From cube-lovers-errors@oolong.camellia.org Mon Jun 9 19:04:20 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id TAA09819; Mon, 9 Jun 1997 19:04:20 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <3.0.1.32.19970609160106.00ade4a0@sdgmail.ncsa.uiuc.edu> X-Sender: mag@sdgmail.ncsa.uiuc.edu X-Mailer: Windows Eudora Pro Version 3.0.1 (32) Date: Mon, 09 Jun 1997 16:01:06 -0500 To: cube-lovers@ai.mit.edu From: Tom Magliery Subject: Re: Designations for the cubes (proposal) In-Reply-To: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" At 11:31 AM 6/8/97 -0400, Nicholas Bodley unabashedly said: > > Peter (Reitan, I think; sorry) (who is not Karen) brought up the >clumsiness of such designations as "5X5X5". I find these downright clumsy >to type (although the Caps Lock key helps). In my private world, I simply >refer to the Pocket Cube as "[the] two", the original Rubik's as "[the] >three", Revenge as "[the] four", and the biggest available as "[the] >five". > > I think that provided we understand that we are referring to the >well-known family of true cubes, it should be OK simply to refer to "the >three", for instance. Granted, these names require more keystrokes, but >numerals should be OK, as in "the 3". I have another suggestion, which might be slightly less likely to require explanation to a newcomer. I know how to *pronounce* it, but I'm not sure how I would recommend *spelling* it. (Considerations include terseness, ease of typing -- which is of course not the same thing!, and likeliness to be mispronounced by a reader.) The pronunciation is three-bye, four-bye, five-bye, ... Possible spellings include: 3by, 4by, 5by, ... 3-by, 4-by, 5-by, ... 3x, 4x, 5x, ... three-by, four-by, five-by, ... mag -- .---o Tom Magliery, Research Programmer (217) 333-3198 .---o `-O-. NCSA, 605 E. Springfield O- mag@ncsa.uiuc.edu `-O-. o---' Champaign, IL 61820 http://sdg.ncsa.uiuc.edu/~mag/ o---' From cube-lovers-errors@oolong.camellia.org Mon Jun 9 19:03:57 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id TAA09815; Mon, 9 Jun 1997 19:03:57 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <339C56DB.5809@hrz1.hrz.th-darmstadt.de> Date: Mon, 09 Jun 1997 21:17:47 +0200 From: Herbert Kociemba X-Mailer: Mozilla 3.0 (Win95; I) MIME-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Re: Detailed explanation of two phase algorithm References: <970608193131.21411978@iccgcc.cle.ab.com> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit SCHMIDTG@iccgcc.cle.ab.com wrote: > > And if we want to show that all depth one nodes will be pruned when > we are at some search depth d where 1 < d < h[0] we would need to show > that: > > 1.9 1 + h[1] > h[0] > Why do you say 1 < d < h[0] and not d = 1? What I wanted to show is, that unter the assumption 2.0 D < h(0) all depth-one nodes will be pruned. As you correctly stated before, for pruning we need 1.8 d + h(d) > D and in the case d=1 this means 1.9a 1 + h(1) > D, which is different from 1.9, because of 2.0 . But 1.9a can be shown easily: In my last message, I tried to explane that 2.1 |h(n-1)-h(n)| <=1, I try to explain it once more in other words. A node at depth n is generated from a node at depth n-1 by applying a single face-turn on it. And as I told, h is defined by h(x,y,z):=max{h1(x,y),h2(x,z),h3(y,z)}, where for example h1(x,y) is the length of the shortest maneuver sequence which transforms (x,y,z) to (x0,y0,z') for any z' (this means the z-coordinate is ignored). And this length can maximal change by one when applying a single move. The same holds for h2(x,z) and h3(y,z). For this reason, h(x,y,z) also can change maximal by one, which implies 2.1 . In the case n=1, from 2.1 follows h(0) <= 1 + h(1), and because of 2.0 we have D < h(0) <= 1 + h(1), which proves 1.9a . > (1) > / \ > (2) (3*) cost = .9 > / > (4*) cost = .7 > > Suppose nodes 3 and nodes 4 were both solutions. Even though node 4 > has a lower cost, phase1 would find node 3 to be our first solution > whereas IDA* wouldn't. I don't think we are far away from each other. Of course, the phase1 (or phase2) algorithm does not claim to be an universal IDA* for any sort of problem. But for a special problem like the cube you can simplify the general IDA* and the simplified algorithm will be equivalent to the one I developed for phase1. Best regards, Herbert From cube-lovers-errors@oolong.camellia.org Mon Jun 9 21:30:42 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA10040; Mon, 9 Jun 1997 21:30:41 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org From: bandecbv@mailhost.rz.ruhr-uni-bochum.de Message-Id: <199706100041.UAA11455@life.ai.mit.edu> Comments: Authenticated sender is To: mgirard@videotron.ca Date: Tue, 10 Jun 1997 02:38:52 +0000 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: Magic Cubes CC: cube-lovers@ai.mit.edu Priority: normal X-mailer: Pegasus Mail for Windows (v2.42a) Mathieu Girard wrote: >I am in a quest to buy both 4x4x4 Rubik's Revenge and also 5x5x5 >cube.. but i just can't find any! Regrettably the 4x4x4 cubes seem to be sold out everywhere in the world. With the 5x5x5 Magic Cubes (in Japan once sold under the name Professor's Cube), the situation is much better: They are still available from me (and as far as I know nowhere else now). Since they will probably never be produced again (the production cost is too high and the general interest too low), they will also be sold out soon. I shall inform the cube-lovers when this is the case. My price of the 5x5x5 cube is 40 DM or 24 USD plus postage. I send my free mail order catalog (containing also many other twisting puzzles like the Magic Dodecahedron , the Skewb, the Pyraminx, Mickey's Chellenge and several books and details how to order) to every cube-lover requesting it and providing a postal address. A few days ago, Joe McGarity complained bitterly about his 5x5x5 cube which fell apart. Fortunately, I did not encounter this problem before, and Joe is not in my files so he has probably not bought his cube from me. On the other hand you will destroy every twisting puzzle by twisting it with force without sufficient aligning the layers before every single move. Since the 5x5x5 cube contains 98 visible little cubies compared to the 26 of the 3x3x3 (and 92 compared to 20 if we only count the freely floating ones), one should accept that it requires a little bit more care. Joe McGarity also mentioned that some orange stickers sometimes do not behave according to there name. I have to admit that this sometimes also happens with my 5x5x5 cubes. Furtunately it happens only to the orange stickers and it can be repaired easily by warm pressure or - better - some glue. Christoph Christoph Bandelow mailto:Christoph.Bandelow@rz.ruhr-uni-bochum.de From cube-lovers-errors@oolong.camellia.org Tue Jun 10 15:15:53 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id PAA11724; Tue, 10 Jun 1997 15:15:53 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Tue, 10 Jun 1997 08:16:53 -0400 (EDT) From: Nicholas Bodley To: Tom Magliery cc: cube-lovers@ai.mit.edu Subject: Re: Designations for the cubes (proposal) In-Reply-To: <3.0.1.32.19970609160106.00ade4a0@sdgmail.ncsa.uiuc.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII I like Tom's version; it is concise, distinctive enough not to be confused (however, I haven't studied all the various math. notations going around), and easy to say and type. Of the ones you gave, I prefer such a form as "3-by". Thanks! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@oolong.camellia.org Tue Jun 10 15:16:08 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id PAA11728; Tue, 10 Jun 1997 15:16:08 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Tue, 10 Jun 1997 13:07:15 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Some Face Turn Numbers In-reply-to: To: Cube-Lovers Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII X-X-Sender: jbryan@PSTCC6.pstcc.cc.tn.us On Thu, 5 Jun 1997, Jerry Bryan wrote: > The following table gives the > best known results for face turns. The results through depth 7 have been > calculated (my message of 19 July 1994). The rest are based on Dan Hoey's > recursion formula PH[n] = 6*2*PH[n-1] + 9*2*PH[n-2] for n>2, where PH[n] > is the number of face turns which are n moves from Start Rats, here is a little correction. I think my meaning was clear from the overall context of the note, but Dan's formula is an upper bound, so it should read PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n>2. For depth 0 through 7, my table provided exact values. As I hope was clear from the context, my table included upper bounds rather than exact values for depths greater than 7. My apologies. