really needed are some scripts to drive and control such a process. I really don't have time right now -- maybe in the future. Also, to go past 10f from Start, the machines working on the project would have to have a good bit of memory, maybe something in the 100MB range would have to be dedicated to the program to calculate 11f or 12f from Start. The existing program is in C. It was suggested in a private E-mail that writing the program in Java might make it easier to run "on the net". Maybe, but I am dubious at this point if Java is ready to handle the size of problem we are talking about. > As a possible strategy, if we could add one level per decade, we could > probably calculate the problem all the way to the end within about 100 > years. Moore's Law (the power of computers doubles about every eighteen > months) suggests that such a schedule might be possible. With respect to the E-mail about waiting for faster machines to deal with exponential problems, my real point was not that waiting on technology is a wonderful way to attack the Cube problem. Rather, I was suggesting that the Cube problem is small enough, even at about 10^19, that it can ultimately be defeated by technology (i.e., by Moore's law). Chess is about 10^75 and Go is about 10^120. Moore's law is therefore pretty well bound to fail before Chess or Go can be solved. (Deep Blue played very good chess against Kasparov, but not perfect chess.) There are two strong local maxima 9f from Start, and they have already been posted to Cube-Lovers. Six more strong local maxima showed up at 10f from Start. Regrettably, my "simple checkpointing" did not include printing out the permutations for the strong local maxima -- I just counted them. I have improved the checkpointing, and am rerunning a part of the program to print out the six strong local maxima. So far, the only one of the six which has been printed turns out to be a 4-H pattern. D B2 L2 B2 D U' R2 F2 R2 U' F2 R2 F2 D' U R2 F2 R2 U D' L' R' D' U' B2 F2 D' U' L' R' L' R' D' U' B2 F2 D' U' R' L' B' F' D' U' L2 R2 D' U' B' F' B' F' D' U' L2 R2 D' U' F' B' D' F2 R2 F2 D' U R2 F2 R2 U B2 L2 B2 D U' R2 F2 R2 U' D L R D' U' B2 F2 D' U' L R L R D' U' B2 F2 D' U' R L B F D' U' L2 R2 D' U' B F B F D' U' L2 R2 D' U' F B B2 F2 L2 R2 U2 B2 F2 L2 R2 U2 B2 F2 L2 R2 D2 B2 F2 L2 R2 D2 L2 D2 B2 F2 R2 B2 F2 R2 U2 R2 R2 D2 B2 F2 R2 B2 F2 R2 U2 L2 B2 D2 F2 L2 R2 F2 L2 R2 U2 F2 F2 D2 F2 L2 R2 F2 L2 R2 U2 B2 ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 28 10:38:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id KAA18123; Tue, 28 Jul 1998 10:38:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Jul 27 19:41:11 1998 Message-Id: <35BD0149.9F7384B5@t-online.de> Date: Tue, 28 Jul 1998 00:38:01 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: cube-lovers@ai.mit.edu Subject: Re: Restoring a 5^3 to solved state (Was: Re: patterns on 5x5x5 cube) References: From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Nicholas Bodley wrote: > Although it's using your mind in a different fashion, you could > disassemble it, sort the pieces (takes a while!) and reassemble it in > the solved state. With any reasonable degree of care, you won't harm a > 5^3, I'm just about sure.* Have a clean work surface. I have done it > maybe 5 or 6 times on mine (from Meffert, ca. 1987). If you have a cat, > don't even think of letting it in the same apartment or house while it's > apart! :) The insides are really quite amazing to see. The internal > "foot" that retains a corner cubie is an amazing shape. I have found an old booklet by Endl (terrible) that contains a Mickey Mouse solution for the 5x5x5 cube. Thank God - I have TWO cats :) > *A 4^3 requires much more care. The center cubies are fragile! There > was a message a while back from someone who's selling parts for 4^3s. Yeah, mine is very flabby. A real cube meister will never disassemble his cube :) Rainer adS From cube-lovers-errors@mc.lcs.mit.edu Wed Jul 29 11:39:38 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id LAA21758; Wed, 29 Jul 1998 11:39:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 28 09:05:00 1998 Message-Id: <35BDCC35.97F9BD3@nadn.navy.mil> Date: Tue, 28 Jul 1998 09:03:49 -0400 From: David Joyner Reply-To: wdj@nadn.navy.mil Organization: Math Dept, USNA To: Cube Mailing List Cc: Rainer.adS.BERA_GmbH@t-online.de Subject: Re: 4*4*4 patterns References: <35BB827C.10B7C9A7@t-online.de> Rainer aus dem Spring wrote: > Dear cube lovers, > > as promised the other day here comes my collection of 4*4*4 patterns. > My favorites are the single twisted rings. I still find it surprising > to see that there is no second ring on the "other" side. > > The maneuvers use all sorts of slice moves which are probably not > accepted as moves by most cubeologists. I am too lazy to rewrite them. > > Does anybody have any idea which format I should post for people > without a TeX system? I have sent Rainier an html conversion of his file. With his permission and approval I'll post on my web page http://www.nadn.navy.mil/MathDept/wdj/rubik.html > ... Does anybody know of a 4*4*4 emulator or even solver? Anything > like a (sub)optimal solver is probably beyond the current PC powers. Yes. MAPLEV5 (Mathematica's main competitor) released a 4x4 Rubik's cube emulator (as well as a masterball emulator and a 3x3 Rubik's cube emulator) in their "share package" included with the software. The share package is actually free but MAPLEV5 is not! Incidently, the emulators work on some older versions of MAPLE as well. The pictured linked to on the bottom of the above-mentioned web page were obtained using this emulator. - David Joyner > Rainer > > PS > I am NOT a LaTeX expert :) hints are welcome ! > > [Moderator's note: ... I wonder if there could be some > simplification with the [X,Y] = X Y X^{-1} Y^{-1} commutator notation > or the X^Y = Y^{-1} X Y conjugate notation, or if this would make the > processes too hard to follow. --Dan] It would be theoretically interesting, IMHO, to have the expressions rewritten using commutators but more confusing in practice to follow. > (Latex file deleted) -- David Joyner, Assoc Prof of Math US Naval Academy, Annapolis, MD 21402 (410)293-6738 wdj@nadn.navy.mil http://web.usna.navy.mil/~wdj/homepage.html ++++++++++++++++++++++++++++++++++++++++++++ "A Mathematician is a machine for turning coffee into theorems." Alfred Renyi From cube-lovers-errors@mc.lcs.mit.edu Wed Jul 29 13:55:23 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA22336; Wed, 29 Jul 1998 13:55:23 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 28 12:10:01 1998 Message-Id: <002001bdba41$af584480$99118bc0@tellus.switchview.com> From: "Michael Swart" To: Subject: Re: Restoring a 5^3 to solved state Date: Tue, 28 Jul 1998 12:06:31 -0400 >A real cube meister will never disassemble his cube :) That's true but it is better than the alternative: the dreaded _sticker peeling_! I took a course in university called "Intro to Public Speaking". In it we had to give a persuasive speech and mine was Called "Why you shouldn't peel stickers of a Cube". Here are some of the reasons I gave. 1. The stickers weren't designed to be peeled. So until the people at 3M come out with a post-it note version of the cube, then peeling will only wear the cube out faster 2. Cheating defeats the purpose of the puzzle. It reduces it to a simple jigsaw puzzle. But if you find this simpler puzzle challenging - an unlikely scenario for cube-lovers - then by all means peel away. 3. Douglas R. Hofstadter once noted that there were two mysteries to the cube: 1. How does one solve the cube and 2. How does the cube stay together. If frustration gets the better of you and you must cheat then disassembling the cube is the preferred way because even though it does nothing to shed light on the first mystery it does give insight to the second mystery. Besides disassembling the cube and reassembling it don't cause (much) damage if you're careful. 4. Chances are greater that you'll leave the cube unsolvable. Kids in my grade school in the 80's used to peel stickers because they were so close to completing two sides that they resorted to peeling one or two stickers to get the job done. This behaviour inevitably left the cube unsolvable. If you disassemble a cube and then assemble it randomly, there is a 1 in 12 chance that you'll be able to solve the cube. But if you take all the stickers off a cube and restick them randomly, then you have a better chance at winning the lotto 6/49 twice in a row on your next two tickets, than your chances and getting a solvable cube. (I'll post my math if anyone asks) Michael Swart Switchview Inc. Michael.Swart@switchview.com From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 30 11:17:20 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id LAA26057; Thu, 30 Jul 1998 11:17:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 28 14:39:58 1998 Message-Id: <35BE13D0.874D8E6D@t-online.de> Date: Tue, 28 Jul 1998 20:09:20 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: Cube Mailing List Subject: 4x4x4 patterns From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Dear cube-lovers, Of course, HTML is the format I should have used. I'll try to get a latex->html converter for NT. Anybody my post the patterns in any format. The R_123 means turn 3 slices - this is the same as turning L and then turn the whole cube R. This notation was inspired by Bandelow's usage of slice moves. The advantage is, it makes sense for any cube. The disadvantage is, most cubologists don't accept these "moves". I have checked all patterns on my (physical) 4x4x4 cube. I don't think using conjugation and commutatotrs is very user-friendly. The maneuvers are not optimized and anybody will be able to figure out how they were constructed using conjugation and commutators, though. As far as I remember the patterns were the last thing I did in those golden cube days :) Does anybody know of a cube simulator for Mathematica ? Rainer adS -- -------------------------------------------------------------------------- Rainer aus dem Spring email Rainer.adS.BERA_GmbH@t-online.de (home) Schimmelbuschstr. 10 email TEEADS@TEE.toshiba.de (business only) 40699 Erkrath tel. +49 (0)02104 35157 (private) Germany tel. +49 (0)02104 936150 (business) --------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 30 12:19:17 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA26710; Thu, 30 Jul 1998 12:19:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Jul 28 21:13:28 1998 Date: Tue, 28 Jul 1998 11:32:48 -0400 Message-Id: <00269731.001706@scudder.com> From: Jacob_Davenport@scudder.com (Jacob Davenport) Subject: Solving the 5^3 To: cube-lovers@ai.mit.edu, Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) I have a fairly straight forward solution to the 5x5x5, and if there is some interest in having me post it, I will. -Jacob From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 30 15:25:22 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id PAA27775; Thu, 30 Jul 1998 15:25:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Jul 29 15:20:36 1998 Date: Wed, 29 Jul 1998 15:17:00 -0400 (EDT) From: David Barr X-Sender: Davebarr@viking.cris.com Reply-To: davidbarr@iname.com To: cube-lovers@ai.mit.edu Subject: Meffert's Challenge Message-Id: I recently bought a "Meffert's Challenge" puzzle, and I see there hasn't been discussion of this puzzle on this list. Maybe it is new. It is a round Skewb, like Mickey's Challenge, but with different markings. When solved, it has four colored rings on it. The triangular pieces each have about a quarter ring on them. Four of the square pieces have two separate quarter ring markings, and the other two square pieces are blank (actually, they say "Meffert's Challenge"). I think it is fun because in addition to solving the puzzle, you can try to make different "snake" patterns on it. It took me a while to figure out how to make a snake that uses all but one of the segments. I threw away the packaging, and I don't remember who makes it. I think it was Pressman. I bought it at a toy store in the Supermall in Auburn, WA, and they also had a normal Skewb made by the same company. David From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 30 18:19:26 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id SAA28303; Thu, 30 Jul 1998 18:19:25 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 30 13:37:06 1998 Message-Id: <35C0AEFD.3D67E038@nadn.navy.mil> Date: Thu, 30 Jul 1998 13:35:57 -0400 From: David Joyner Reply-To: wdj@nadn.navy.mil Organization: Math Dept, USNA To: Rainer.adS.BERA_GmbH@t-online.de Cc: Cube Mailing List Subject: Re: 4x4x4 patterns References: <35BE13D0.874D8E6D@t-online.de> Rainer aus dem Spring wrote: > Dear cube-lovers, > > Of course, HTML is the format I should have used. I'll try to get a > latex->html converter for NT. > Anybody my post the patterns in any format. Rainer's patterns (with some pictures) are on the web page http://web.usna.navy.mil/~wdj/4x4patterns_b.htm - David Joyner -- David Joyner, Assoc Prof of Math US Naval Academy, Annapolis, MD 21402 (410)293-6738 wdj@nadn.navy.mil http://web.usna.navy.mil/~wdj/homepage.html ++++++++++++++++++++++++++++++++++++++++++++ "A Mathematician is a machine for turning coffee into theorems." Alfred Renyi From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 31 10:28:00 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id KAA29897; Fri, 31 Jul 1998 10:27:59 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 30 14:17:23 1998 Message-Id: <199807301816.OAA14197@life.ai.mit.edu> Date: Thu, 30 Jul 98 14:16:38 EDT From: Nichael Cramer To: cube-lovers@ai.mit.edu Cc: nichael@sover.net Subject: "Hints" for Solving the 5X Since you 1] posted to this list and 2] have a 5X and have solved it the past, I'm going to make the assumption that really want is not necessarily a cook-book for solving the 5X, but enough hints to get you started in the right direction. If I'm wrong about this, you can stop here. ;-) If not, below is high-level description of a scheme for solving the 5X that assumes 1] that you're 3X3X3-literate and 2] leaves unspecified the details of the three other simple operations that you'll need. (Now, this is far from elegant; and certainly not anything like a maximal solution. But at least it will 1] get your cube solved and 2] at least you thinking about these things again.) --- Step 1: First, ignore everything except the corner cubies, the center (face) cubies and the center-edge cubes. Now, paying attention to those cubies only, pretend that you're dealing with a 3X and "solve" it. --- Step 2: Solve the non-center edge-cubies [NCEC]. First devise an operator that allows you to cyclically-swap three of the NCECs (i.e. without messing with any of the previously solved cubies). With a little clever permutation, this single operation will allow you to complete this step (but see Step 2A below). (Note that since a NCEC _has_ to be in its correct orientation when it is in its corrected location --i.e. a correctly placed NCEC can't be simply flipped as can a center-edge cubie in a 3X-- you can complete this step using this single operation.) Step 2A: There is one wrinkle at this point. It is possible to be in an "orbit" in which you can apparently "solve" all of the NCECs except for two. If you reach this point, leave the two unsolved NCEC alone for the moment. They will be easier to solve after completing the next step. --- Step 3: Solve the remaining non-center face cubies [NCFC]. Similar to the above, devise two operators: one that allows you to cyclically-swap three "corner-like" NCFCs and one that allows you to cyclically-swap three "edge-like" NCFCs (i.e. without in either case disturbing the previously-solved cubies). Again with a little clever permutation you should be able to complete this step with this single operation. (Note that since, for a given color, all four "center-like" NCFCs are pretty much interchangable --as are all four "edge-like" NCFCs. This means that that any "bogus" symetries are invisible. What this means is that, by saving this step 'til last, you don't risk getting all the way to the and finding out you're in some non-standard "orbit" that you have to back out of.) --- Step 2A Continuted: Assuming that you've got this far, you should now be in a state where the entire cube is solved except for --at most-- two NCECs. In short, this state of affairs means that your cube is a wrong "orbit"; i.e. there is a "hidden" symmetry among that cubies that allows your cube to appear to be more nearly solved than it is. The quickest way to get your cube in the "correct orbit" is as follows: Choose one of the "internal" slices that contains one of non-solved NCECs (by "internal", I mean a slice that is neither a face slice nor one that contains a center cubie). Now rotate that internal slice by a quarter turn (i.e. by 90 degrees) in either direction. Now what you want to do is solve remaining cubies from its current situation. The tricky part here is keeping everything straight. There are a couple of things that you can do help this. 1] If possible, you can first manipulation the NCECs in such a way that the two unsolved NCEC share the same slice and are on the same face. Then it will be possible --when performing the quarter-turn as described above-- to bring one of the unsolved NCEC into its correct location. Once that is done, you will now have exactly three unsolved NCECs. Since these must necessarily be a cyclic permuation, you should be able to solve these without further ado. Now, all that remains is solving the remaining newly scrambled face-cubies. 2] If you are unable to position the two unsolved NCEC as described above, proceed as follows: >From the current state, (i.e. after performing the quarter-turn on the internal slice) re-solve the newly shuffled face-cubies *while*being*sure*not*to*disturb*any*other*cubies! Once you have all the faces solved, you should now have four NCEC in the wrong locations, along the single internal slice. Using the operation that you developed in Step 2 above, this should be easy to solve. --- Step 4: Place your newly-solved cube in a prominent place on your desk and assume a smug demeanor when asked about it. Hope this helps Nichael -- Nichael Cramer work: ncramer@bbn.com home: nichael@sover.net http://www.sover.net/~nichael/ From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 31 11:16:22 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id LAA00122; Fri, 31 Jul 1998 11:16:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 30 17:34:13 1998 Message-Id: <35C0C750.9D29907D@t-online.de> Date: Thu, 30 Jul 1998 21:19:44 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: Cube Mailing List Subject: Corrections References: <35BE13D0.874D8E6D@t-online.de> From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Sorry: \subsubsection*{2 Twisted Rings} $L_{12}^2 F_{12}^{-1} L_{12}^{-1} F_{12} L_{12}^{-1} U_{12} B_{12}^{-1} U_{12} B_{12} U_{12}^2 (B^{-1} U^2 B R^{-1} D^2 R)^2$ not ... U^2 R)^2$ Rainer adS U was "unten", i.e., German for down. From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 31 12:04:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA00301; Fri, 31 Jul 1998 12:04:32 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 30 17:52:36 1998 Message-Id: <35C0D64D.B84386E@t-online.de> Date: Thu, 30 Jul 1998 22:23:41 +0200 Reply-To: Rainer.adS.BERA_GmbH@t-online.de Organization: BERA Softwaretechnik GmbH To: Cube Mailing List Subject: The hunt is up From: Rainer.adS.BERA_GmbH@t-online.de (Rainer aus dem Spring) Dear cube lovers, I am glad to see that my patterns started such a long thread. The cube is alive :) Mike Reid sent some improvements. I am sure he doesn't mind me to send them to the mailing list. What about other improvements ? Anybody mad enough to search other patterns ? Mike's improvements: \subsubsection*{Exchanged 2x3x3 Bricks} $D_{12}^2 L^2 B^2 D_{12}^{-1} R^2 D_{12} R^2 U_{12}^{-1} R^2 U_{12} R^2 B^2 U_{12} L^2 D_{12} L^2 D_{12}$ (Michael Reid) improves my ????? :) $U_{12}^2 R^2 B^2 D_{12} L^2 D_{12}^{-1} L^2 U_{12} L^2 U_{12}^{-1} L^2 B^2 U_{12}^{-1} R^2 U_{12} R^2 U_{12}$ (Michael Reid) improves the 1x1x2 Brick pattern $U R^{-1} U^{-1} F_2 D_{12}^{-1} F^2 D_{12} F_2^{-1} D_{12}^{-1} F^2 D_{12} U R U^{-1}$ (Michael Reid) improves "1 Small Twisted Ring" Rainer -- -------------------------------------------------------------------------- Rainer aus dem Spring email Rainer.adS.BERA_GmbH@t-online.de (home) Schimmelbuschstr. 10 email TEEADS@TEE.toshiba.de (business only) 40699 Erkrath tel. +49 (0)02104 35157 (private) Germany tel. +49 (0)02104 936150 (business) --------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 31 13:39:50 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA01321; Fri, 31 Jul 1998 13:39:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jul 30 18:29:08 1998 From: Douglas Zander Message-Id: <199807302228.RAA17538@solaria.sol.net> Subject: Meffert's Challenge (fwd) To: cube-lovers@ai.mit.edu (cube) Date: Thu, 30 Jul 98 17:28:21 CDT David Barr wrote: > I recently bought a "Meffert's Challenge" puzzle, and I see > I threw away the packaging, and I don't remember who makes it. I > think it was Pressman. I bought it at a toy store in the Supermall > in Auburn, WA, and they also had a normal Skewb made by the same > company. > David yes, I just happened to buy both of these puzzles myself today. They are made by Pressman. It is interesting to note that the copyright says, "(copyright) 1997 Uwe Meffert patent #5,358,247" for *both* puzzles. (same mechanism) They also are selling the Pyraminx in the same type packaging. Is the Pyraminx also a Skewb mechanism? (I didn't buy one since I already had one) Now I wish they would bring back some other puzzles! :-) -- Douglas Zander | dzander@solaria.sol.net | Shorewood, Wisconsin, USA | From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 31 16:44:32 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id QAA01768; Fri, 31 Jul 1998 16:44:31 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jul 31 16:24:59 1998 Date: Fri, 31 Jul 1998 16:24:49 -0400 From: michael reid Message-Id: <199807312024.QAA10616@euclid.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: new optimal solver lately i've been working on a new optimal solver. this is similar to the previous program, but uses different subgroups. let H be the subgroup in which the four edges FR, FL, BR and BL are all in place, and are correctly oriented and the four U corners are on the U face (and thus the four D corners are on the D face, and they are oriented so that the U [respectively D] facelet is on the U [respectively D] face. then the cosets H \ G are described by triples (e, cl, ct) where e describes the location and orientation of the four edges FR, FL, BR and BL, cl describes the location of the four U corners, and ct describes the orientation of the eight corners. there are 24 * 22 * 20 * 18 = 190080 different e coordinates, / 8 \ \ 4 / = 70 different cl coordinates, and 3^7 = 2187 different ct coordinates. all combinations are possible, so there are 190080 * 70 * 2187 = 29099347200 cosets. the subgroup H has 16-fold symmetry; it is invariant under any symmetry of the cube that preserves the U-D axis. therefore the coset space H \ G also has this symmetry. up to symmetry, there are 12094 e coordinates. thus, we can reduce the coset space to 12094 * 70 * 2187 = 1851470460 configurations. store each configuration in half a byte of memory (storing its distance from start). the whole thing can be stored in a tiny array of 925735230 bytes, approximately 883 megabytes. the number of cosets (actual numbers, not reduced by symmetry) at each distance is distance quarter turns face turns 0 1 1 1 8 12 2 76 162 3 696 2044 4 6418 25442 5 57912 316290 6 514318 3899553 7 4496206 46650252 8 38304572 517476714 9 308312232 4480840746 10 2142297548 16776040760 11 9789496784 7259620140 12 14800845359 14475084 13 2014724044 14 291026 i have this running on one processor of a sun ultra enterprise 450, configured with 1024Mb of RAM. startup time is significant: it takes about 85 minutes for quarter turns, 125 minutes for face turns, to exhaustively search the coset space. some rough estimates are that it is 6.7 times faster than my previous optimal solver for quarter turns, 3.4 times faster for face turns. this is not nearly as good as i'd hoped. there seems to be some performance issue with this machine. it appears to be significantly slower when accessing large amounts of memory at random, despite the fact that it is all real memory, so no swapping is occurring. the performance drop off starts at about 256Mb. my program runs slower by a factor of 3 or maybe even 4 because of this. my sysadmin has reproduced the same behavior on a small test program, so the problem is unlikely to be caused by my code. i'm told that it is probably some gross inefficiency in the cache paging system of the operating system (solaris). the os seems to have plenty of options, so perhaps one of them will fix this problem and speed up my program by a factor of 3 or maybe 4. it seems ridiculous to me that things work this way, but apparently they do. nevertheless, the program is already fast enough for the tasks at hand. mike From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 1 23:44:12 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id XAA05471; Sat, 1 Aug 1998 23:44:12 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jul 31 22:28:10 1998 Date: Fri, 31 Jul 1998 22:28:02 -0400 From: michael reid Message-Id: <199808010228.WAA11081@euclid.