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. .__._____._____.__.__._____._____.__. | \ : / : \ | \ : | : \ | +-. `-+-' .-+-. `-+-. `-+--+--+-. `-+ |..\..:../..:..\..|..\..:..|..:..\..| | | : / : / | / : / : | | +--+--+-' .-+-' .-+-' .-+-' .-+--+--+ |..|..:../..:../..|../..:../..:..|..| | \ : | : \ | \ : / : \ | +-. `-+--+--+-. `-+-. `-+-' .-+-. `-+ .__._____._____.__|__\__:__|__:__\__|__\__:__/__:__\__| | \ : / : \ | \ : | : \ | +-. `-+-' .-+-. `-+-. `-+--+--+-. `-+ |..\..:../..:..\..|..\..:..|..:..\..|