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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@oolong.camellia.org Tue Jun 10 15:15:37 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id PAA11720; Tue, 10 Jun 1997 15:15:37 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <339CB8B7.1B8D@snowcrest.net> Date: Mon, 09 Jun 1997 19:15:19 -0700 From: Joe McGarity Reply-To: joemcg3@snowcrest.net X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: "Mailing List, Rubik's Cube" Subject: 5x cubes Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Yes, the cube I purchased was not from Cristoph Bandelow. He is off the hook. It was purchased in San Francisco at the Game Gallery. Although I am considering buying my next one from him. And no, I don't force it either. I treated it with kindness and love, yet it betrayed me. From cube-lovers-errors@oolong.camellia.org Wed Jun 11 00:38:42 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id AAA12658; Wed, 11 Jun 1997 00:38:41 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org From: SCHMIDTG@iccgcc.cle.ab.com Date: Wed, 11 Jun 1997 0:35:55 -0400 (EDT) To: cube-lovers@ai.mit.edu Message-Id: <970611003555.21417ec3@iccgcc.cle.ab.com> Subject: Re: Detailed explanation of two phase algorithm Herbert Kociemba wrote: >SCHMIDTG@iccgcc.cle.ab.com wrote: >> >> And if we want to show that all depth one nodes will be pruned when >> we are at some search depth d where 1 < d < h[0] we would need to show >> that: >> >> 1.9 1 + h[1] > h[0] >> > >Why do you say 1 < d < h[0] and not d = 1? Oops, I think that should have been 'D' and not 'd'. >[...slightly different restatement of earlier proof omitted...] After examining this once again, I have now satisfied myself that it is correct. It's just that for some reason, I seem to find the result rather counter-intuitive. But that makes the result all the more interesting. So I think this may yet me another case where the phase1 algorithm differs slightly from IDA*, but the difference is not significant since, in this case, one is able to prove a special property of the heuristic that demonstrates that the number of nodes explored by the two algorithms is comparable. At this point, I think we can wind down this thread, (I do hope others on this list have found it interesting) and I will still continue to think of possible ideas for improving the algorithm. I do have one last question regarding the pruning tables. While the three tables used in phase1 are clear, I do not recall reading a description of the tables that are used in phase2. I examined Dik Winter's program and he seems to have a few more "maximum move" (i.e. "mm" tables) than I expected, namely: phase1 ------ mm_twists[] mm_flips[] mm_choices[] /* and the following "mixed" tables */ mm_tf[][] /* twist & flip */ mm_tc[][] /* twist & choice */ mm_fc[][] /* flip & choice */ phase2 ------ mm_eperms[] /* edge perms */ mm_cperms[] /* corner perms */ mm_sperms[] /* slice orderliness */ /* "mixed" tables follow */ mm_cs[][] /* corner perms & slice orderliness */ mm_es[][] /* edge perms and slice orderliness */ Are you using the same tables? Or are the "mixed" tables ones that Dik added to the algorithm? It appears that Dik was able to use them because he had a machine with more memory at his disposal than your 1MB Atari ST. His program can be built with or without the "mixed" tables and is 11MB with them. He also mentions that the small program finds a reasonable solution in 30 minutes whereas the large program finds it in only a few seconds. I have also been studying his code to try to understand how he generates these tables. He does not seem to be using breadth-first-search to fill in these tables as Korf does. I will be interested in looking at your new program when it becomes available. Thanks again for your patience. Best regards, -- Greg From cube-lovers-errors@oolong.camellia.org Wed Jun 11 14:05:23 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA14044; Wed, 11 Jun 1997 14:05:23 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 11 Jun 1997 08:39:17 -0700 From: "Jason K. Werner" Message-Id: <9706110839.ZM926@isdn-rubik.corp.sgi.com> In-Reply-To: Joe McGarity "5x cubes" (Jun 9, 19:15) References: <339CB8B7.1B8D@snowcrest.net> Reply-to: "Jason K. Werner" X-Mailer: Z-Mail-SGI (3.2S.3 08feb96 MediaMail) To: cube-lovers@ai.mit.edu, Mark Pilloff , Mathieu Girard , joemcg3@snowcrest.net Subject: Re: 5x cubes Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Here's the place where I got my 5x cube, which has been extremely solid and nearly indestructible: Game Gallery 2855 Stevens Creek Blvd. Santa Clara, CA 95050 USA 408-241-4263 408-241-5945 FAX http://www.gamegallery.com From cube-lovers-errors@oolong.camellia.org Wed Jun 11 14:04:50 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA14037; Wed, 11 Jun 1997 14:04:50 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 11 Jun 1997 08:32:27 -0400 (EDT) From: der Mouse Message-Id: <199706111232.IAA00315@Twig.Rodents.Montreal.QC.CA> To: cube-lovers@ai.mit.edu Cc: Mathieu Girard Subject: Re: ... MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-MIME-Autoconverted: from 8bit to quoted-printable by life.ai.mit.edu id IAA16376 > I am in a quest to buy both 4x4x4 Rubik's Revenge and also 5x5x5 > cube.. but i just can't find any! > I live in Qu=E9bec, Canada, Where? Montr=E9al, or Qu=E9bec City, or what? I'm in Montr=E9al; I got = my 5-Cube at Valet de Coeur, on the west side of St-Denis, somewhere a bit south of Mont-Royal. I don't know whether they still have them; this _was_ back in 1993 (December 15, according to my records). > By the way.. if it is not asking too much... could u please give me > an aproximate of the prices of thoses cubes... in canadian or us > dollar... I paid $45.01, including tax, for my 5-Cube. But as I remarked above, that _was_ three and a half years ago. der Mouse mouse@rodents.montreal.qc.ca 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B From cube-lovers-errors@oolong.camellia.org Wed Jun 11 16:11:56 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA14368; Wed, 11 Jun 1997 16:11:56 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org To: Cube-Lovers@AI.MIT.EDU From: Wei-Hwa Huang Subject: Re: 5x5x5 Stuctural Integrtity Date: 11 Jun 1997 19:01:30 GMT Organization: California Institute of Technology, Pasadena Message-ID: <5nmsma$t63@gap.cco.caltech.edu> References: NNTP-Posting-Host: blend.ugcs.caltech.edu X-Newsreader: NN version 6.5.0 #2 (NOV) David Litwin writes: > The orange sticker problem seems to be with all of them. Mine had >some small documentation with it mentioning that putting a piece of paper >on the orange side and ironing it a bit will help fix the stickers on the >cube. It worked well for me and I've not had any problems with them >anymore. I must say that the first sticker I lost on my 5x5x5 was red. (And I don't know where it went! ARGH!!!) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ ------------------------------------------------------------------------------- Inspiration strikes suddenly, so be prepared to defend yourself. From cube-lovers-errors@oolong.camellia.org Wed Jun 11 16:11:24 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA14364; Wed, 11 Jun 1997 16:11:24 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <339EF3BF.766D@hrz1.hrz.th-darmstadt.de> Date: Wed, 11 Jun 1997 20:51:44 +0200 From: Herbert Kociemba X-Mailer: Mozilla 3.0 (Win95; I) MIME-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Re: Detailed explanation of two phase algorithm References: <970611003555.21417ec3@iccgcc.cle.ab.com> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit SCHMIDTG@iccgcc.cle.ab.com wrote: > I do have one last question regarding the pruning tables. While > the three tables used in phase1 are clear, I do not recall reading > a description of the tables that are used in phase2. In phase2, the state of the cube also is described by a triple (x,y,z), in this case 0<=x<8! describes a permutation of the 8 corners, 0<=y<8! describes a permutation of the 8 UD-slice edges and 0<=z<4! describes a permutation of the middleslice edges. Because the overall permutation must be even, only half of the triples belong to physical cubes. We could correct this, by defining the z coordinate to describe one of the 12 possibilities for the locations of two middleslice edges - the other two edges will then be corrected automatic. But there are good reasons not to do so (which I think is not necessary to explain here). > I have also been studying his code to try to understand how he generates > these tables. He does not seem to be using breadth-first-search to > fill in these tables as Korf does. > I only use the "mixed" tables. How to generate the tables is quite obvious and though I don't know how Dik does it I'm sure it is similar: 1. On initialisation set all elements of the table to 0xf (we use four bits per entry), only the element belonging to (x0,y0,z0) is set to 0. Set L=0, n_done=1, n_old=1 (n_done denotes the number of valid tableentries). 2. Check all elements of the table one after the other. If an entry is 0xf, do nothing. If the entry is L, compute the the 18 possible "child nodes" and check, if the corresponding tableentry is 0xf. Only in this case set it to L+1 and increment n_done. 3. Check if n_done=n_old. In this case we are ready. Else increment L, set n_old=n_done and goto 2. > I will be interested in looking at your new program when it becomes > available. I'm writing too much to this mailing list and do not work at my windows-help! The program itself is ready. I did a two hours run on each of Rich Korfs 10 random cubes on a Pentium133 with 16MB RAM and the result were really pleasing: The generated maneuver lenghts were on the average less than 1 move away from Rich Korfs optimal solutions (exactly: 9 moves more for the 10 cubes). Best regards, Herbert From cube-lovers-errors@oolong.camellia.org Wed Jun 11 20:44:06 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id UAA15754; Wed, 11 Jun 1997 20:44:06 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org From: SCHMIDTG@iccgcc.cle.ab.com Date: Wed, 11 Jun 1997 20:42:11 -0400 (EDT) To: cube-lovers@ai.mit.edu Message-Id: <970611204211.2141971d@iccgcc.cle.ab.com> Subject: Re: Detailed explanation of two phase algorithm Herber Kociemba wrote: [...much additional detailed explanation deleted...] Thank again, I found this information helpful, especially when my only other option is to examine code in great detail in order to extract out the general principles. >> I will be interested in looking at your new program when it becomes >> available. > >I'm writing too much to this mailing list and do not work at my >windows-help! Sorry about that. I'll stop with my questions. In fact, no need to even answer this response! I'm sure your program will be well worth the wait :). > The program itself is ready. I did a two hours run on each >of Rich Korfs 10 random cubes on a Pentium133 with 16MB RAM and the >result were really pleasing: The generated maneuver lenghts were on the >average less than 1 move away from Rich Korfs optimal solutions >(exactly: 9 moves more for the 10 cubes). Very impressive. And if you perform some longer runs and find optimal solutions, please be sure to let us know the run times. Best regards, -- Greg From cube-lovers-errors@oolong.camellia.org Thu Jun 12 13:23:55 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA17772; Thu, 12 Jun 1997 13:23:54 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Thu, 12 Jun 1997 01:03:11 -0400 (EDT) From: Nicholas Bodley To: Wei-Hwa Huang cc: Cube-Lovers@ai.mit.edu Subject: Re: 5x5x5 Structural Integrtity (Stickers) In-Reply-To: <5nmsma$t63@gap.cco.caltech.