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: all 24q maneuvers for superflip with my new optimal solver, i've calculated all 24q maneuvers for superflip. there are three transformations we can apply to a maneuver for superflip, none of which change its length. we may conjugate by any cube symmetry. we may cyclically permute the maneuver, i.e. replace sequence_1 sequence_2 by sequence_2 sequence_1 we may invert the maneuver. in a previous message (august 7, 1997), i showed that, using these three transformations, any maneuver for superflip can be transformed into one that begins with one of the ten sequences U R2 U D' R U D R U D R' U R F U R F' U R' F U R' F' U' R F' U' R' F' my program took 101 hours to exhaustively search these ten cases. there are four inequivalent maneuvers; two were previously known: R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q*) U R2 F' R D' L B' R U' R U' D F' U F' U' D' B L' F' B' D' L' (24q*) the two new ones are: U D' R F U' D' L D' F R U' R U' D' F U' F L B' U F' B' L B' (24q*) U D' R F' D L' B L' U' R' D' B' U' D L' F D' R B' R U L D B (24q*) this last one can be written as (U D' R F' D L' B L' U' R' D' B' R_rl)^2 where R_rl denotes reflection through the R-L plane. we can also count the total number of 24q maneuvers for superflip. note that U2 = U U also is U' U' , so can be cyclically shifted in an extra way. similarly, U D' = D' U , so this also accounts for an extra cyclic shift. and the same is true for U' D'. the total number of maneuvers therefore is 28 * 24 * 2 + 28 * 48 * 2 + 28 * 48 * 2 + 26 * 24 * 2 = 7968 where the first factor is the number of cyclic shifts, the second factor is the number of cube symmetries we can apply, and the third factor is 2, for inversion. the first and last maneuvers only get a factor of 24 for the number of cube symmetries, because a cyclic shift by 12q gives the same maneuver in a different orientation. the total number of 24q sequences is 274575811926317204506464. the total number of even positions is 21626001637244928000. so even positions have an average of 12696.56 different 24q maneuvers. mike From cube-lovers-errors@mc.lcs.mit.edu Sun Aug 2 17:46:24 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA08130; Sun, 2 Aug 1998 17:46:24 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Aug 2 08:47:54 1998 Date: Sun, 2 Aug 1998 08:47:44 -0400 From: michael reid Message-Id: <199808021247.IAA08734@cauchy.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: superflip composed with four spot with my new optimal solver, i can show that the position superflip composed with four spot is exactly 26 quarter turns from start. this gives a new lower bound for the diameter of the cube group. the previous lower bound, 24q, was from the position superflip, and was first established by jerry bryan. let F2 B2 U D' R2 L2 U D' be our choice of orientation of four spot. although four spot is not central, the position F2 B2 U D' R2 L2 U D' C_U2 moves only face center cubies: (F, B) (R, L). (here C_U2 denotes whole cube rotation by 180 degrees about the U-D axis.) since quarter turns do not move face center cubies, we see that the sequence above commutes with any sequence of quarter turns. the same is also true for superflip . four spot . C_U2 in terms of singmaster's fixed face model, this means that we can cyclically shift a maneuver for superflip composed with four spot, but the part that is cyclically shifted gets conjugated by the cube rotation C_U2. for example: (B U2 L) (U' D L2 F2 R2 B U2 R' L' D R2 D F2 U R2 D B) creates this position. if we cyclically shift the first three twists to the end, we get another maneuver for this position: (U' D L2 F2 R2 B U2 R' L' D R2 D F2 U R2 D B) (F U2 R) this observation about cyclic shifting enables us to prove proposition 1. superflip composed with four spot is a local maximum in the quarter turn metric. proof. we need to show that any quarter turn takes us closer to start. the 12 different twists split up into two different types under the symmetry of this position: {U, U', D, D'} and {R, R', F, F', L, L', B, B'}. we claim that any maneuver for superflip composed with four spot must contain twists of both types. a maneuver consisting only of twists in {U, U', D, D'} clearly cannot produce this position. also, a maneuver consisting only of twists in {R, R', F, F', L, L', B, B'} cannot flip any edges. thus both twist types must occur. now consider a minimal maneuver for superflip composed with four spot. we may cyclically shift (and apply symmetry) so that the last twist is U'. thus, applying U cancels this last twist and brings us closer to start. similarly, we can cyclically shift to get a minimal maneuver ending with R', so applying R also brings us closer to start. since any twist is equivalent to U or R , we have proved local maximality. qed the significance of this proposition is that this is the first case beyond the hoey-saxe local maxima in which we can prove local maximality without computer searching. (please correct me if i'm wrong about this.) dan hoey noted (a long time ago) that the position four spot is a local maximum. however, i don't see that this can be proved without computer search. the sticking point is that four spot can be achieved using only {R, R', F, F', L, L', B, B'}. however, no minimal maneuver consists only of these twists, a fact determined by computer search. similar to the transformations for superflip, we have three transformations to apply to maneuvers for superflip composed with four spot. we may conjugate by any of the 16 cube symmetries that fix the U-D axis. we may cyclically shift the maneuver, as described above. we may invert the maneuver. proposition 2. by using the three transformations above, any maneuver for superflip composed with four spot can be transformed into one that begins with one of the six sequences R U R' U D R' U F' R' U R' R' U B' R' U L' proof. as shown in prop. 1, any sequence for superflip composed with four spot contains both types of twists. thus, the two types occur as consecutive twists. by cyclic shifting, and applying symmetry, we may suppose that the first two quarter turns are either R U or R' U. (this would already be enough reduction for my program). we can cut down the case R' U further. there are eleven possibilities for the third quarter turn; only U' is not allowed. the case R' U U = R' U2 is equivalent under symmetry to R U2, which is part of the case beginning with R U. the case R' U D' is equivalent under symmetry to R D' U = R U D', again part of the case beginning with R U. the case R' U B inverts to B' U' R, and this is equivalent to R U B', which is part of the case beginning with R U. similarly, the cases beginning with R' U R , R' U F and R' U L invert to R U R' , R U F' and R U L', respectively. this leaves only the sequences listed above. qed my program exhaustively searched the positions superflip. four spot . R U through 22q and superflip. four spot . R' U D \ superflip. four spot . R' U F' \ superflip. four spot . R' U R' > all through 21q superflip. four spot . R' U B' / superflip. four spot . R' U L' / and found no maneuvers. thus superflip composed with four spot requires more than 24 quarter turns. the total search time was about 153 hours. to see that superflip composed with four spot can be achieved in 26 quarter turns, use U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f) it might be reasonable to ask for all 26q maneuvers. this is probably out of reach for now. however, i suspect that there will be so many different 26q maneuvers that it would not be of much use to see a long list of maneuvers. (i have a bunch already.) superflip composed with four spot also requires 20f. proposition 3. any maneuver for superflip composed with four spot of length <= 20f can be transformed to one that begins with one of the sequences U2 R , R2 F or R2 U . the proof is very similar to the reductions for superflip in the face turn metric. using this, a complete search for 20f maneuvers is straightforward. there are two inequivalent 20f maneuvers for superflip composed with four spot: F U2 R L D F2 U R2 D F2 D F' B' U2 L F2 R2 B2 U' D (20f*, 28q) F U2 R L D F2 U R2 D F2 D F' B' U2 L U' D R2 B2 L2 (20f*, 28q) this also shows that no maneuver is simultaneously minimal in both metrics. mike From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 3 13:19:38 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA11416; Mon, 3 Aug 1998 13:19:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Aug 3 02:10:51 1998 Date: Mon, 3 Aug 1998 00:24:35 -0400 Message-Id: <00273C38.001706@scudder.com> From: Jacob_Davenport@scudder.com (Jacob Davenport) Subject: Web address for solving the 5^3 To: cube-lovers@ai.mit.edu I have created a page with a description of how I solve the 5x5x5. Please check out www.wunderland.com/wts/jake. Although I did spend a fair amount of time on this page, I certainly consider it a first draft, and I would appreciate any comments about it, either those involving clarity of the explanation, or even better moves that would perform the same functions. Be warned that it is a long page, although I'm sure you expected that. One person wrote to me and said that he had all but two cubies solved. I suspect his difficulty was "parity" which I cover in my Sixth Step. I'm sure there are many good solutions to the 5x5x5, just as there are for the 3x3x3, so if you have a half-solved cube you may need to scrap your work if you want to use my solution. Good luck. -Jacob From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 6 11:20:35 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id LAA21031; Thu, 6 Aug 1998 11:20:34 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Aug 5 13:41:35 1998 Date: Wed, 05 Aug 1998 13:41:10 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: superflip composed with four spot In-Reply-To: <199808021247.IAA08734@cauchy.math.brown.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Sun, 02 Aug 1998 08:47:44 -0400 michael reid wrote: > with my new optimal solver, i can show that the position > > superflip composed with four spot > > is exactly 26 quarter turns from start. this gives a new lower bound > for the diameter of the cube group. the previous lower bound, 24q, was > from the position superflip, and was first established by jerry bryan. Nobody has said so yet on the list, but I think this is exciting news for Cube-Lovers, both the fact that a new lower bound has been discovered for the diameter of the cube group, and the fact that a new (and very long) local maximum has been found by means other than computer search. It seems to me that Mike's proof might provide an outline for a method for looking for other local maxima. I have not at this point been able to use his method to find other local maxima, but here is how it might work. > proposition 1. superflip composed with four spot is a local > maximum in the quarter turn metric. > > proof. we need to show that any quarter turn takes us closer to > start. the 12 different twists split up into two different > types under the symmetry of this position: {U, U', D, D'} > and {R, R', F, F', L, L', B, B'}. we claim that any maneuver > for superflip composed with four spot must contain twists of > both types. a maneuver consisting only of twists in > {U, U', D, D'} clearly cannot produce this position. also, > a maneuver consisting only of twists in > {R, R', F, F', L, L', B, B'} cannot flip any edges. thus > both twist types must occur. More generally, for any position x, calculate Symm(x)=K, where K is as usual the subgroup of all k in M such that x=k'xk, and where M is the group of 48 symmetries of the cube. Conjugation by K and grouping the elements of Q into conjugate equivalence classes induces a partition on Q, the set of twelve quarter turns. In Mike's case, the partition is {Q1,Q2} where Q1={U, U', D, D'} and Q2={R, R', F, F', L, L', B, B'}. The process I am going to describe is much simpler if we confine ourselves to 2-way partitions of Q, such as Mike's case. I think the process I am descibing can be generalized to more than 2-way partitions of Q, but some of the steps get more complicated. So for now we confine ourselves to subgroups K of M which induce at most a 2-way partition of Q. Roughly speaking, this means that we need to find positions that are fairly symmetric. I have been meaning for a long time to calculate a table of partitions of Q for each of the possible 98 subgroups of M. Perhaps Mike's new result will provide sufficient motivation to perform the calculations. The next hurdle is that we must find positions x such that x is not in or , so that a maneuver for x must contain quarter turns from both Q1 and Q2. Mike's position certainly satisfies this criterion. Notice that if we get this far, we can say that a maneuver for x must contain at least one element from Q1 and at least one element from Q2, but the elements from Q1 and Q2, respectively, need not necessarily appear at the end of the maneuver. Also, by the definition of Q1 and Q2, *any* maneuver from Q1 and Q2 can appear in a maneuver for x by K-conjugation. So far, so good. I would go about this type of a search by determining which subgroups K of M induce a 2-way partition of Q, and then by thinking about what a K-symmetric position must look like. But here's the rub -- the part I cannot figure out *in general". In order to get the elements of Q1 and Q2 to the end of the maneuver for x, we need positions which may be cyclically shifted, either in the normal sense or in Mike's new sense where the part of the maneuver that is shifted is conjugated by K. There is a good bit of discussion in the archives about cyclical shiftiness. I'm going to go back and re-read that discussion to see if it helps with this problem. But any position x whose symmetry group induces a 2-way partition {Q1,Q2} on Q, where x is not in or , and where x is cyclically shiftable (possibly with K-conjugation of the shifted part) is a local maximum in the quarter turn metric. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 6 12:19:33 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA21968; Thu, 6 Aug 1998 12:19:32 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Aug 5 21:22:00 1998 Message-Id: <35C8FF13.A6997C43@frontiernet.net> Date: Wed, 05 Aug 1998 20:55:47 -0400 From: John Bailey To: Submissions Cube-Lovers Subject: Four dimensional cube solution and two dimensional cube simulator Earlier this year I announced: http://www.frontiernet.net/~jmb184/Nteract4.html a four dimensional Rubik Cube (2x2x2x2) While that post referenced a sketch of a solution, it seemed that a clearer, more explicit solution was needed to show that the tesseract was indeed a tractable cubing problem. An explicit solution of the four dimensional analog of the Rubik cube is posted at: http://www.frontiernet.net/~jmb184/solution.html This page includes extensive graphics which are intended to make the solution clear and visible. Also, during the process of developing a detailed explaination, I realized that by using similar display techniques, a 2D analog of the cube provided an interesting model of cube solutions. This 2 dimensional 3X3 cube simulator is at: http://www.frontiernet.net/~jmb184/3x3cube.html All of these are written in Javascript, which means they do not require extended interaction with the server to manipulate. They are read in directly and then can be kept for running off-line. John Bailey From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 12 10:43:50 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id KAA16420; Wed, 12 Aug 1998 10:43:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Aug 6 23:33:59 1998 Date: Thu, 6 Aug 1998 23:33:52 -0400 From: michael reid Message-Id: <199808070333.XAA03183@chern.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: superflip composed with six spot another position to consider is superflip composed with six spot, since its maneuvers also have a corresponding cyclic shifting property. until recently, i didn't know any 24q maneuvers for this position, so i had planned to do an exhaustive search through 24q. however, by looking at the new 24q maneuvers for superflip, i was able to modify one to get a 24q maneuver for superflip composed with six spot: D' R L' F L' F B U' B L' F' U F' U D R' U R' F' D L' U D F' (24q*) as a result, i only did a partial 24q search, namely for maneuvers that contain a half turn. up to cyclic shifting and symmetry, the only such maneuvers are R' U D R' U F' D R' B U' L' U' F' D F' B' D' R' F D F D' R2 (24q*) and a suitable reorientation of the inverse maneuver. (the inverse position is the same pattern, but in a different orientation.) superflip composed with six spot also requires 20f: U2 F B' R F L2 F2 D B2 D2 R2 B' L2 F' D2 R2 D' B R B2 (20f*) no maneuver is simultaneously minimal in both metrics; this is a consequence of the partial 24q search. mike From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 17 14:50:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA05577; Mon, 17 Aug 1998 14:50:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Aug 16 11:40:37 1998 Message-Id: From: Noel Dillabough To: "'cube-lovers@ai.mit.edu'" Subject: New puzzle simulator Date: Sun, 16 Aug 1998 11:32:28 -0400 I wrote up a puzzle simulator called Puzzler, containing the cubes (2x2x2, 3x3x3, 4x4x4, 5x5x5), a pyramid, a sphere, a skewb, and a dodecahedron. While not the same as a physical puzzle, its still pretty fun to use. Its just a beta version, just compiled yesterday, so there are bound to be problems, and features that should/could be implemented. Anyone interested should download the program at http://www.mud.ca/puzzler/puzzler.html, and let me know of any problems, enhancements etc. One of my friends asked me to implement the square 1 in the same program. In order to do so I would have to change a lot of the backend code so I won't do so unless there is enough interest. -Noel P.S. I forgot, the program is for win95, win98 and winNT. From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 18 17:36:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA10973; Tue, 18 Aug 1998 17:36:34 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Aug 18 16:28:43 1998 Date: Tue, 18 Aug 1998 16:28:28 -0400 From: michael reid Message-Id: <199808182028.QAA24353@euclid.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for X symmetric positions X is the subgroup of the cube symmetry group which preserves the U-D axis. there are 128 positions which have X symmetry: the UR edge can go in any of the 8 positions UR, RU, DR, RD, UL, LU, DL, LD; this determines the location of the edges UB, UL, UF, DR, DB, DL, DF. the FR edge can go in any of the 4 positions FR, RF, BL, LB; this determines the location of the edges FL, BR, BL. the UFR corner can go in any of the 4 positions UFR, UBL, DRF, DLB; this determines the location of all the corners. any combination of these is possible, which gives 128 positions. 4 of the positions have more symmetry, namely M symmetry. (these positions are start, superflip, pons asinorum, and pons asinorum composed with superflip.) minimal maneuvers for the other positions are: 1. F2 R L' D2 F2 D2 R' L F2 D2 (16q*, 10f*) 2. U F' B' R2 U' D' F' B U D R2 F B D' (16q*, 14f) U F2 U2 F2 R L F2 U2 F2 U2 R' L' U (13f*, 20q) 3. U F B R2 U D F B' U' D' R2 F' B' D' (16q*, 14f) U F B U2 R2 U2 R2 F' B' R2 U2 R2 U (13f*, 20q) 4. F2 B2 U D' R2 L2 U' D (12q*, 8f*) 5. F2 R2 F2 B2 R2 B2 (12q*, 6f*) 6. U F' B' R' L' F' B' R L F B R L U' (14q*, 14f) U F2 U2 F2 R L B2 D2 B2 U2 R' L' U (13f*, 20q) 7. U F B R L F B R' L' F' B' R' L' U' (14q*, 14f) U F B U2 R2 D2 R2 F' B' L2 U2 L2 U (13f*, 20q) 8. F R' U B2 L' F U D' L' B R2 U' F L' U' D (18q*, 16f*) 9. F U' F R' D F' D F' R L B' U B' U L' B D' B U D (20q*, 20f) F2 R F B' D B2 D' F2 B2 U F2 U' F B' R' B2 (16f*, 22q) 10. F R F D' F' B R F' U' B' R L' F U' D' F' B' R2 U (20q*, 19f) U F' B' R F2 U D' L2 F' U' D' F2 R' U' D R L' D2 (18f*, 22q) 11. F R D R' F' U B' L U' D L' F D' B R U' R' B' (18q*, 18f*) 12. F R' B R' L U' R L' U B R L' D' B' L F' (16q*, 16f) F R2 F2 U' D R' U2 D2 L' F2 L2 U' D F (14f*, 20q) 13. F R' L' U' F B' R' L F2 U2 F B D' R L U2 B' (20q*, 17f) F U2 F2 B2 R' L F2 U F2 B2 U F2 R L' D2 F (16f*, 24q) 14. U F B' U' R F' R' B R' U2 R' F R' B' R U' F' B U (20q*, 19f) U F2 R' L' F D2 R' L B2 D' F' B' R' U D' R' L U2 (18f*, 22q) 15. F B R F2 U' D R2 B' U' D' L' U D' R' L U R2 (20q*, 17f*) 16. F B U D R2 L2 U D F B (12q*, 10f*) 17. U2 D2 (4q*, 2f*) 18. U F B R2 U D F' B U' D' R2 F' B' D' (16q*, 14f) U F B D2 R2 D2 R2 F' B' R2 D2 R2 U (13f*, 20q) 19. U F' B' R2 U' D' F B' U D R2 F B D' (16q*, 14f) U F2 D2 F2 R L F2 D2 F2 D2 R' L' U (13f*, 20q) 20. F2 R2 F2 B2 R2 B2 U2 D2 (16q*, 8f*) 21. F2 B2 U D' R2 L2 U D' (12q*, 8f*) 22. U F B R' L' F B R' L' F B R' L' U' (14q*, 14f) U F B D2 R2 U2 R2 F' B' L2 D2 L2 U (13f*, 20q) 23. U F' B' R L F' B' R L F' B' R L U' (14q*, 14f) U F2 D2 F2 R L B2 U2 B2 D2 R' L' U (13f*, 20q) 24. F R' L' U' L2 B2 R L F' B R' L D' R L U2 F' (20q*, 17f) F U2 F2 B2 R L' B2 U' R2 L2 U F2 R L' D2 B' (16f*, 24q) 25. F R2 D F' U' R U2 D2 L' U B D' L2 B' (18q*, 14f*) 26. F U R' U L' U' L D R L U R D' R' D L' D B (18q*, 18f) F B R F2 U' D R2 B' U' D' R' U' D R L' D L2 (17f*, 20q) 27. F R' F' B L' F U' F B U2 F B U' R' B R L' F R' (20q*, 19f) U F' B' R F2 U D' L2 F' U' D' F2 R' U' D R L' U2 (18f*, 22q) 28. F U' R B R F B' U' F U' D L' D R L' B2 R' U F' (20q*, 19f) F2 R F B' D B2 U' R2 L2 D B2 U' F' B L' F2 (16f*, 22q) 29. F R2 U' R U F' U D' L U' F' U F2 L' U D' (18q*, 16f*) 30. F R' F' B L' F U F B U2 F B U L' F R' L B L' (20q*, 19f) F B' R F B D L2 F B' U2 L' U D R2 F2 B' R L' (18f*, 22q) 31. F B' U R D B' U D' R D' F' D' R B2 U' D' R' L' U (20q*, 19f) U F2 R' L' F D2 R' L B2 D' F' B' R' U D' R' L D2 (18f*, 22q) 32. U D F2 B2 U D' F2 B2 D2 (14q*, 9f) F2 R2 F2 B2 R2 F2 R2 L2 (8f*, 16q) 33. F2 B2 U D' F2 B2 U' D (12q*, 8f*) 34. F B R L F B R' L' F' B' R' L' U' F2 B2 R2 L2 D' (22q*, 18f) F B R F2 B2 U2 D2 L' F' B' U2 R' L' F2 R' L' U2 (17f*, 24q) 35. F R F L D F' L' F B' R' L F' B R F U L' B' L F (20q*, 20f) F2 U F B D2 R2 B2 D2 L2 F' B' U2 R2 D F2 (15f*, 24q) 36. U2 F2 B2 R2 L2 D2 (12q*, 6f*) 37. F2 B2 R2 L2 (8q*, 4f*) 38. F U' B' D' F2 U' F' U' D2 F U2 F' D' F U' B' L2 D' (22q*, 18f) F2 R2 L2 U R L F2 U D L2 U D B2 R L U B2 (17f*, 24q) 39. F2 U F B D2 F B R2 L2 D' B2 U' D' R2 U' D' (22q*, 16f*) 40. U F R U F' B2 R F D' R' F' L' F2 R D R L2 D2 (22q*, 18f) U2 F U2 R' L F2 U' B2 R' L D2 B' U2 D2 B D2 (16f*, 24q) 41. F U2 R' L F2 U' B2 R' L D2 B' U2 D2 B (20q*, 14f*) 42. F B R F2 U D' L2 F' U D L U' D R L' U' L2 (20q*, 17f*) 43. F B R F B' R' L B' U' D' R' U2 B2 U2 R L F B' U (22q*, 19f*) 44. F U D F B' U2 R' L B' R' L F B' R' U' D' F' (18q*, 17f) F R2 U D R U' D F2 R L' D R L' U2 R2 F' (16f*, 20q) 45. F R L U B2 R2 U D' R' L' U D' R U' D' L2 F' (20q*, 17f) F R2 U D' B2 R2 L F2 B2 L' B2 U' D R2 F (15f*, 22q) 46. F B R F B' R' L B' R2 U' D' R2 L U2 F B' R2 U (22q*, 18f*) 47. F R' F R2 L' B' D R' U' B L F' R2 B2 L2 D' F2 (22q*, 17f*) 48. F2 B2 U D' F2 B2 U D' (12q*, 8f*) 49. U D F2 B2 U D' F2 B2 U2 (14q*, 9f*) 50. F B R F2 B2 U2 R L B2 R L U2 L F B R2 (22q*, 16f) F B R F2 R2 B2 U2 L2 F2 R2 U2 L' F B L2 (15f*, 24q) 51. F B R' L' F B U F2 B2 R2 L2 D R L F' B' R L (22q*, 18f) F B R F2 B2 U2 D2 L' F' B' U2 R' L' F2 R' L' D2 (17f*, 24q) 52. U2 F2 B2 R2 L2 U2 (12q*, 6f*) 53. F B R F2 R' L' D2 F2 B2 R L B2 L F B L2 (22q*, 16f) F2 U F2 R U2 F2 B2 R2 L2 U2 L B2 D B2 (14f*, 24q) 54. F U' F L' F' D F' R' L U' D F R' L D R' D L' F' L U' L (22q*, 22f) F B R F2 B2 U2 D2 L' U2 R2 U2 F B U2 R2 U2 R2 (17f*, 28q) 55. F U2 R' L B2 U F2 B2 U' F2 R' L U2 B' (20q*, 14f*) 56. F U D R U' D R' L U F' B R2 F' B R' L U2 B' (20q*, 18f) F R F2 R U L D B2 U' R' D F2 B2 D2 L2 B' (16f*, 22q) 57. F R L U R' D' F' B' U L F' B' R F' B' D' F R U R (20q*, 20f) F U' L U L F2 D B2 U' D B D' R' B' L2 U' B L2 (18f*, 22q) 58. F B R F B' R' L B' U' D' L F B' D2 F2 L2 F2 U' (22q*, 18f*) 59. F R2 L2 F' U2 R' L B2 D F2 R L' U2 B' (20q*, 14f*) 60. F U D F B' U2 R' L B' R' L F B' R' U' D' F' U2 D2 (22q*, 19f) U2 F R2 L2 F' U2 R' L B2 D F2 R L' U2 B' D2 (16f*, 24q) 61. F B R F B' R' L B' U' D' L F B' D2 B2 L2 B2 U' (22q*, 18f*) 62. F B R F B' R' L B' R2 U' D' L D2 F B' L2 U' (20q*, 17f*) 63. F B U D R L F B U D R L (12q*, 12f*) 64. F B U D R L F' B' U' D' R' L' U2 D2 (16q*, 14f*) 65. U F2 R L' F B' U2 B2 R L D2 R2 D' (18q*, 13f*) 66. U F2 R' L F' B D2 B2 R' L' U2 R2 D' (18q*, 13f*) 67. F B U2 F B R L' F B D2 R' L' F2 U' D' (18q*, 15f) F B' R L F' B R2 B2 L2 U' D' L2 F2 R2 (14f*, 20q) 68. F B U D R L F B U D' R2 L2 D2 R' L' (18q*, 15f) F B' R L F' B R2 F2 R2 U D R2 B2 R2 (14f*, 20q) 69. U F2 R L F B R L U D F2 U' (14q*, 12f*) 70. U F2 R' L' F' B' R' L' U' D' F2 U' (14q*, 12f*) 71. F R' F' B R' B' L' F B U R' L D B U L D L D' R' (20q*, 20f) F U R F' B U2 D F' U' D R F U' D L' U' F' R2 L2 (19f*, 22q) 72. F R B D F U B R' L F' D' F B' R' B R2 D (18q*, 17f*) 73. U F' U D' F' B D F B R' U2 F' B L2 D' R' L' U (20q*, 18f) F B' R U2 R D2 R' U D' F R2 D2 F R' L' U2 D' (17f*, 22q) 74. F R2 U' D B2 L U' D' F B R2 B U D' F B' U R' L' (22q*, 19f*) 75. F U L U F' R' L' U B' L' F' U' D B' U' D' R2 U' D' (20q*, 19f*) 76. U F R L' F' B L' U' D' F D2 R L' F2 D R L U (20q*, 18f) U F B R' U2 F2 R' U D' B L2 F' R2 B' R' L D2 (17f*, 22q) 77. F2 R L' U2 B R' L' B2 D' R L' U' D R F B D (20q*, 17f*) 78. F B U D R L F' B' U' D' R' L' (12q*, 12f*) 79. F B U D R L F B U D R L U2 D2 (16q*, 14f*) 80. U F2 U D' F' B' R' L' F' B' D2 B2 U' (16q*, 13f*) 81. U F2 U D' F B R L F B U2 B2 U' (16q*, 13f*) 82. F U' B' L' B' U F U D' L D R' B' R' D' L D2 (18q*, 17f) F B' R L F' B R2 F2 R2 U' D' L2 F2 L2 (14f*, 20q) 83. F B U D R L F B U2 F2 B2 U D' R' L' (18q*, 15f) F B' R L F' B R2 B2 L2 U D R2 B2 L2 (14f*, 20q) 84. U F B R' L' U D F' B' U D F' B' U' (14q*, 14f) U F2 R' L' F' B' R' L' U' D' F2 U D2 (13f*, 16q) 85. U F B R L U' D' F B U' D' F' B' U' (14q*, 14f) U F2 R L F B R L U D F2 U D2 (13f*, 16q) 86. F R2 U' D B2 L F' B' U' R L' U' D R' B2 R L U' D' (22q*, 19f*) 87. F R2 U' D B2 L U' D' F B R2 F U' D F' B D R' L' (22q*, 19f*) 88. U F' R U L U' R F' U' R U R L' B' R' L' F' D (18q*, 18f) F B' R U2 R D2 R' U D' F R2 D2 F R' L' D (16f*, 20q) 89. F R' F' R U D' F' D' F' R' U' L' B' U L' F' D' L (18q*, 18f) U F B R' U2 F2 R' U D' F U2 F' D2 F' R' L U2 (17f*, 22q) 90. F U L U F' R' L' U B' L' F' U' D B' U' D' R2 U D (20q*, 19f) F U R D' R2 U2 F U D B D2 L2 U' L D B R L (18f*, 22q) 91. U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f) F U2 R L D F2 U R2 D F2 D F' B' U2 L F2 R2 B2 U' D (20f*, 28q) 92. F B' R F2 L B2 R' U' D F L2 U2 F R' L' U (20q*, 16f*) 93. F B' U2 L F B L2 U F B' U' D B' R' L' U R2 (20q*, 17f*) 94. F B U D R' L' U2 F' B' U' D' R L U2 (16q*, 14f*) 95. F B U D R' L' U2 F B U D R' L' D2 (16q*, 14f*) 96. F B R L F B R2 U' F2 B2 R2 L2 D' R2 U' D' (22q*, 16f*) 97. F R' B L U B R U D L F D R F L' B R2 L2 (20q*, 18f) F2 R L F B R L U D F2 U' F2 B2 R2 L2 D' (16f*, 22q) 98. F B U D R' L' F' B' U D R L U2 D2 (16q*, 14f*) 99. F B U D R' L' F B U' D' R' L' (12q*, 12f*) 100. F R L B' R' L' B U' D' F' B' U' D' F' R' L' F R L B' (20q*, 20f) F B R L F B U D' F2 U' F2 B2 R2 L2 D' F2 U2 (17f*, 24q) 101. F B R F' B' R' L F B R L U D F