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII One cure might be to take a solved cube (more convenient...) and remove all the red stickers; carefully clean off the adhesive, perhaps with 99% isopropyl alcohol, and paint the surfaces neatly with the type of paint used for plastic model kits. I have also thought of removing the stickers, cleaning all the adhesive off both the stickers and the cubies, and then reattaching the stickers with a different type of adhesive. These are just ideas, and I hope no source of trouble. My best to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@oolong.camellia.org Wed Jun 18 16:19:15 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA04432; Wed, 18 Jun 1997 16:19:15 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org To: cube-lovers@ai.mit.edu Date: Wed, 18 Jun 1997 15:05:54 -0500 Subject: Square One Message-ID: <19970618.150557.11350.0.shaggy34@juno.com> X-Mailer: Juno 1.38 X-Juno-Line-Breaks: 0-2 From: Josh D Weaver Does anyone know how to solve one of those "Square One" puzzles? Josh From cube-lovers-errors@oolong.camellia.org Wed Jun 18 18:43:38 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id SAA04738; Wed, 18 Jun 1997 18:43:37 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <33A84C0E.94@hrz1.hrz.th-darmstadt.de> Date: Wed, 18 Jun 1997 22:58:54 +0200 From: Herbert Kociemba X-Mailer: Mozilla 3.0 (Win95; I) MIME-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Windows95 program now available Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit The Windows95 program which implements my algorithm to solve Rubik's Cube is now availabe at http://home.t-online.de/home/kociemba/cube.htm It not only solves Rubik's cube, but also does a few other nice things... Herbert [ Moderator's note: This program is also available in the Cube-Lovers Archive. See: ftp://ftp.ai.mit.edu/pub/cube-lovers/contrib/cubexp10.zip - Alan ] From cube-lovers-errors@oolong.camellia.org Thu Jun 19 01:16:47 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id BAA05551; Thu, 19 Jun 1997 01:16:47 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org From: Benjamin Wong To: Josh D Weaver Date: Thu, 19 Jun 1997 11:37:43 +1000 (EST) X-Sender: chi@pipe02.orchestra.cse.unsw.EDU.AU cc: cube-lovers@ai.mit.edu Subject: Re: Square One In-Reply-To: <19970618.150557.11350.0.shaggy34@juno.com> Message-ID: On Wed, 18 Jun 1997, Josh D Weaver wrote: ._@_.Does anyone know how to solve one of those "Square One" puzzles? ._@_. http://www.cfar.umd.edu/~arensb/Square1/ is the only page on the net (that i can find) which describe how to solve square 1 however, either i can not follow instruction, or error in it's instructuion i just can not solve it with their algorimthm I bought square 1 mess them up, only manage to solve them 2 times. (beginner luck) but the page does not help very much. ._@_.Josh ._@_. o------------------------------------------------------o |Error: Reality.sys Corrupt? Reboot Universe [Y,N,Q] | +---------------o--------------------------------------o | Benjamin Wong | E-mail: chi@cse.unsw.edu.au | | | or benjaminwong@hotmail.com | | | http://www.cse.unsw.edu.au/~chi | o---------------o--------------------------------------o |=A1u=C2=E5=A5=CD=A1I=BD=D0=B0=DD=A1y=BA=B5=BF=DF=B2=B4=A1z=AA=BA=A6=A8=A6]=ACO=AC=C6=BB=F2=A1H=A1v | |Quick Quiz: Describe Universe ? Give Three Example. | o------------------------------------------------------o From cube-lovers-errors@oolong.camellia.org Thu Jun 19 12:31:32 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id MAA06811; Thu, 19 Jun 1997 12:31:31 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <33A8DBC4.2A8F@snowcrest.net> Date: Thu, 19 Jun 1997 00:12:04 -0700 From: Joe McGarity Reply-To: joemcg3@snowcrest.net X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: Josh D Weaver CC: "Mailing List, Rubik's Cube" Subject: Re: Square One References: <19970618.150557.11350.0.shaggy34@juno.com> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit After three months of agony and wondering if Square One was actually the puzzle box from Clive Barker's Hellraiser, I managed to come up with a solution that covered all the bases, i.e. worked every time. I have never tried to write it out in a step by step form however. I will try to cover the basics. First the puzzle looks like a Rubik's Cube when solved in that it is a cube with a solid color on each face, but the similarity ends there. Square One more closely resembles the Orb, Masterball and Smart Alex in the ways that it moves. If you can solve any of those you will be a step closer to the Square One. I see the square one as nearly identicle to the Smart Alex. The shapes of the pieces are different, but they move as a disk divided into sectors (exactly like the Masterball). There are six pieces on each face if you count the small sectors as half pieces. Count the pieces and you will see what I mean. The little ones are half the the size of the angle of the big ones. The idea then for me was to get the little ones paired up like the picture in the instruction booklet. Once they were paired correctly I could solve it just like the Masterball or Smart Alex making it look like it did when it was new in the package (remember it came in a slightly scrambled state with instructions on how to solve it from there in about six moves). Then I could just follow the booklet for the final part. Like I said it took three months and scores of note paper to finally get it. When I did, the walls opened up and the Cenobites took me away, but it was worth it. I hope I haven't caused more confusion. It is difficult to describe without having one in my hands to show you. This is just a sketchy overview of how I solve it. If I get a chance to document this solution I will send you a copy, but it probably won't be for a while. I'm sure that someone has a better solution and I'd be interested in seeing what others have come up with. My solution takes about twenty minutes to do and there must be a faster way. The ones I have trouble with are the Sqewb and the Alexander's Star. Anybody got a good solution for any of these? Joe McGarity From cube-lovers-errors@oolong.camellia.org Thu Jun 19 12:32:15 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id MAA06816; Thu, 19 Jun 1997 12:32:15 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: From: "joyner.david" To: "'Josh D Weaver'" Cc: "'cube-lovers@ai.mit.edu'" Subject: RE: Square One Date: Thu, 19 Jun 1997 08:06:58 -0400 X-Mailer: Microsoft Exchange Server Internet Mail Connector Version 4.0.994.63 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit >---------- >From: Josh D Weaver[SMTP:shaggy34@juno.com] >Sent: Wednesday, June 18, 1997 4:05 PM >To: cube-lovers@ai.mit.edu >Subject: Square One > >Does anyone know how to solve one of those "Square One" puzzles? There's a paper on my web page which indirectly explains how. (It's actually a math paper written with a student of mine explaining the group theory of the puzzle.) What's useful are some of the moves which we give. If you can't print it out (It's a dvi file) I'll mail it to you if you give me your postal address. http://www.nadn.navy.mil/MathDept/wdj/rubik.html The idea, if I remember, is 1. get into a square form, 2. use the special moves we give (moves which permute 3 pieces only and leave the others alone, for example) to solve the puzzle as one solves the Rubik's cube. - David Joyner > >Josh > > From cube-lovers-errors@oolong.camellia.org Thu Jun 19 12:32:39 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id MAA06820; Thu, 19 Jun 1997 12:32:39 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Thu, 19 Jun 1997 09:19:01 -0400 Message-Id: <199706191319.JAA27307@bso.newvision.com> From: Carl Woolf To: chi@cse.unsw.edu.au CC: shaggy34@juno.com, cube-lovers@ai.mit.edu In-reply-to: (message from Benjamin Wong on Thu, 19 Jun 1997 11:37:43 +1000 (EST)) Subject: Re: Square One Square One is a great puzzle! I think there is an instruction booklet, published in Massachusetts or thereabouts, and available from Puzzlets (mgreen@puzzletts.com). I developed a set of techniques that let me solve the thing, but I haven't worked my notes into a form intelligible by other humans (or by me on a bad day). -- -- Carl ----------------------------------------------------- Business: woolf@newvision.com Personal: woolf@ccs.neu.edu http://www.ccs.neu.edu/home/woolf From cube-lovers-errors@oolong.camellia.org Thu Jun 19 21:55:51 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA07813; Thu, 19 Jun 1997 21:55:51 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199706200151.AA15092@world.std.com> To: "cube-lovers@ai.mit.edu" Subject: Square One Solution Date: Thu, 19 Jun 97 21:53:07 -0500 From: Mike Masonjones X-Mailer: E-Mail Connection v2.5.03 Cube Lovers, This is long, but complete solutions are long. I even left quite a bit of the boring and obvious stuff out, and it still took me 3 hours to write it. Any errors, let me know. I hope this satisfies. Apparently I have the least extracurricular life of all of you, since I know how to solve Square One much quicker than any reports I've seen here. Quite a dubious honor, I suspect. Anyway, here goes my solution, which with a little practice, guarantees a solution within 75-100 seconds for someone who can do The Cube in about 55-60 seconds (assuming there's a correlation in hand speed from puzzle to puzzle). The solution is completely my own, as is the notation. Sorry if it offends anyone. 1. Start by getting to the cube state for the top and bottom faces. Ignore the middle slice til the very end. This can be most efficiently done by memorizing a table. A pretty good (but not error-free) one is on the only square-one web site in existence that I know of. Sorry, you'll have to find the site yourself with a browser, since I can't look it up right now. I have a scheme written down somewhere that is quite a bit easier to memorize, but why should I take the fun away from any of you looking for the solution yourselves. If there is a big response to this letter, then I will dig out my easy table. Tables are difficult to memorize, so I usually just try to get to a six pointed star on one side and all the little wedges plus the remaining 2 big wedges on the other. It is easy to get to one of the five possible states that result, thus requiring memorization of only 5 solutions to get back to a cube. This method takes about 5-10 seconds longer, an average, than the table technique. Using a notation where L represents a large wedgie, and S a small wedgie, the five possible states can be written as: 1) bottom = LLLLLL, top = LLSSSSSSSS 2) bottom = LLLLLL, top = LSLSSSSSSS 3) bottom = LLLLLL, top = LSSLSSSSSS 4) bottom = LLLLLL, top = LSSSLSSSSS 5) bottom = LLLLLL, top = LSSSSLSSSS For cases 1),3),5), rotate the top face so that it will be sliced symmetrically between the two L pieces when the center is flipped. The next move in each of these cases involves moving the top face one way and the bottom face the other, when looking from the front. (Front will be the term used from now on to denote the end nearest you of the central cut through which flipping occurs (a 180 turn of one half of the cube)). After a flip, cases 1) and 5) should give two barrel shapes (LLSSLLSS), top and bottom. You should aim in case 3) for two tomahawk (LLSSLSLS) shapes. Any self- respecting cubist should be able to get home from here. Cases 2 and 4 are a little more complicated. For both cases align the top so that the left half of the top face reads, going clockwise, SLSSS. Flip right side. Now rotate the bottom so that when you flip with the right hand , the top will read SlSSSSSLL starting from the front and going clockwise. Now rotate the top 1/12 turn counterclockwise and the bottom so it reads LLLLSSL going clockwise from the front and flip again. Now you're in an easy state to get home from (LLLLSSSS on top and LLSLSSLS on bottom). 2. Now that you're in a cube state top and bottom, get all the wedgies on their correct side (top and bottom face all the same color, respectively). This is very straightforward and intuitive. I usually start with one large wedgie, and sequentially put in one at a time next to it going around a face until you get down to one S wedgie stuck on the wrong side. Sometimes it is easier to do LSL on one half of the top, and then do LSL on the other, and then putting in the second to last S between the groups. Now position the top face so that the Odd S wedgie (O) is positioned as LSLOLSLS going clockwise from the front. Put the bottom odd wedgie in front with the bottom square skewed from the top (bottom should read LSLSLSLO going clockwise from the front). Now do FT4B1FT-4B-1FT4B1FT-3F, where F = flip with right hand, Bx = turn bottom face clockwise x/12 of a turn, B-x = same thing counterclockwise, Tx, T-x mean similar things. 3. Now get L's positioned. Case 1. No L's are correctly adjacent to each other. Position top and bottom (top = LSLSLSLS, bottom = SLSLSLSL, each going clockwise from front). Now go FB3FT-3B-3FT3F, turn the whole puzzle 180 degrees so that the back of the central cut is now the front, and repeat the move. Case 2. Two sets of adjacent pairs are out of whack, one on top, and one on bottom. Do the move for case 1 once, with the components of the pairs in question all nearest the front. Case 3. Only one adjacent pair correct. Position the top so that the correctly adjacent pair (denoted as A) is positioned as ASASLSLS, and the bottom reads LSLSLSLS (same conventions as before). Now do FB-3FB3FB-3FB3F. Case 4. Only one pair incorrect. Position the top (with the incorrect pair) so that the correctly adjacent pair is positioned as LSASASLS, and the bottom reads LSLSLSLS. Do the move in Case 3 twice with a T3 between instances. Case 5. One side is OK, the other has no correct adjacent pairs. Bad side = top. top = LSLSLSLS, bottom = LSLSLSLS. Do the move in Case 3 twice with T6 between instances. 4. Now check for parity. With the L's in place it is easy to identify whether you need to change the parity of the system. It should take an even number of switches to right the S's at this stage. A cycle of three is even, since it would take two switches to fix it. A cycle of two or four is odd. If the overall parity is odd, do the following: starting with top = LSLSLSLS, bottom = LSLSLSLS, go FT3B3FT1B2FT2B2FT- 2FT2B2FT3B2FT-3B-3T-3B-1FT-2B-2F This may not be the optimum way, but it preserves the corners, and it's easy to remember the path. (Try it) 5. Place the S's (they should already be on their correct face). The most useful moves are the below: All permutations of S's can be solved with application of a maximum of three of these short moves in sequence, combined with the appropriate turns in between to set things up. Move 1. Start with top = LOLSLSLO, bottom = LOLSLSLO, where O = pairs that will be switched on a given side. Do FT-3FT1B1FT2B-1F. Repeated twice with a T3 between makes a three-cycle on the top side. Move 2. top = LSLOLSLO, bottom = LSLOLSLO, O definition same as Move 1. Do FT1B1FT6FT-1B-1F. There may be quicker solutions than applying these moves for a 4-cycle/2- cycle combination or a 4-cycle/4-cycle combination, where you have to apply 3 moves in succession. I'd like to hear about suggestions. I haven't investigated it too much since these modes come up so rarely. 6. Fix middle slice. If square shaped but wrong, do FT6B6F. Otherwise, position the bad half on the right, and do FB6FB6F. Congratulations, you have a solved Square One. Happy cubing. Mike Masonjones From cube-lovers-errors@oolong.camellia.org Thu Jun 19 21:55:37 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA07809; Thu, 19 Jun 1997 21:55:36 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <2.2.32.19970619180050.0068b11c@uclink4.berkeley.edu> X-Sender: mdp1@uclink4.berkeley.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 19 Jun 1997 11:00:50 -0700 To: cube-lovers@ai.mit.edu From: Mark Pilloff Subject: Re: Square One At 12:12 AM 6/19/97 -0700, J.M. wrote: >The ones I have trouble with are the >Sqewb and the Alexander's Star. Anybody got a good solution for any of >these? > >Joe McGarity I finally came up with a solution to the Alexander Star last year. I haven't ever written out all of the details, but here are some helpful hints. First of all, the star is almost identical to the Megaminx (aka, magic dodecahedron, etc.) with all of the corners pieces removed. The only reason I say "almost" is that on the star, every individual piece is doubly degenerate. This sometimes leads to a problem wherein using the Megaminx moves seems to leads to an insoluble position. The trick in this case is two swap two of the degenerate pieces while disturbing as little of the rest of the star as possible. This has always worked perfectly for me. As for the rest of the star, I usually find that I can solve most of the star (except for the uppermost regions) just by inspection. From there, very slight modifications of Rubik's cube manipulations are useful. It's worthwhile to note that locally, the star (and the megaminx) are identical to the cube (except, perhaps, for the missing corners on the star). Thus, cube moves which only affect small portions of the cube will often be successful on the star or megaminx. In any event, I'm not going to write out explicit moves because I believe solutions to the megaminx are floating on the net as well, but I hope this is somewhat helpful. If there is really strong demand for explicit solutions, I'll see what I can do about that. Good luck, Mark ************************************ ** Mark D. Pilloff ** ** mdp1@uclink4.berkeley.edu ** ************************************ From cube-lovers-errors@oolong.camellia.org Fri Jun 20 11:36:32 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id LAA09262; Fri, 20 Jun 1997 11:36:32 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Fri, 20 Jun 1997 07:05:50 +0200 (MET DST) Message-Id: <1.5.4.16.19970620070547.2e6f908a@mailsvr.pt.lu> X-Sender: geohelm@mailsvr.pt.lu X-Mailer: Windows Eudora Light Version 1.5.4 (16) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: Carl Woolf From: Georges Helm Subject: Re: Square One Cc: cube-lovers@ai.mit.edu At 09:19 19/06/1997 -0400, you wrote: >Square One is a great puzzle! > >I think there is an instruction booklet, published in Massachusetts or >thereabouts, and available from Puzzlets (mgreen@puzzletts.com). I >developed a set of techniques that let me solve the thing, but I >haven't worked my notes into a form intelligible by other humans (or >by me on a bad day). > There is a solution in Puzzle World. Check out details at http://ourworld.compuserve.com/homepages/Georges_Helm/cubeold.htm Georges geohelm@pt.lu http://ourworld.compuserve.com/homepages/Georges_Helm http://www.geocities.com/Athens/2715 From cube-lovers-errors@oolong.camellia.org Fri Jun 20 11:36:18 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id LAA09258; Fri, 20 Jun 1997 11:36:17 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Fri, 20 Jun 1997 07:03:46 +0200 (MET DST) Message-Id: <1.5.4.16.19970620070348.2e6f1bf6@mailsvr.pt.lu> X-Sender: geohelm@mailsvr.pt.lu X-Mailer: Windows Eudora Light Version 1.5.4 (16) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: joemcg3@snowcrest.net From: Georges Helm Subject: Re: Square One Cc: cube-lovers@ai.mit.edu At 00:12 19/06/1997 -0700, you wrote: >The ones I have trouble with are the >Sqewb and the Alexander's Star. Anybody got a good solution for any of >these? I have solutions for both. Check out my list of solutions at http://ourworld.compuserve.com/homepages/Georges_Helm/cubeold.htm Alexander's book on his star Flettermann has a good solution for the skewb (but I realize I don't have him listed) I can send copies, though Georges geohelm@pt.lu http://ourworld.compuserve.com/homepages/Georges_Helm http://www.geocities.com/Athens/2715 From cube-lovers-errors@oolong.camellia.org Sat Jun 21 14:03:33 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA13787; Sat, 21 Jun 1997 14:03:33 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199706201853.TAA09093@mail.iol.ie> From: Goyra To: cube-lovers@ai.mit.edu Subject: Re: Square One Date: Thu, 19 Jun 1997 19:00:58 +0100 X-MSMail-Priority: Normal X-Priority: 3 X-Mailer: Microsoft Internet Mail 4.70.1161 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit > >Does anyone know how to solve one of those "Square One" puzzles? Can anyone tell me what this looks like so I can put up a Java version? David Byrden From cube-lovers-errors@oolong.camellia.org Mon Jun 23 21:05:29 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA03450; Mon, 23 Jun 1997 21:05:28 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Mon, 23 Jun 1997 18:33:40 -0400 From: "Jonathan R. Ferro" Message-Id: <199706232233.SAA51996@knave.ece.cmu.edu> Organization: Electrical and Computer Engineering, CMU X-Disclaimer: This disclaimer is not required by Leader Kibo. This article does not necessarily represent the opinions of Leader Kibo. Have a nice day! X-Exclaimer: Yow! To: cube-lovers@ai.mit.edu Subject: An art project... http://www.wunderland.com/EBooks/Window/Pages/SUTW-JD.html -- Jon From cube-lovers-errors@oolong.camellia.org Tue Jun 24 20:45:27 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id UAA06941; Tue, 24 Jun 1997 20:45:27 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <33B0668C.97B@ibm.net> Date: Tue, 24 Jun 1997 17:30:04 -0700 From: Jin "Time Traveler" Kim Organization: The Fourth Dimension X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: cube-lovers@ai.mit.edu Subject: Re: An art project... References: <199706232233.SAA51996@knave.ece.cmu.edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Jonathan R. Ferro wrote: > > http://www.wunderland.com/EBooks/Window/Pages/SUTW-JD.html > > -- Jon Very impressive. How many cubes and were they altered in any way except turning them? -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa From cube-lovers-errors@oolong.camellia.org Wed Jun 25 13:29:10 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA08775; Wed, 25 Jun 1997 13:29:10 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 25 Jun 1997 18:11:31 BST From: David Singmaster Computing & Maths South Bank Univ To: cube-lovers@ai.mit.edu Message-ID: <009B651B.AC285B40.328@vax.sbu.ac.uk> Subject: Pilloff's query about 5^3; Korf's method. I've been away and have just seen email for late May and early June. Pilloff asks about getting the parity of the edges correct on the 5^3. As he notes, the commutators cannot solve this. Examining the basic moves, one sees that the rotation of a inner face changes the parity of the edges while conserving the parity of some pieces, but not of others. However, the pieces whose parity is not conserved are the pieces next to the centre and there are four apparently identical copies of these, so one can simulate an exchange by a 3-cycle with two pieces the same. Hence one wants to apply a rotation of an inner face. What one does is to move the two edges into the same inner face. Then rotate the face. Then make a 3-cycle of the edges. This produces an exchange of edges - and rather messes up the centre pieces. Then put things back. The edges go A B C D to D A B C to B A C D. Because this is relatively easy to do, but messes up the centres, I normally do the edges and corners first and then put centres in place. Re: Korf. Someone has said it reminds them of alpha-beta pruning. It reminds me of branch and bound search. Both are older names for the general process of using information about the remainder of the problem to estimate the number of steps for a solution via a partial solution. Going back to Pilloff's query, I have several methods in my files for exchanging two edges without moving anything else (I think, but that seems to contradict what I said about parities??) 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@oolong.camellia.org Wed Jun 25 13:28:50 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA08771; Wed, 25 Jun 1997 13:28:50 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 25 Jun 1997 09:53:21 -0500 To: cube-lovers@ai.mit.edu From: Kristin Looney Subject: Re: An art project... Cc: jake@wunderland.com >> http://www.wunderland.com/EBooks/Window/Pages/SUTW-JD.html > > Very impressive. How many cubes and were they altered in any > way except turning them? 100 cubes, scrambled - not altered in any other way. This is, I believe, the sixth cube sculpture that Jacob has done in the window of my gameroom. A seventh is currently in progress. Previous sculptures include: "Rubik's Cube", "Merry Xmas" with a picture of a Christmas tree, a bizarre (and not too successful) abstract thingamabob, a pacman with several ghosts, and the Apple logo. I think the pacman is probably my favorite, it's a hard choice. They are all very much worth a look... I have pictures of them, and I will encourage Jake to put them on his web page for all to see. The one he is currently working on is, believe it or not, TWO SIDED. Unlike most which he just sorta fiddles with while playing games at my gameroom table, this one was carefully planned out in advance with graph paper. -K. 5th fastest hands in the nation (at least back in 1981) kristin@wunderland.com www.wunderland.com/kristin ------------------------------------------------------------- "I'm really angry that I, a superior human being in every way, have less money than my neighbor, who's wife I would love to nail, if only I weren't so busy sleeping and eating pork chops." -- George "Cannonball" Carlin, on the 7 deadly sins From cube-lovers-errors@oolong.camellia.org Wed Jun 25 17:18:25 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id RAA09126; Wed, 25 Jun 1997 17:18:25 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 25 Jun 1997 22:14:32 BST From: David Singmaster Computing & Maths South Bank Univ To: cube-lovers@ai.mit.edu CC: notation@vax.sbu.ac.uk, for@vax.sbu.ac.uk, 4^3@vax.sbu.ac.uk, and@vax.sbu.ac.uk, 5^3@vax.sbu.ac.uk Message-ID: <009B653D.9EE57540.331@vax.sbu.ac.uk> Subject: notation for 4^3 and 5^3 David Barr has described some moves for the 5^3 using his own notation. In the early 1980s, I extended my 3^3 notation to the 4^3 and this can be used on the 5^3 as well. I described this in my Cubic Circular but perhaps it would be useful to give it here. I will describe the situation for the 4^3. On the 5^3, we don't ever have to make a slice move of just a central layer. Further, a combination of 4^3 and 3^3 processes will solve the 5^3, so we don't really need to label the central layers. Consider the four layers from L to R. I denote the inner layers by l and r. So the four layers are: LlrR. Similarly we have four layers: FfbB and UudD. To describe a piece requires more effort than before, but each piec lies in three layers and we can describe the piece by these layers. E.g. FUR is a corner piece; FUr is an edge piece, lying on the FU edge - but there are two of these and they are distinguished as being FUr and FUl (I usually give the layers in clockwise order, but it is not essential and there are times when it is more informative to use the other order.); Fur is a face-centre piece, the one in the upper right corner of the inner four cells of the F face. If you have been paying attention, you will ask about fur. This is one of the body-centre cells, invisible to you unless you make a transparent cube! Using the standard notation of [F, R] = FRF'R', we find a number of easy 3-cycles. [[F,R],L] = (FLU,ULB,RFU) [[F,R],l] = (FlU,UlB,DfR) (I've copied this from my Circular, but I wonder if it's right as I thought there'd be some symmetry with the preceding??) [[F,r],l] = (Flu,Ulb,Drb) In theory, these and a careful consideration of