B R F B L2 (20q*, 19f) F2 U F B' U2 R L F B' L2 F2 U' D' B2 D' F2 (16f*, 22q) 102. F U D' F U D' R2 U2 L' F2 U D' L2 F R L U B' (22q*, 18f*) 103. F R B L D F' B L' U' D F' B2 U D' B' D' R' B' U' (20q*, 19f*) 104. F R' U' D' R D' F' D' R U' D' L U' B' U' L U' D' L' B (20q*, 20f) F B' R D2 L F2 B2 R' F2 R' U' D B U2 F2 R' L D' (18f*, 24q) 105. F B' U2 R F B R2 U' D2 F B' U' D B' R' L' U L2 (22q*, 18f*) 106. F R U' D2 F' B R' B' R L' D B R L' U' L' F' (18q*, 17f*) 107. F U2 R' L F2 U R L F' B' D2 F' R' L F B' R' U D (22q*, 19f*) 108. F R L U' R U2 B R' F L B' L' B L D' F B' U' F D' F' (22q*, 21f) F B' R U D' B2 R2 B' R L' D F2 D F2 B2 U' L2 D' (18f*, 24q) 109. F B' U2 L F' B' D' F' B U' D B U2 R L D L2 (20q*, 17f*) 110. F B U D R' L' U2 F B U D R' L' U2 (16q*, 14f*) 111. F B U D R' L' U2 F' B' U' D' R L D2 (16q*, 14f*) 112. F2 U R L' D2 F B R L' B2 R2 U' D' L2 D' B2 (22q*, 16f*) 113. F B R L F B R2 U' F2 B2 R2 L2 D' R2 U D (22q*, 16f*) 114. F B U D R' L' F B U' D' R' L' U2 D2 (16q*, 14f*) 115. F B U D R' L' F' B' U D R L (12q*, 12f*) 116. F B R F B R L' F' B' R' L' U D F' B' L F B R2 (20q*, 19f*) F2 U F B' U2 R L F B' L2 F2 U D F2 D' B2 (16f*, 22q*) 117. F B R L F B U D' F2 U' F2 B2 R2 L2 D' F2 D2 (24q*, 17f*) 118. F B R F B' R' L F L2 F B U' D' L D2 F B' L2 D (22q*, 19f*) 119. F B R2 F B U' F B' D' L' U' B R' L' D' B L B D (20q*, 19f*) 120. F B' U2 R F B R2 U F' B U' D F' R' L' U R2 (20q*, 17f*) 121. F R L' U' R2 D B' L' B' D B' U' F D B' D R U R' U' D (22q*, 21f) F B' R U D' F L2 F R2 L2 B' D2 F' R L' U L2 U2 (18f*, 24q) 122. F R L' B R L' U2 R2 D' F2 R' L D2 F U D R F' (22q*, 18f*) 123. F B' U2 R F B R2 D F B' U D' F2 B R' L' U R2 (22q*, 18f*) 124. F R' D' R F R' L2 U B U' R' U R' L' U' D' F' B' D F2 (22q*, 20f) F B R' F2 U2 R' U' D B R2 B' U2 F' U2 D2 R L' D (18f*, 24q) as usual, when there are two maneuvers are given, this means that no maneuver is simultaneously minimal in both metrics. superflip composed with four spot is #91. many of these are locally maximal. local maxima in the quarter turn metric: #1, 4, 8, 9, 10, 15, 20, 21, 24, 25, 27, 29, 30, 31, 33, 34, 38, 39, 40, 41, 42, 43, 46, 47, 48, 50, 51, 53, 55, 58, 60, 61, 62, 64, 67, 68, 71, 72, 73, 75, 76, 77, 79, 82, 83, 86, 90, 91, 92, 93, 94, 95, 96, 98, 102, 103, 105, 107, 108, 110, 111, 112, 113, 114, 117, 118, 119, 121, 122, 123, 124. (strong) local maxima in the face turn metric: #8, 10, 11, 16, 29, 30, 31, 34, 38, 39, 43, 46, 51, 54, 58, 61, 64, 71, 72, 75, 77, 79, 86, 91, 93, 94, 95, 98, 100, 103, 104, 107, 110, 111, 114, 117, 118, 119, 121. mike From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 19 22:57:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id WAA17267; Wed, 19 Aug 1998 22:57:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Aug 19 17:10:15 1998 Date: Wed, 19 Aug 1998 17:09:54 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: minimal maneuvers for X symmetric positions In-Reply-To: <199808182028.QAA24353@euclid.math.brown.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Tue, 18 Aug 1998 16:28:28 -0400 michael reid wrote: > X is the subgroup of the cube symmetry group which preserves > the U-D axis. there are 128 positions which have X symmetry: > > the UR edge can go in any of the 8 positions UR, RU, DR, > RD, UL, LU, DL, LD; this determines the location of the > edges UB, UL, UF, DR, DB, DL, DF. > > the FR edge can go in any of the 4 positions FR, RF, BL, LB; > this determines the location of the edges FL, BR, BL. > > the UFR corner can go in any of the 4 positions UFR, UBL, > DRF, DLB; this determines the location of all the corners. > > any combination of these is possible, which gives 128 positions. > 4 of the positions have more symmetry, namely M symmetry. > (these positions are start, superflip, pons asinorum, and > pons asinorum composed with superflip.) > > minimal maneuvers for the other positions are: > > 1. F2 R L' D2 F2 D2 R' L F2 D2 (16q*, 10f*) > > 2. U F' B' R2 U' D' F' B U D R2 F B D' (16q*, 14f) > U F2 U2 F2 R L F2 U2 F2 U2 R' L' U (13f*, 20q) > I don't think Mike has said so explicitly, but he appears to have adopted a very useful convention from Herbert Kociemba's Cube Explorer 1.5. To wit, Cube Explorer 1.5 flags the length of a maneuver with an asterisk when the length has been shown to be minimal. Cube Explorer 1.5 operates only in face turns, so it omits the q or f designation of units. But for example, Cube Explorer 1.5 might show the length of a cube upon which it is operating as (13) meaning 13f, then later in the search show the length as (12), and still later show the length as (12)* to show that 12 face moves have been proven to be minimal. The only difference between Mike's style and Cube Explorer's style is that Cube Explorer puts the asterisk outside the parentheses. I loaded Mike's E-mail into Cube Explorer to take a quick look at the X symmetric positions. Many of them are familiar to readers of this list, and all of them are quite pretty. (Loading Mike's E-mail into Cube Explorer "just worked". I didn't have to edit it at all to remove extraneous text. Cube Explorer's maneuver reader seems to have a remarkable ability to extract maneuvers in BFUDLR notation which are imbedded in other extraneous text.) If you have Cube Explorer 1.5 (and you should!), I would encourage you similarly to load Mike's X symmetric patterns into it and take a look. The patterns look as expected for patterns which preserve the U-D axis. The U and D faces are the same pattern. The F, R, B, and L faces are the same pattern and may be described as being in the same orientation with respect to rotations of the square. For positions #1 through #62, the U and D faces may be described as being symmetric with respect to the symmetries of the square. They range from being solid, to having one dot, to being a +, to being an X, etc. All are glyphs. Positions #63 through #124 are essentially the first 62 positions composed with superflip. I had never noticed it, and I don't *think* it has been described on the list, but for every symmetry group, half of the corresponding positions can be described as "basic" positions and the other half can be described as the basic positions composed with superflip. That is, if Symm(x)=K, then Symm(xf)=Symm(fx)=K, where x is any position and f is the superflip. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 19 23:44:32 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id XAA17423; Wed, 19 Aug 1998 23:44:31 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Aug 19 23:34:51 1998 Date: Wed, 19 Aug 98 23:34:34 EDT Message-Id: <19Aug1998.231259.Hoey@AIC.NRL.Navy.Mil> From: Dan Hoey To: jbryan@pstcc.cc.tn.us Cc: reid@math.brown.edu, cube-lovers@ai.mit.edu In-Reply-To: (message from Jerry Bryan on Wed, 19 Aug 1998 17:09:54 -0400 (Eastern Daylight Time)) Subject: Re: minimal maneuvers for X symmetric positions Jerry Bryan writes: > Positions #63 through #124 are essentially the first 62 > positions composed with superflip. I had never noticed it, and > I don't *think* it has been described on the list, but for every > symmetry group, half of the corresponding positions can be > described as "basic" positions and the other half can be > described as the basic positions composed with superflip. That > is, if Symm(x)=K, then Symm(xf)=Symm(fx)=K, where x is any > position and f is the superflip. This is easy to see if we consider that Symm(x) is the set of all m in M that commute with x, because m' x m = x if and only if x m = m x. At some times since 1981 I've wondered whether symmetry discussions are better done with commutativity rather than conjugacy. So if c is any element of the center of G* -- i.e., c commutes with all elements of M and G -- then Symm(x)=Symm(x c). As is well known to cube-lovers, the center of the usual cube group consists of the identity and the superflip. In the supergroup, we may also compose these with Big Ben (all face centers rotated 90 degrees) and Noon (Big Ben squared). Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 20 14:36:11 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA20857; Thu, 20 Aug 1998 14:36:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Aug 20 11:37:31 1998 Date: Thu, 20 Aug 1998 11:37:14 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: minimal maneuvers for X symmetric positions In-Reply-To: <199808182028.QAA24353@euclid.math.brown.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Tue, 18 Aug 1998 16:28:28 -0400 michael reid wrote: > local maxima in the quarter turn metric: > > #1, 4, 8, 9, 10, 15, 20, 21, 24, 25, 27, 29, 30, 31, 33, 34, 38, > 39, 40, 41, 42, 43, 46, 47, 48, 50, 51, 53, 55, 58, 60, 61, 62, > 64, 67, 68, 71, 72, 73, 75, 76, 77, 79, 82, 83, 86, 90, 91, 92, > 93, 94, 95, 96, 98, 102, 103, 105, 107, 108, 110, 111, 112, 113, > 114, 117, 118, 119, 121, 122, 123, 124. > > (strong) local maxima in the face turn metric: > > #8, 10, 11, 16, 29, 30, 31, 34, 38, 39, 43, 46, 51, 54, 58, 61, 64, > 71, 72, 75, 77, 79, 86, 91, 93, 94, 95, 98, 100, 103, 104, 107, 110, > 111, 114, 117, 118, 119, 121. I am curious how the local maxima were determined. 4-spot composed with superflip was based on sort of an "extended symmetry" argument, but what about all the others? If I had to guess, I would suspect that you found all minimal maneuvers for each position and observed that there was a maneuver terminating with each quarter (respectively, face) turn for each position. Or equivalently, perhaps you found all minimal maneuvers unique to symmetry for each position and observed that conjugation of the maneuvers would yield a maneuver terminating with each required kind of turn. Was it something like this? (All you would really need for the conjugation argument, since you already know that the maneuvers in question preserve the U-D axis, would be to find at least one minimal maneuver ending with any of {U, U', D, D'} and to find another minimal maneuver ending with any of {R, R', F, F', L, L', B, B'}.) It is interesting that you found strong local maxima in the face turn metric, rather than just "plain" local maxima. In my experience, finding strong local maxima with a computer search is easier than finding "plain" local maxima. Finding "plain" local maxima includes finding weak local maxima (where at least one face turn does not change the distance of the position from Start). If my guess about how you are identifying local maxima is correct, then your method would not identify weak local maxima. Finally, I have mused previously to Cube-Lovers that strong local maxima in the face turn metric may be extremely rare. I think I might be wrong. My God's algorithm searches in the face turn metric have already turned up more strong local maxima than I expected, and your search of the X-symmetric positions turned up more strong local maxima than I would have expected. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 20 18:08:04 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id SAA21659; Thu, 20 Aug 1998 18:08:03 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Aug 20 18:06:49 1998 Date: Thu, 20 Aug 98 18:06:30 EDT Message-Id: <9808202206.AA13431@sun28.aic.nrl.navy.mil> From: Dan Hoey To: jbryan@pstcc.cc.tn.us Cc: reid@math.brown.edu, cube-lovers@ai.mit.edu Subject: Group centers (oops) I wrote: > So if c is any element of the center of G* -- i.e., c commutes > with all elements of M and G -- then Symm(x)=Symm(x c). As is > well known to cube-lovers, the center of the usual cube group consists > of the identity and the superflip. In the supergroup, we may also > compose these with Big Ben (all face centers rotated 90 degrees) and > Noon (Big Ben squared). In short, I should not have included Big Ben in that paragraph, only Noon. The long explanation is that both of these are in the center of the usual supergroup, as is any position that differs from Solved only by face center orientation. But for the Symm(x)=Symm(x c) argument to work, c must be in the center of the group generated by the union of the supergroup with M. This is equivalent to saying that c must be in the center of the supergroup and be M-symmetric. Big Ben is only C-symmetric. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 21 23:33:56 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id XAA27682; Fri, 21 Aug 1998 23:33:44 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Aug 21 23:09:47 1998 Date: Fri, 21 Aug 1998 23:07:24 -0400 From: michael reid Message-Id: <199808220307.XAA10899@cauchy.