parity are sufficient to solve the 4^3 and the 5^3. However, the parity problem is a bit awkward. In my original approach, I got the corners in place and then all edges except leaving the four edges along the FU and BU edges. Examine the parity of these carefully. If they are in an odd permutation, apply r^2 D^2 l' D^2 r^2 which 4-cycles these four edges and moves some centres. Once the parity is corrected, there is little difficulty restoring the rest of the cube. For the 5^3, once you have paired up the edges, one can solve the central edges by treating the 5^3 as a 3^3 with fat slices. To correct a single pair of edges, one can use the following. rrDDl'DDrr rrD'RR [[R,U],l'] RRDrr = rrDDl'DR'UR'U'l'URU'lRDrr = (UBl,UBr) (Ful,Ubl,Bdl,Ufr) (Fdl,Ufl,Bul,Ubr). This messes up some centres, but they are not too hard to restore. Indeed applying rrUUr (uurrll)^2 r'UUrr wil correct the F and B centres disturbed by the above, leaving a 180 degree rotation of the four U centres. After I had developed the notation and solution, a Peter Lees pointed out an unexpected feature. The exchange of upper and lower case letters is a duality. The dual of URF is urf, while the dual of URf is urF. This gives us a Pricniple of Duality: The dual of a sequence of moves is the same process on the dual pieces. E.g., we had [[F,R],l] = (FUl,UBl,DRf), so [[f,r],L] = (fuL,ubL,drF). This duality allows one to transfer a number of 3^3 processes to 4^3 processes and to solve the invisible interior part of the cube! By always moving an outer layer with its inner layer, one is obviously simulating the 2^3. However, if one always moves, say R and l together, one is also simulating the 2^3 in eight copies! Ooops, one wants to move R and l' together. If one moves R and l together, I think you get eight versions of the 2^3, but each is a reflection of its neighbours! If you are tired of thinking about God's Algorithm on the 3^3, try the 4^3. I'm not even sure how to count moves. E.g., to do r, one normally does Rr and then R', so does r count as one move or two? Likewise, does Rr count as one move or two? Enough for now. 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@oolong.camellia.org Wed Jun 25 18:29:12 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id SAA09246; Wed, 25 Jun 1997 18:29:12 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 25 Jun 1997 23:28:16 BST From: David Singmaster Computing & Maths South Bank Univ To: cube-lovers@ai.mit.edu Message-ID: <009B6547.EBD77100.328@vax.sbu.ac.uk> Subject: 4^3 and 5^3 I've just seen a comment about the 5^3 saying the writer had problems with the four pieces at distance 1 from the center. My approach treated both these and the pieces ate distance 2 from the center in the same way. We know that the commutator of two slice moves on the 3^3 produces two 3-cycles of the centres. Applying this idea to the 4^3, we find that [f,r] gives two 3-cycles of central pieces, one on each face. By turning one face and then inverting, we get a 3-cycle of central pieces, two being in one face. E.g. [[r,b],U] = (Fur,Ubr,Ubl). A similar result holds if we combine any two inner moves, so we can replace the b above by a central slice on the 5^3 to obtain a 3-cycle of the pieces at distance 1 from the central piece, while the process directly gives us a 3-cycle of the pieces at distance 2 from the central piece. Although tedious, these moves mean that once one has the corners and edges in place, the rest of the problem is easy - though very tedious - it generally took me about an hour to do the 5^3, assuming I could get the corners and edges correct without making too many mistakes. 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@oolong.camellia.org Wed Jun 25 23:51:13 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id XAA00509; Wed, 25 Jun 1997 23:51:13 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Thu, 26 Jun 1997 04:44:18 BST From: David Singmaster Computing & Maths South Bank Univ To: cube-lovers@ai.mit.edu Message-ID: <009B6574.123497C0.321@vax.sbu.ac.uk> Subject: names for cubes What's wrong with 2^3, 3^3, 4^3, 5^3, pronounced 2 cube, 3 cube, 4 cube, 5 cube. If you have superscripts available, you can use them instead of the uparrows. 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@oolong.camellia.org Thu Jun 26 15:40:18 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id PAA06801; Thu, 26 Jun 1997 15:40:17 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Thu, 26 Jun 1997 07:44:43 -0400 (EDT) From: Nicholas Bodley To: David Singmaster Computing & Maths South Bank Univ cc: cube-lovers@ai.mit.edu, notation@vax.sbu.ac.uk, for@vax.sbu.ac.uk, 4^3@vax.sbu.ac.uk, and@vax.sbu.ac.uk, 5^3@vax.sbu.ac.uk Subject: Hidden cubies; Spaceball In-Reply-To: <009B653D.9EE57540.331@vax.sbu.ac.uk> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Reading David Singmaster's recent posts, it seems almost obvious to me that (for the ambitious, which I'm definitely not!), computer simulations that are unhindered by real-world mechanical constraints and opacity can permit manipulation of the normally-hidden (and physically-nonexistent) internal cubies. It's very likely that a new collection of maneuvers would need to be developed for this. I haven't thought much about coloring the internal faces of the outer cubies... Incidentally, if this weren't the cube-lovers' List, I would have split that first sentence into a few shorter ones. Sorry if "cubie" is not the most-preferred term; should be no great problem. On another topic, it seems to me that an ideal device for controlling a computer-simulated Cube (or other similar puzzle) would be the Spaceball, a ball that you can grip. It senses torque around all three mutually- orthogonal axes, as well as "translational" force along those axes. It's not a consumer item; not sure it's still being made. I'm reasonably sure of the tradename. It was/is used with workstations. "Spaceball" sounds much like the name of a puzzle. (I expect some astute reader to tell me that the MIT Media Lab did just this thing 5 years ago!) My best to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@oolong.camellia.org Thu Jun 26 16:35:00 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA06927; Thu, 26 Jun 1997 16:35:00 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <33B2D235.3A77@ibm.net> Date: Thu, 26 Jun 1997 13:33:57 -0700 From: Jin "Time Traveler" Kim Organization: The Fourth Dimension X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: Nicholas Bodley CC: cube-lovers@ai.mit.edu Subject: Re: Hidden cubies; Spaceball References: Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Nicholas Bodley wrote: > > On another topic, it seems to me that an ideal device for controlling a > computer-simulated Cube (or other similar puzzle) would be the Spaceball, > a ball that you can grip. It senses torque around all three mutually- > orthogonal axes, as well as "translational" force along those axes. It's > not a consumer item; not sure it's still being made. I'm reasonably sure > of the tradename. It was/is used with workstations. "Spaceball" sounds > much like the name of a puzzle. > > (I expect some astute reader to tell me that the MIT Media Lab did just > this thing 5 years ago!) > > My best to all, > Actually, the Spaceball that you talk about is still in existence of sorts. I have three Spaceballs. Actually, they were known as the Spacetec Spaceball Avengers. Those are no longer produced. They've been replaced by the newer model, the SpaceOrb 360. Not to get too far off subject, but the SpaceOrb is used by some people to play Quake. You can't beat a good Mouse and Keyboard for Quake, but the SpaceOrb's multiple axes of movement does allow for some interesting possibilities. Due to its 3D nature, I think the SpaceOrb would be a natural extension for the solving of 3d puzzles in graphical environments. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa From cube-lovers-errors@oolong.camellia.org Thu Jun 26 16:55:53 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA06984; Thu, 26 Jun 1997 16:55:52 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Thu, 26 Jun 1997 13:53:47 -0700 From: "Jason K. Werner" Message-Id: <9706261353.ZM3850@neuhelp.corp.sgi.com> In-Reply-To: Nicholas Bodley "Hidden cubies; Spaceball" (Jun 26, 7:44) References: Reply-to: "Jason K. Werner" X-Mailer: Z-Mail-SGI (3.2S.2 10apr95 MediaMail) To: Nicholas Bodley , cube-lovers@ai.mit.edu Subject: Re: Hidden cubies; Spaceball Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii On Jun 26, 7:44, Nicholas Bodley wrote: > Subject: Hidden cubies; Spaceball > On another topic, it seems to me that an ideal device for controlling a > computer-simulated Cube (or other similar puzzle) would be the Spaceball, > a ball that you can grip. It senses torque around all three mutually- > orthogonal axes, as well as "translational" force along those axes. It's > not a consumer item; not sure it's still being made. I'm reasonably sure > of the tradename. It was/is used with workstations. "Spaceball" sounds > much like the name of a puzzle. In case anyone is interested: http://www.spacetec.com/ http://www.spaceorb.com/ -Jason -- Jason K. Werner Email: mrhip@sgi.com Systems Administrator Phone: 415-933-6397 USFO I/S Technical Services Fax: 415-932-6397 Silicon Graphics, Inc. Pager: 415-317-4084, mrhip_p@sgi.com "Winning is a habit"-Vince Lombardi;"These go to eleven"-Nigel Tufnel From cube-lovers-errors@oolong.camellia.org Fri Jun 27 16:39:01 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA09468; Fri, 27 Jun 1997 16:39:01 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Fri, 27 Jun 1997 16:28:04 -0400 From: Corey Folkerts Subject: First corners, then sides To: Cube-Lovers@ai.mit.edu Message-ID: <199706271628_MC2-1961-C7B8@compuserve.com> During the last 5 months or so I have been fiddling around with a Rubik's Cube that I got for Toys R Us. I have become pretty good at solving it with the layers method (90 seconds is my record so far.) My question is this : Is the method of solving the cube corners first and then sides any faster (once one becomes good at it) then solving it by layers? If so, could someone please reply with a description of that method. I don't know the names of any fancy moves, so if possible please use F B U D L R for the faces during specific moves and expain the other parts of the strategy clearly. I realize this is asking alot. Thanks in advance to anyone who replies! Corey Folkerts From cube-lovers-errors@oolong.camellia.org Sat Jun 28 14:03:46 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA13053; Sat, 28 Jun 1997 14:03:46 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Sat, 28 Jun 1997 05:50:29 -0400 (EDT) From: Jiri Fridrich X-Sender: fridrich@bingsun2 To: Corey Folkerts cc: Cube-Lovers@ai.mit.edu Subject: Re: First corners, then sides In-Reply-To: <199706271628_MC2-1961-C7B8@compuserve.com> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII It appears that both systems are good if worked out into sufficient detail. As far as I know, the fastest speed cubists can achieve an average of 16-17 seconds irrespective! of whether they use corners-edges or by-slices systems. You can look at a system which I have developed some time ago and which enables me to solve the cube in 17 sec. on average. It is described in detail here: http://ssie.binghamton.edu/~jirif/. I also recommend the section on speed cubing tips. Good luck! Jiri On Fri, 27 Jun 1997, Corey Folkerts wrote: > > > During the last 5 months or so I have been fiddling around with a > Rubik's Cube that I got for Toys R Us. I have become pretty good at solving > it with the layers method (90 seconds is my record so far.) My question is > this : Is the method of solving the cube corners first and then sides any > faster (once one becomes good at it) then solving it by layers? If so, > could someone please reply with a description of that method. I don't know > the names of any fancy moves, so if possible please use F B U D L R for > the faces during specific moves and expain the other parts of the strategy > clearly. I realize this is asking alot. Thanks in advance to anyone who > replies! > > Corey Folkerts > > ********************************************************************** | Jiri FRIDRICH, Research Associate, Dept. of Systems Science and | | Industrial Engineering, Center for Intelligent Systems, SUNY | | Binghamton, Binghamton, NY 13902-6000, Tel.: (607) 797-4660, | | Fax: (607) 777-2577, E-mail: fridrich@binghamton.edu | | http://ssie.binghamton.edu/~jirif/jiri.html | ********************************************************************** ...................................................................... Remember, the less insight into a problem, the simpler it seems to be! ---------------------------------------------------------------------- From cube-lovers-errors@oolong.camellia.org Sun Jun 29 22:09:51 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (oolong.camellia.org [206.119.96.100]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id WAA16484; Sun, 29 Jun 1997 22:09:51 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199706292239.CAA27387@telecom.lek.ru> From: Alex Joukov To: Cube-Lovers@ai.mit.edu Subject: Newcomer: Algoritms. How to solv U-slice if D-slice and UD-slice have been solvd yet? Date: Mon, 30 Jun 1997 03:11:05 +0400 X-MSMail-Priority: Normal X-Priority: 3 X-Mailer: Microsoft Internet Mail 4.70.1155 MIME-Version: 1.0 Content-Type: text/plain; charset=KOI8-R Content-Transfer-Encoding: 7bit Dear Cube-Lovers, How to solv U-slice if D-slice and UD-slice have been solvd yet? I remember that in about 1994 when I was a child I used 3 or 5 algoritms (with mirrow variants). But now I don't remember the way. I read cube-lovers arhives (from #0) and step by step find out some algoritms. But before I will be able to solve Cube I lose a lot of interesting in archive messages. I just can't try a lot not having solved Cube! Please help me. A need just "1. U2DFTU'RD' 2. UR'D2L'D2 3..." without comments. Or, may be somebody have created such type FAQ which is accesible for ftp? lllykob@telecom.lek.ru Sasha From cube-lovers-errors@oolong.camellia.org Mon Jun 30 15:09:03 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id PAA18257; Mon, 30 Jun 1997 15:09:02 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Mon, 30 Jun 1997 14:08:40 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Inverses of Local Maxima To: cube-lovers@ai.mit.edu Reply-to: Jerry Bryan Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII X-X-Sender: jbryan@PSTCC6.pstcc.cc.tn.us One of the oldest unsolved problems on Cube-Lovers (aside from God's Algorithm itself) has to do with inverses of local maxima. It seems obvious that the inverse of a local maximum also ought to be a local maximum. But is it necessarily so? In Symmetry and Local Maxima, Jim Saxe and Dan Hoey suggest that it may not be. Their example is UFF, which can end with F or F' because we can write it as UF'F'. But the inverse is F'F'U', which can only end with U' Hence, there are very simple positions where the number of q-turns with which the position can end is different than the number of q-turns with with the inverse of the position can end. If the same thing should happen with a local maximum, then the inverse would not be a local maximum. On the other hand, for all known local maxima in G, the inverse is also a local maximum. What are we to think? I have some small progress. I can report that for the corners-only group, there are local maxima for which the inverse is not a local maximum. The results were obtained with my new Shamir program. For each position x, we define E(x) to be the set of all quarter-turns with which a minimal process for the position can end. As an example, if x=UFF, then E(x)={F,F'}. E(x) is a subset of Q, the set of twelve quarter-turns, or equivalently it is an element of P(Q), the power set of Q. As such, it is conveniently represented in my program as a bit-string of twelve bits. In this notation, we would say that a position x is a local maximum if E(x)=Q or if |E(x)|=12. We also define S(x) to be the set of all quarter-turns with which a minimal process for a position can start. In this notation, for x=UFF we would say that |S(x)|=1 and |E(x)|=2. So the general question for local maxima becomes the following: if |E(x)|=12, does it necessarily follow that |S(x)|=12? My program calculates S(x) and E(x) as follows. Any breadth-first search may be characterized as calculating products of the form z=xy for suitable choices of x and y. Most typically, x comes from Q[n], the set of all quarter-turns of length n, and y comes from Q[1], the set of all quarter-turns of length 1. But in my more general Shamir program, x comes from Q[m] and y comes from Q[n] to form products of length m+m. In any case, S(z) is the union of S(x) over all x which can be a part of a product which produces z, and E(z) is the union of E(y) over all y which can be a part of a product which produce z. For each q in Q, we initialize with S(q)=E(q)={q} and go from there. Here is a portion of a printout from my program. |x| |E(x)| |S(x)| M-Conjugacy Positions Classes 0 0 0 1 1 1 1 1 1 12 2 1 1 2 96 2 2 2 3 18 3 1 1 12 576 3 1 2 3 96 3 2 1 3 96 3 2 2 4 96 3 3 3 2 60 As you can see, the effect pointed out by Saxe and Hoey first shows up three moves from Start, where there are six positions unique up to M-conjugacy where |S(x)| is not equal to |E(x)|. (Actually, three of the six positions are just the inverses of the other three.) The first local maxima are six moves from Start in the corners-only group. |x| |E(x)| |S(x)| M-Conjugacy Positions Classes 6 12 12 8 114 As you can see, there are 114 local maxima of which 8 are unique up to M-conjugacy. However, for all 8 of them, the inverse is also a local maximum so we discover nothing new. The new discovery occurs 7 moves from Start. |x| |E(x)| |S(x)| M-Conjugacy Positions Classes 7 12 8 4 120 7 12 10 3 144 7 12 12 14 336 As you can see, there are 21 local maximu unique up to M-conjugacy. For 14 of them, the inverse is also a local maximum. But for the other 7, the inverse is not a local maximum. In 4 cases, we have |S(x)|=8, and in 3 cases we have |S(x)|=10. Here follow summaries for local maximum up to a distance of 11 moves from Start. |x| |E(x)| |S(x)| M-Conjugacy Positions Classes 8 12 6 14 576 8 12 8 12 576 8 12 10 86 4128 8 12 11 13 624 8 12 12 272 12012 9 12 4 26 1152 9 12 6 31 1344 9 12 8 24 1152 9 12 10 14 576 9 12 12 131 5976 10 12 2 14 576 10 12 4 88 4032 10 12 6 218 10368 10 12 8 144 6336 10 12 10 168 8064 10 12 12 140 5664 11 12 4 384 18432 11 12 6 2687 128688 11 12 8 5550 264192 11 12 10 5014 240576 11 12 12 3617 166224 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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@oolong.camellia.org Mon Jun 30 19:21:04 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id TAA18780; Mon, 30 Jun 1997 19:21:03 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <19970630230531.25512.rocketmail@send1.rocketmail.com> Date: Mon, 30 Jun 1997 16:05:31 -0700 (PDT) From: Bill Webster Subject: Hi To: cube-lovers@ai.mit.edu MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Hi, I first encountered the cube