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for E symmetric positions E is the subgroup of cube symmetries consisting of rotations (no reflections) that preserve the tetrad of corners UFR, UBL, DFL and DBR. of course it preserves the other tetrad as well. there are 72 positions that have E symmetry: each corner must remain in place, but can be twisted. corners in the same tetrad must be twisted in the same direction; therefore, by conservation of twist, adjacent corners are twisted in opposite directions. the UR edge can go in any location in any orientation. this determines the location and orientation of all edges. this gives 3 * 24 = 72 positions. if the UR edge remains in the F-B slice, then the position has more symmetry, namely H symmetry (at least). this accounts for 24 of the 72 positions; 20 of which are H symmetric, and 4 of which are M symmetric. E is a normal subgroup of M; in fact, it's the commutator subgroup. therefore, any M conjugate of an E symmetric position is also E symmetric. the 48 remaining positions form 12 equivalence classes under M conjugacy, of 4 positions each. minimal maneuvers for these are 1. F' B' U R' U' D R L' U R' D' L U' D R L' D' L F' B' (20q*, 20f) F R L F U' D' F R2 L2 U2 D2 B R' L' B U D B (18f*, 22q) 2. F' R2 U D R' L' U' F B R2 F' U' D' F' B U' D' B' (20q*, 18f*) 3. F' B' R' L' F B U D R' L' U' D' (12q*, 12f*) 4. F U2 R F B R' U2 B R L B U' D' R' L U' L2 (20q*, 17f*) 5. F R' F L U D' L' U R U' D F' D' B U D' B' R' F R' (20q*, 20f) F2 U D2 L2 F U D F' L2 U F B U R' L' F' B R (18f*, 22q) 6. F B R' L' F B U' D' R L U' D' (12q*, 12f*) 7. F R U2 F L B U D' F L D F' B D R B D (18q*, 17f*) 8. F U2 F' B D B' L B L D' B2 R' D F D F' R D2 F' (22q*, 19f) F' R2 D2 F U' D F2 R B2 L F2 R D2 F B' U' F' B' (18f*, 24q) 9. F' R F U R2 U F' L' D F U2 R' F' B2 R' D' R' B' U' (22q*, 19f*) 10. F R2 B' L' U' L' F' B D B R B R F U' R U2 D2 F' (22q*, 19f*) 11. F B R2 U' F L F U' F B' R F' U' B' R L' B' R L (20q*, 19f) F R2 U D' F2 L' U D L2 B U D' F B' U' D2 F B (18f*, 22q) 12. F R' D B U D F' L B D R' F R' L B' U' F' B' L F' D L (22q*, 22f) F B U F L2 D2 F2 B R F B R F' L2 D2 R2 U2 B U (19f*, 26q) as usual, i give a maneuver that is simultaneously minimal in both metrics, unless one does not exist. some of these are local maxima. local maxima in the quarter turn metric: #1, 4, 7, 8, 9, 10, 11, 12. (strong) local maxima in the face turn metric: #10, 11, 12. mike From cube-lovers-errors@mc.lcs.mit.edu Sun Aug 23 01:09:13 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id BAA00571; Sun, 23 Aug 1998 01:09:03 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Aug 22 20:16:23 1998 Date: Sat, 22 Aug 1998 19:27:50 -0400 From: michael reid Message-Id: <199808222327.TAA13228@cauchy.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: Re: minimal maneuvers for X symmetric positions jerry asks > I am curious how the local maxima were determined. 4-spot > composed with superflip was based on sort of an "extended > symmetry" argument, but what about all the others? > > If I had to guess, I would suspect that you found all minimal > maneuvers for each position and observed that there was a > maneuver terminating with each quarter (respectively, face) > turn for each position. Or equivalently, perhaps you found all > minimal maneuvers unique to symmetry for each position and > observed that conjugation of the maneuvers would yield a > maneuver terminating with each required kind of turn. Was it > something like this? yes, this is essentially what i did. i added automatic symmetry reduction to my program (this was a challenge to program, but it makes things so much more convenient). so now the program finds all minimal maneuvers up to symmetry, from which local maxima can be spotted easily. i did not find all minimal maneuvers for #91 (superflip composed with four spot) nor for #117 in the quarter turn metric, because these are too far from start (26q, 24q respectively). so for these positions, which are locally maximal, it suffices to find minimal maneuvers ending with each quarter turn. as you see, symmetry is helpful here. also, all the X symmetric positions have order 2, so any maneuver can be inverted. this is also helpful. > It is interesting that you found strong local maxima in the face > turn metric, rather than just "plain" local maxima. In my > experience, finding strong local maxima with a computer search > is easier than finding "plain" local maxima. Finding "plain" > local maxima includes finding weak local maxima (where at least > one face turn does not change the distance of the position from > Start). If my guess about how you are identifying local maxima > is correct, then your method would not identify weak local > maxima. yes, this is exactly correct. i will leave it to someone who's more interested in "weak" local maxima to determine those. mike From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 24 14:00:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA02163; Mon, 24 Aug 1998 14:00:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Aug 23 14:41:17 1998 Message-Id: From: Noel Dillabough To: "Cube-Lovers (E-mail)" Subject: New version of Puzzler Date: Sun, 23 Aug 1998 14:30:24 -0400 The latest version of Puzzler can be found at http://www.mud.ca/puzzler/puzzler.html. In addition to the bug fixes that were put in, I have added the most requested features, the ability to take back moves, and the ability to enter move macros for cube puzzles in standard cubist (UDFBLR) notation. For those who haven't used the program before, Puzzler is a collection of sequential movement puzzles including the cubes (all sizes from 2x2x2 to 5x5x5), the pyraminx, the impossiball, the skewb, and the megaminx puzzle. Enjoy, -Noel From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 24 14:37:46 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA02331; Mon, 24 Aug 1998 14:37:46 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Aug 22 20:18:25 1998 Message-Id: <35DF587E.6C0D@ameritech.net> Date: Sat, 22 Aug 1998 18:47:10 -0500 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: designs from Rubik's cubes Hi, fellow cube-lovers I am a recent member of the cube-lovers mailing list. I would like to help me answer this question: has any of you constructed, or does anyone of you know someone who has constructed, a composite, pleasant, geometrical design on a set of Rubik's cubes? The design is a three dimensional {but not necessarily cubical} structure that exhibits some symmetry on all its faces. Such designs are quite different from the picture-like structures built by Jacob Davenport. I saw his pictures when I consucted an ongoing web search to answer the above question. So far I was not successful, and so I seek your help. I am the author of these designs. I know they can be done because I have done them. I do not have a web page of my own yet, but a friend of mine kindly offered to put three of these designs on his web page. They may be seen at http://www.ssie.binghamton.edu/~jirif/cube.html. This should open my friend's speed cubing page. My designs are there under the heading "Hana Bizek's cube art." Yes, Jirka Fridrich is a speed cubist, which is an art in itself. You will find other interesting things there, including a signature of Erno Rubik. A photo of a design has one flaw; you can only see three faces of the design. What does the rest of the design look like? Answer: sometimes opposite faces of the design are exactly identical, both in color and geometrical pattern. One of my designs in Jirka's page, the so-called ctyrsprezi design, is such a design. It has four colors only on its six faces. Why this should be so is a cornerstone of the design theory. The reason is explained in my book,"Mathematics of the Rubik's cube design," published last year. amazon.com has it online. Well, O better end this message, or it will itself deberate into a book. Any help you can offer in my search for a "cube sculptor" will be gratefully appreciated. And of course I stand by these designs and will answer any questions. My name is Hana M. Bizek and my email address is hbizek@ameritech.net. Thank you very much. I will be looking forward to hearing from you. Best regards, Hana From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 24 15:32:46 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id PAA02566; Mon, 24 Aug 1998 15:32:45 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Sun, 23 Aug 1998 23:19:35 -0400 (EDT) From: Jerry Bryan Subject: "Basic" vs. Superflipped Positions To: Cube-Lovers Message-Id: I recently commented on the fact that half of Mike Reid's X-symmetric positions were "basic" and the other half were superflipped versions of the first half. I further commented that this was true for all positions associated with any symmetry group -- half are "basic", and the other half are superflipped versions of the first half. Well, I am not quite sure that this is true in general. Or more correctly, I am not sure you can always tell the "basic" version apart from the superflipped version. Consider any two positions x and xf, where f is the superflip. We would say that x is the "basic" position and xf is the superflipped position. But if we define y=xf, then y is the "basic" position and yf is the superflipped position, and it is also true that x=yf. So which is the "basic" position, x or y? It appears that there is no way to tell. Yet when you look at X-symmetric positions, it is trivial for the eye to see which ones are "basic", and which ones are superflipped. So what is going on here? I briefly (*very* briefly) hoped to find a unique subgroup H of G with index 2 which did not contain superflip. Then, it would have been natural to call H the "basic" positions and Hf the superflipped positions. But it is well known to Cube-Lovers that the only subgroup of G with index 2 is the subgroup consisting of those positions where are an even number of quarter turns from Start. And this subgroup does contain the superflip. Therefore, there seems to me to be little possibility of a general way to distinguish between "basic" positions and superflipped positions. Upon further reflection, it seems to me that there is a natural way to tell "basic" positions apart from superflipped positions for some symmetry groups but not for others. I have not examined all 98 symmetry groups (33 symmetry classes) of the cube in this regard, but I have looked at a few of them, and can give a few examples. Before looking at examples, we need to look at a subtle but important point. We may think of a position x as consisting of corners and edges separately, so that x=x[c]*x[e]. Similarly, we may look at the symmetry of the corners and the edges separately, as in Symm(x[c]) and Symm(x[e]). The equation that relates the symmetries is Symm(x)=Symm(x[c]*x[e])=Symm(x[c]) intersect Symm(x[e]). But because Superflip affects only the edges, we need consider only Symm(x[e]) when we compare "basic" positions to superflipped positions. Example 1. Suppose Symm(x[e])=M. Then it seems natural to view the position as "basic" if all four edge facelets on each face are the same color, and to view the position as superflipped otherwise. The eye sees this distintion very clearly. Example 2. Suppose Symm(x[e])=X1. X1 is the symmetry group in Dan Hoey's taxonomy which preserves the U-D axis. X2 and X3 are conjugate subgroups preserving the F-B and R-L axes, respectively. X is the symmetry class consisting of X1, X2, and X3. All of Mike Reid's X-symmetric positions are in particular X1-symmetric. For X1, it seems natural to view the position as "basic" if all four edge facelets on the U and D faces are the same color, and to view the position as superflipped otherwise. For X2, the same rule would apply to the F and B faces. FOr X3, the same rule would apply to the R and L faces. The eye sees this distinction very clearly. Example 3. Suppose Symm(x[e])={i,v}, where v is the central inversion. For such a position, any particular edge cubie could be placed anywhere, but each edge cubie would have to be placed diametrically opposite its diametrically opposed edge cubie. For example, if cubie uf were placed in the rd cubicle, then cubie db would have to be placed in the lu cubicle, etc. Also, for Symm(x[e]) to be {i,v} the edges could not have any additional symmetry. In this case, I don't think there is any natural way to distinguish between a "basic" position and a superflipped position. Example 4. Suppose Symm(x[e])=I={i}. In other words, the edges have no symmetry. In this case, I don't think there is any natural way to distinguish between a "basic" position and a superflipped position. Example 5. Suppose Symm(x[e])={i,c_u2}, where c_u2 is a 180 degree whole cube rotation around the U-D axis. In this case, the position would be "basic" if opposite edge facelets on the U face were the same color and if opposite edge facelets on the D face were the same color, and would be superflipped otherwise. The eye sees this distinction very clearly. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 24 17:55:26 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA04071; Mon, 24 Aug 1998 17:55:26 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Mon, 24 Aug 1998 10:05:52 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: minimal maneuvers for E symmetric positions In-Reply-To: <199808220307.XAA10899@cauchy.math.brown.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Fri, 21 Aug 1998 23:07:24 -0400 michael reid wrote: > E is the subgroup of cube symmetries consisting of rotations > (no reflections) that preserve the tetrad of corners UFR, > UBL, DFL and DBR. of course it preserves the other tetrad as > well. there are 72 positions that have E symmetry: > > each corner must remain in place, but can be twisted. > corners in the same tetrad must be twisted in the same > direction; therefore, by conservation of twist, adjacent > corners are twisted in opposite directions. > > the UR edge can go in any location in any orientation. > this determines the location and orientation of all edges. > There are generally several different (equivalent) ways to characterize a subgroup of the cube symmetries. For example, of the 48 symmetries, 24 of them are even and 24 of them are odd, and 24 of them are rotations and 24 of them are reflections. The E symmetries may be characterized as the intersection of the even symmetries with the rotational symmetries, and hence consist of the 12 even rotations. The 12 even rotations consist of the identity, the three 180 degree rotations around the face axes (c_u2 around the U-D axis, c_f2 around the F-B axis, and c_r2 around the R-L axis), and the eight 120 degree rotations around the four major diagonal axes (c_urf and c_ufr; c_ufl and c_ulf; c_ulb and c_ubl; and c_ubr and c_urb). It is the eight major axis rotations which give E its tetradic nature. In addition to the characterizations of the E positions which Mike gave (the corners must stay home, perhaps twisted, etc.), we can describe the E positions informally by the appearance of the faces. Each face must have the same pattern as its opposite face, and each pattern must have the 180 rotational symmetry of the square. The hardest part (to me, at least) in thinking about what a position x with Symm(x)=E must look like is to subtract out or ignore those positions which are E-symmetric but which have more symmetry. Indeed, many of the Symm(x)=E positions look very much like slightly broken versions of positions with stronger symmetry. For example, #3 and #6 look like slightly broken 6-spots. #7, #10, and #12 look like slightly broken 6-H's. #1, #2, and #4 look like slightly broken Pons Asinorums. Etc. This visual effect is the strongest if your cube adopts the "opposite faces differ by yellow" convention, so that white is opposite yellow, green is opposite blue, and red is opposite orange. Your eye will then tend to identify white with yellow, green with blue, and red with orange. With these identifications having taken place, most (if not all) of the Symm(x)=E positions look exactly like positions with more symmetry. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 26 12:59:48 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id MAA15433; Wed, 26 Aug 1998 12:59:48 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Wed, 26 Aug 1998 13:20:41 +0100 From: David Singmaster Organization: Computing & Maths South Bank Univ To: cube-lovers@ai.mit.edu Message-Id: <009CB47C.C9309B92.16@ice.sbu.ac.uk> Subject: depicting a cube With reference to Hana Bizek's reference to how one can show all six faces of a cube, I found the following the most satisfactory. View the F, U and R faces along the diagonal. Now imagine the back faces 'exploded' out, i.e. moved outward along the axes. When they are moved far enough, they can be seen. The effect is that the cube seems to be suspended in front of a corner and the three back seem to have been projected onto the walls and floor. I'll try to make a drawing. /| |\ / | / \ | \ / | / \ | \ | / / \ \ | | / |\ /| \ | |/ | \ / | \| | \ / | \ | / \ | / \|/ / \ / \ / \ \ / \ / \ / This is a bit crude, but it may be better when printed? if one puts in more horizontal space, it might look better. /| |\ / | / \ | \ / | / \ | \ | / / \ \ | | / |\ /| \ | |/ | \ / | \| | \ / | \ | / \ | / \|/ / \ / \ / \ \ / \ / \ / Well, that's a bit better, but one can't get it perfect on an orthogonal grid. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 27 21:08:23 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id VAA26187; Thu, 27 Aug 1998 21:08:23 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <35E5E400.6033@ameritech.net> Date: Thu, 27 Aug 1998 17:56:00 -0500 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: Re: depicting a cube References: <009CB47C.C9309B92.16@ice.sbu.ac.uk> David Singmaster wrote: > With reference to Hana Bizek's reference to how one can show > all six faces of a cube, I found the following the most > satisfactory. View the F, U and R faces along the diagonal. Now > imagine the back faces 'exploded' out, i.e. moved outward along the > axes. When they are moved far enough, they can be seen. The effect > is that the cube seems to be suspended in front of a corner and the > three back seem to have been projected onto the walls and floor. A mirror can be placed on those walls and floor so that the design's B, L and D faces can be reflected off those surfaces. The design would need to stand on a glass-topped table, so that the D face can be reflected off the mirrorred floor. The whole setup could be photographed. Unfortunately, I do not have resources to implement this. I don't even own a glass-topped table! Here is a challenge for the programmers out there. Can you write an applet that will slowly rotate my design in order for a viewer to see F, B, R, L and U faces, then tilt it upward to expose the D face? Do these moves any way you want, just make sure a viewer can see it all.Thank tou very much. You can find three of my designs at http://www.ssie.binghamton.edu/~jirif/cube.html. Two designs there are cubical. Opposite faces are identical, both in color and geometrical pattern. To wit: e. g. F face is exactly identical to B face, etc. This property holds for a majority of these designs, but there are exceptions. Hana Bizek {hbizek@ameritech.net} From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 27 22:49:16 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id WAA26478; Thu, 27 Aug 1998 22:49:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <006a01bdd226$d0495920$551a2bcb@mercury> Reply-To: "Bill Webster" From: "Bill Webster" To: Subject: Re: Depicting a cube Date: Fri, 28 Aug 1998 11:54:36 +1000 Hana Bizek wrote: >Unfortunately, I do not have resources to implement this. I don't even >own a glass-topped table! >Here is a challenge for the programmers out there. Can you write an >applet that will slowly rotate my design in order for a viewer to see F, >B, R, L and U faces, then tilt it upward to expose the D face? Do these >moves any way you want, just make sure a viewer can see it all.Thank you >very much. If static, generated images are acceptable, (i.e. if the pattern is more important to your sculpture than its physical realisation in plastic), you could achieve this reasonably easily with a ray-tracer - build the cube model and situate it in a scene with three plane mirrors, or perhaps models of some other reflecting objects for enhanced aesthetics - perhaps even a glass topped table! A free ray tracer is available at www.povray.org I have some source and sample images for cube models if you are interested. Cheers, Bill Webster (haddock@bluep.com) From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 28 07:25:33 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id HAA27336; Fri, 28 Aug 1998 07:25:32 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <3.0.32.19980828092911.009738f0@mail.spc.nl> Date: Fri, 28 Aug 1998 09:29:12 +0200 To: cube-lovers@ai.mit.edu From: Christ van Willegen Subject: Re: Depicting a cube Bill Webster (haddock@bluep.com) wrote: >If static, generated images are acceptable, (i.e. if the pattern is more >important to your sculpture than its physical realisation in plastic), >you could achieve this reasonably easily with a ray-tracer.... Hi, my first mailing on this list. I've seen some stuff, and thought it too hard (at the moment). _This_ discussion, however, I can handle! Another way to do this is in VRML. It's quite easy to build up a model of a cube in 3D, including colors. For a sample of what can be done with VRML in combination with a computer program to generate the stuff, go to: http://www.iaehv.nl/users/richtofe/ Follow the link about 'Triplets'. These are 3D models, inspired by Douglas Hofstadter. I have included some examples on that page, as well. It wouldn't be hard to do the same for a cube model. Writing a text file with LRUDTB and ' in it to describe the model, then and generating VRML is not too hard. This has the advantage over PovRAY that you can really rotate the model in space, and look at it in all directions. The mirror idea is nice, but it will mess up the design (Left-Right swap). More thoughts/ideas? Christ van Willegen From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 31 17:11:38 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA10141; Mon, 31 Aug 1998 17:11:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <35E8A9ED.72BA@ameritech.net> Date: Sat, 29 Aug 1998 20:25:01 -0500 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: Re: Depicting a cube References: <006a01bdd226$d0495920$551a2bcb@mercury> Bill Webster wrote: > > If static, generated images are acceptable, (i.e. if the pattern is more > important to your sculpture than its physical realisation in plastic), Oh no! Please remember that those are Rubik's cubes. Their "physical realization" is usually that they are stacked together to form larger cubes. If *that* was all to the design problem, I wouldn't have the nerve to make a posting to the cube-lovers msiling list. Some of its members are first-class mathematicians. I feel that an explanation of what I call the design problem is in order. Ther goal of this problem is to create, by conventional cube manipulation, a composite pleasant geometrical design on a set of Eubik's cubes. The basic algorithm consists of three simple steps: 1} construct patterns on individual cubes 2} make sure that the colors match properly from cube to cube {color control} 3} stack the cubes together. You start with a set of solved cubes. If you have scrambled cubes, you need to solve them. That is just one excellent reason why you *must* solve the Rubik's cube comopletely. Being able to solve only one side is woefully inadequate. Don't forget color control. Without it you don't have a design. This unavoidable aspect of the design problem further complicates the design algorithm. It is a little bit like chess. You try to consider two or three moves ahead of your opponent to achieve a winning strategy..,. or create a viable design from a set of Rubik's cubes. The last step is easy. It is sort of like a three-dimensional jigsaw puzzle. The patterned cubes you constructed are part of this jigsaw. Here, in a nutshell, is a description of the design problem. Please, get our your Rubik's cubes and start twiddling. Good luck, Hana From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 1 10:26:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id KAA13649; Tue, 1 Sep 1998 10:26:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Tue, 01 Sep 1998 09:59:50 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Strong Local Maxima 9f and 10f from Start To: Cube Lovers Message-Id: #1. D2 L2 B2 F2 U2 B2 F2 U2 R2 U2 (10f*) #2. D2 F2 L2 D' U L2 F2 D' U' (9f*) #3. U2 B2 L2 D U' R2 B2 D' U' (9f*) #4. B2 D' U' L2 B2 L2 D' U' B' F' (10f*) #5. L2 U2 F2 L2 D2 F2 U2 R2 B' F' (10f*) #6. D' U' B2 R2 D2 L2 D U B' F' (10f*) #7. D U L2 D2 R2 F2 D' U' B' F' (10f*) #8. B2 F2 D U' B' F L R' D U' (10f*) This completes the list of strong local maxima 9f and 10f from Start in the face turn metric. I posted #1, #2, and #3 previously, but the rest are new. 9f is the shortest strong local maximum. I continue to think that all eight of these positions share a special kind of symmetry that is related to the fact that they are strong local maxima, but I can't quite get my arms around a good description for this symmetry. Generally speaking, they look more symmetric if you look at corner cubies and edge cubies separately than if you look at them in combination. Also, they look more symmetric if you look only at the colors of the facelets (looking at two dimensional 3x3 faces) rather than if you look at the location of entire cubies. They do all share the following in common. Looking just at the colors of the facelets, all pairs of opposed 3x3 faces have the same pattern for all eight positions. Hence, there are (up to) three different face patterns for each position. Also, if the cube is colored according the "opposite faces differ by yellow" convention, then the pairs of opposed face patterns for all eight positions are the "yellow complements" of each other. Finally, all the face patterns (and some of them are fairly complicated, having as many as four colors) are symmetric with respect a reflection across either a vertical or horizontal axis of the 3x3 square making up the face. Even though none of these strong local maxima are q-transitive in the classic Saxe-Hoey sense, the "face symmetry" they all share seems too unusual to me to be just a coincidence. I think #8 is an especially interesting position. All six faces have the same face pattern, sort of a three colored checkerboard (if that is not a contradiction in terms). The position is basically Pons Asinorum with the edge and corner cubies rotated as a unit along a major diagonal axis relative to the fixed face centers. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 9 13:14:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA28911; Wed, 9 Sep 1998 13:14:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Sep 1 12:41:00 1998 Date: Tue, 1 Sep 1998 12:36:17 -0400 (Eastern Daylight Time) From: Dale Newfield Reply-To: DNewfield@cs.virginia.edu To: cube-lovers@ai.mit.edu Subject: Re: Depicting a cube In-Reply-To: <35E8A9ED.72BA@ameritech.net> Message-Id: On Sat, 29 Aug 1998, Hana Bizek wrote: > Here, in a nutshell, is a description of the design problem. Please, > get our your Rubik's cubes and start twiddling. My supply of spare cubes seriously dwindled as I constructed a series of "Bandaged Cubes" (XXX XXX XXX). Do you have a suggestion XXX XXX XXX XXXXX , and XXXXXXX XXX XXX XXX XXXXXXX XXX XXX XXX XXX regarding the acquisition of a large enough supply to create interesting art without going broke? -Dale Newfield Dale@Newfield.org From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 9 13:47:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA29037; Wed, 9 Sep 1998 13:47:35 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Sep 6 23:27:56 1998 Message-Id: <35F350B1.626F@ameritech.net> Date: Sun, 06 Sep 1998 22:19:13 -0500 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: Rubik's cube kingdom Hello, cube-lovers, the following miniessay talks about that realm of human knowledge, where the rubik's cube reigns supreme. The gates of this kingdom are open to anybody, but only those who love the cube, venture beyond its gates. The rest of humanity are either unaware of it, or ignore its existence. Those folks are truly missing a lot. After finishing {if indeed you choose to finish} this epistle, you may want to do several things: a} emit a few chuckles thinking what an insane idea this is b} pause to think about the contents, debating whether all this is worth your precious time c} email your ideas, commenrts, etc, to me at hbizek@ameritech.net Thank you. WELCOME TO THE RUBIK'S CUBE KINGDOM. This is not a kingdom of people, it is a kingdom of ideas. Its king is the Rubik's cube. We pay homage to our king by trying to learn more about him and see if this knowledge could be extended to other areas of human pursuit. I am now going to tell you what I think those areas are and point to a couple of web sites where results may be found. This is by no means a finite list. As new ideas occur to all of us, they should be added to the kingdom. The Rubik's cube kingdom is there for anyone to benefit from, just as any other field. Please feel welcome to sample and browse. Initially, of course, one should master the Rubik's cube solution. The ability to solve one side should be the absolute minimum. It is far better and morre satisfying to be able to solve the cube completely, that is, get all six sides a solid color. I have seen numerous solution algorithms on the web. Someone might consider collecting those varied algorithms in a handy volume for cube lovers and others to use. I have seen some solutions on the web, in which you are presented with a solved cube in a little square field. You are instructed to press 's' to scramble the cube and 'r' to restore {i. e. solve} it. That is not solving the cube! You have to understand the steps of the solution algorithm. The areas where the cube has any impact are art, mathematics and science. Let me look at art first. Quite recently there was a small item in TIME magazine stating that the cube has entered Hollywood and is the subject of some movies. I heard that there are also songs about the cube. By the above definition those human expressions too belong in the Rubik's cube kingdom. However, I am going to zero on two aspects: pictures and sculptures. At http://www.wunderland.com/WTS/Jake/CubeArt one may see creations by the people at Wunderland company {the spelling is correct} that depict mostly 2-dimensional picture-like creations from a set of many Rubik's cubes. They just show the pictures, thwy do not describe their method in any book, as far as I can determine {of course, I can be wrong}. But from my observation it seems to me they need to be worried about continuity from one side of the cube to the one side of the next cube, which is not too complicated. The sculptures are 3-dimensional structures and require some symmetry on all the faces of the cube, simultaneously. In this case, the complete solution of the cube is a must. The required algorithm is described in the book, "Mathematics of the Rubik's cube design," written by me and published last year. As far as I know, I am the author of these designs, described in my previous postings to the cube lovers. I have stated a web site where three of these designs may be seen. I repeat it here for completeness: http://www.ssie.binghamton.edu/~jirif/hana1.html Next, I am going to talk about science. I have to warn you: those ideas are, as far as I know, unknown and undeveloped, as is, indeed, the design Problem itself. First on the agenda is fractals and fractal design prototypes. Some of the designs in my book, such as the Menger sponge, are such fractals. One can think of the Rubik's cube as a three dimensionl version of a Cantor set, which is a {one dimensional} line. Actually these fractals are neither three nor one dimensional; they have fractional dimensions. But the 0th iteration are. One can formulate rules for geometrical fractal iteration. Remember that iteration preserves fractal dimension. By the same token, for integer dimension, it really doesn't matter if you subdivide by m or 10000n; the dimension is always the same. Between the integer dimension there are fractals of fractional dimension. These fractals can be reached by breaking up of integer dimension or by some other manipulation. Fractal design made from cubes suggest one such manipulation, as witnessed by box fractal. But this is supposed to be a miniessay, nor a book. Another idea consists of computerizing the design algorithm, in some comprehensive way so that the Rubik's cubes are used as 3-dimensional cellular automata. It would be deliciously complicated game, perhaps employing some of the patterns in the design theory. Finally there is the question of what happens if internal combined faces of a design that touch are colored the same? I will let you figure that one out. It is not too hard. Personally, I had much more difficulty to properly formulate the question than to provide the answer. Finally, there is math. In this respect the ideas were formulated partially by mathematicians in the 1980s during the heyday of the cube, partially by cube lovers today. I would like to include my book in this category. At least its title indicates that there is some math, if that has to be the only reason. And, as every cube lover knows, all possible elements of a Rubik's cube form a mathematical group. A visitor to the Rubik's cube kingdom will surely encounter some joys of group theory on his travels. Any objections, criticism, etc are welcome. Free speach prevails in the Rubik's cube kingdom. Suppress free speach and not much is left. Hana [ Moderator's note: I am somewhat concerned at the low information-to-woowoo ratio of Hana's "miniessay", but I think there are enough real ideas there that I've passed it on to the list. I must, however, note that while "free speach" may prevail on the Internet, the contents of the cube-lovers mailing list is subject to editing for topicality, format, sensibility, and content. Which is to say that if the silliness level gets too high, you may have to find somewhere else to make your "kingdom". I encourage guidance from the readership on where to draw the line; send your opinions to cube-lovers-request@ai.mit.edu. --Dan ] From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 9 17:02:21 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA00591; Wed, 9 Sep 1998 17:02:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Sep 8 21:02:47 1998 Message-Id: From: Noel Dillabough To: "Cube-Lovers (E-mail)" Subject: Dogic Date: Tue, 8 Sep 1998 20:53:22 -0400 A while back we heard about the puzzle "Dogic", an icosahedron puzzle. After playing around with it for a while, I decided to model it in Puzzler (http://www.mud.ca/puzzler/puzzler.html), since I dare not mix it up until I have some moves to work with :) That brings me to the question, has anyone made up a notation for moves with the Dogic puzzle? Perhaps similar to the Megaminx (R++, R+- R-- etc) moves. I have yet to seriously sit down and try to solve it, but eventually I will find some time and a few rudimentary moves would be very helpful. Also, if anyone wants a physical puzzle to play with, Hendrik Haak (mailto:HendrikHaak@t-online.de) still has some available (that's where I got mine) -Noel P.S. To those using the puzzler version, moves can be made along any of the 12 axis in both minor (just the tip pieces) or major (the entire slice) by dragging a cubie from one place to its eventual destination (I didn't bother putting an entry for it in the helpfile). Also, I had a few requests for more detailed information on the puzzles and solving them. I have very nice solutions for the Megaminx, Pyraminx and Cubes, but nothing written down for the Skewb, Masterball, or Dogic puzzles (the skewb and masterball are quite easy so perhaps a solution is unnecessary). Any notes or information on these puzzles would be appreciated. From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 9 17:32:25 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id RAA00879; Wed, 9 Sep 1998 17:32:24 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Wed, 9 Sep 1998 14:59:27 -0400 Message-Id: <002BFC91.001706@scudder.com> From: Jacob_Davenport@scudder.com (Jacob Davenport) Subject: Re: Rubik's cube kingdom To: cube-lovers@ai.mit.edu While it is true that most of the cube art on our web pages (http://www.wunderland.com/WTS/Jake/CubeArt) is two dimensional and therefore pretty easy to make, I have made a few designs that were bloody difficult. I'm rather proud of writing "WTS" on both sides of a hundred some cubes. I'm particularly happy with the chessboard made of four 5x5x5 cubes with a symmetrical design on the sides. Some of my failed experiments were still tough to make, even if they didn't look very good. If anyone has any good 3d design suggestions, I'd like to hear them. Hana, here is my favorite pattern for a single 3x3x3 cube. There is no easy set of twists from solved to this pattern. I had fun doing this pattern on a 5x5x5 cube, and you should be able to create an analogous pattern with all of your cubes: ------- |\ * * *\ | \ o o *\ |x \ x o *\ | * \------ |x o|* x o| | * | | |x *|x x o| \x | | \x|o o o| \|------ The same pattern should be on the other three faces with the other three colors. My ASCII art isn't the greatest but I hope this is clear enough. -Jacob From cube-lovers-errors@mc.lcs.mit.edu Sun Sep 13 16:29:34 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id QAA13820; Sun, 13 Sep 1998 16:29:34 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Sat, 12 Sep 1998 13:01:52 -0400 (EDT) From: Jerry Bryan Subject: Weak Local Maxima, 6f from Start To: Cube-Lovers Reply-To: Jerry Bryan Message-Id: I finally have had enough time to add support to my God's Algorithm program to calculate weak local maxima. The shortest strong local maximum in the face turn metric is 9f, but the shortest weak local maximum has not previously been verified. It has long been known that Pons Asinorum is a weak local maximum at 6f from Start. I have been curious to know if Pons is the shortest, and if there are any other short weak local maxima. It turns out that 6f is indeed the shortest. There are two such positions unique to symmetry which are 6f from Start, the Pons and one other. The other one is quite pretty: L2 R2 D2 U2 B' F (6f*) The eighteen neighbors are as follows. L2 R2 D2 U2 F2 B' (6f*) L2 R2 D2 U2 F (5f*) B' F L2 R2 D2 U' (6f*) B' F L2 R2 D' U2 (6f*) B' F D2 U2 L2 R' (6f*) B' F D2 U2 L' R2 (6f*) L2 R2 D2 U2 B' (5f*) L2 R2 D2 U2 B2 F (6f*) B' F L2 R2 D2 U (6f*) B' F L2 R2 D U2 (6f*) B' F D2 U2 L2 R (6f*) B' F D2 U2 L R2 (6f*) L2 R2 D2 U2 B' F' (6f*) L2 R2 D2 U2 B F (6f*) B' F L2 R2 D2 (5f*) B' F L2 R2 U2 (5f*) B' F D2 U2 R2 (5f*) B' F D2 U2 L2 (5f*) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 14 11:21:10 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id LAA16790; Mon, 14 Sep 1998 11:21:10 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <35FC8F60.60A5521D@ibm.net> Date: Sun, 13 Sep 1998 20:37:04 -0700 From: "Jin 'Time Traveler' Kim" Reply-To: chrono@ibm.net To: cube-lovers@ai.mit.edu Subject: Rubik's Cube-type puzzles FAQ References: <9705131050.AA12905@mentda.me.ic.ac.uk> Back in February or so I became fairly active in the bulletin board at www.rubiks.com. I found the same questions being repeated over and over so with the help of some people I took it upon myself to write a FAQ for people. Due to the type of questions answered I refrained from talking about it on the cube lovers list, but someone told me to give it a try anyway. So here it is: http://www.slamsite.com/chrono or more specifically, http://www.slamsite.com/chrono/other/rcfaq006.txt It's just a very big text file. Kept it simple to give it that "old school" flavor. Corrections, additions, comments, & criticisms are welcome. After all, what's a FAQ if it provides the wrong answers. -- Jin "Time Traveler" Kim chrono@ibm.net http://www.slamsite.com/chrono '95 PGT - SCPOC