sometime in the early 80s when the fad hit Australia. Solving it absorbed all my spare time for two weeks along with several wads of A4 and $15 for a second cube which I used to develop what I called 'sequences'. My experiments on the second cube were conducted with great care, as I had not yet discovered that Satan's Algorithm could be had for the price of a small screwdriver. I never became a speed-freak, or even employed operators outside my own meagre discoveries, but could solve the cube comfortably inside three minutes. I solved corners, then edges because the first reasonable sequences I discovered were edge-disrupting corner operators. My operators were short and scant so my method incurred a lot of short term memory overhead, manipulating faces into susceptible positions, applying the sequence, then inverting the prior manipulation. A friend of mine aquired a cube at about the same time and much to my chagrin, solved it in less than a day, without paper, without explicitly developing any operators. In fact, he couldn't give a satisfactory account of exactly how he'd solved it. He may have just got *extremely* lucky and stumbled on something close to START, but I don't think so. I handed him a scrambled cube a couple of weeks later and he was quite taken aback - he wasn't going through all that again, he'd done it hadn't he?. I always felt that my own solution was somewhat contrived after witnessing this feat. Does anyone else have examples of GestaltCube? I have coded a C++ class which represents cube states and operators and which includes methods to manipulate the cube. I would like to implement overloaded C++ operators in a manner which is consistent with (and perhaps extends) the appropriate mathematical notation *if this is feasible*. Is there anyone out there familiar with both grammars and willing to make suggestions? I am aware and prepared to accept that the use of some (C++) operators may introduce inefficiencies in the form of temporary objects created during expression evaluation - such operators will not be used in time critical code. I have been using a freeware ray-tracer, POV-Ray to produce 'photo-realist' cube images. I intend to extend my software to export animation scripts, so that I can produce (externally rendered) animated solutions. These take forever to trace, so their value is aesthetic rather than practical. I have been experimenting with solid gold cubes inlaid with coloured marble etc., but still prefer the platonic form. I have the POV source for a static cube if anyone is interested. The POV team are true heroes - details... "The internet home of POV-Ray is reachable on the World Wide Web via the address http://www.povray.org and via ftp as ftp.povray.org." "POV-Ray can be used under MS-DOS, Windows 3.x, Windows for Workgroups 3.11, Windows 95, Windows NT, Apple Macintosh 68k, Power PC, Commodore Amiga, Linux, UNIX and other platforms." Regards, Bill Webster _____________________________________________________________________ Sent by RocketMail. Get your free e-mail at http://www.rocketmail.com From cube-lovers-errors@oolong.camellia.org Mon Jun 30 21:31:40 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA19022; Mon, 30 Jun 1997 21:31:39 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Mon, 30 Jun 1997 20:01:16 -0400 (EDT) From: Jerry Bryan Subject: Re: Inverses of Local Maxima In-reply-to: To: cube-lovers@ai.mit.edu Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII On Mon, 30 Jun 1997, Jerry Bryan wrote: > > I have some small progress. I can report that for the corners-only > group, there are local maxima for which the inverse is not a local > maximum. > There is a minor interesting point that might be added. When we find a local maximum x for which |S(x)|<12, we can form a new, longer local maximum qx for suitable q in Q. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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@oolong.camellia.org Mon Jun 30 21:57:52 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA19066; Mon, 30 Jun 1997 21:57:52 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707010132.VAA26778@life.ai.mit.edu> Date: Mon, 30 Jun 1997 21:37:39 -0400 From: michael reid To: cube-lovers@ai.mit.edu, jbryan@pstcc.cc.tn.us Subject: example of a local maximum whose inverse is not a local maximum jerry bryan asks if the inverse of a local maximum is necessarily a local maximum. the following example shows that this need not be the case. the interesting "six-two-one" pattern is produced by the sequence B U2 F2 R U' R' B' R' U F2 U2 (15q) this position has six symmetries, generated by the cube rotation C_UFR and central reflection. therefore we also have the maneuvers L F2 R2 U F' U' L' U' F R2 F2 D R2 U2 F R' F' D' F' R U2 R2 F' D2 B2 L' D L F L D' B2 D2 R' B2 L2 D' B D R D B' L2 B2 U' L2 D2 B' L B U B L' D2 L2 for the same position. it is not hard to check (by computer) that these are minimal maneuvers. note that for each quarter turn, we have a maneuver that ends with that quarter turn. thus, from this position, any quarter turn brings us closer to start, so our position is a local maximum. consider now the inverse position; it is produced by U2 F2 U' R B R U R' F2 U2 B' (15q) it is not hard to check (by computer) that applying the quarter turn B' to this moves us further from start (16q), so this position is not locally maximal. note that this is already in the archives; i first reported it on april 20, 1995 in my message "correction and an interesting example" mike From cube-lovers-errors@oolong.camellia.org Sat Jul 5 20:21:49 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id UAA00582; Sat, 5 Jul 1997 20:21:48 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707052054.QAA25375@life.ai.mit.edu> Date: Sat, 5 Jul 1997 16:58:20 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: optimal cube solver the recent work by rich korf on finding optimal solutions has prompted me to try my hand at writing an optimal cube solving program. so far, i've done this for the face turn metric. a description of my program follows. let H denote the intermediate subgroup which we've seen before. we'll use distances to this intermediate subgroup for our pruning tables (or "pattern databases"). calculating these distances involves doing a breadth first search on the coset space H \ G , and storing these distances in memory. (i've written this as a right coset space, rather than a left coset space.) this search has been done several times, by dik winter and by myself. some review. positions in H are characterized by the following. corners cannot change orientation; their U or D facelet remains on the U or D face. similar, edges cannot change orientation. furthermore, the four U-D slice edges remain in the U-D slice. therefore, cosets in H \ G are described by triples (c, e, l), where c denotes corner orientation, e denotes edge orientation and l denotes the location of the four U-D slice edges. there are 3^7 = 2187 possible corner orientations, 2^11 = 2048 possible edge orientations / 12 \ and \ 4 / = 495 possible U-D slice edge locations. all combinations are possible, so there are 2187 * 2048 * 495 = 2217093120 cosets. since this is too many configurations to store in memory, we use symmetry to to reduce this number. there are 16 symmetries of the cube that preserve the U-D axis, and therefore the intermediate subgroup H. rather than store all the cosets, we'll just store one of each up to symmetry. actually, this is slightly more complicated than necessary; instead, we could just divide the corner coordinate by symmetry. this is what i did in my message of january 7, 1995. however, i encountered a pitfall along the way. i discovered (very late in the development stage) the need for very large transformation tables. although i continued with the same approach at that time, i gave two options for overcoming this problem: > i) only use the 8 symmetries that preserve my choice of > 12 edge facelets. > > ii) combine the two coordinates edge and location into a single > coordinate and divide this coordinate by the 16 symmetries. of these, clearly the second is the better choice, since it utilizes more symmetry. this new edge coordinate has 2048 * 495 = 1013760 possibilities. up to symmetry, there are 64430 possibilities. we need room for 64430 * 2187 = 140908410 cosets in memory. for each of these, we store its distance to the identity coset. this is an integer between 0 and 12 (inclusive), so each is stored in half a byte. thus the whole table requires 67 megabytes. essentially, what we're doing here is changing coordinates from (c, e, l) to (c, e', s), where e' is our new edge coordinate, and s is a symmetry coordinate. some cosets have multiple coordinates in this new system, but that causes no harm. a breadth first search of this space takes under 11 minutes. the increase in speed is partially due to a more powerful computer, and partially due to switching to "backward searching" (or "bidirectional search") at the optimal time. we'll also use distances to the intermediate subgroups and . we don't need to store additional coset spaces, since we can derive that information from our first coset space. note that the cube rotation C_UFR takes the subgroup to the subgroup . therefore it transforms the first coset space into the second coset space. furthermore, it preserves distances, so the one pattern database suffices for all three applications. an attractive feature of this approach is that it uses the 16 symmetries to reduce the size of the pattern database, and then uses the remaining symmetry of the cube in applying it in different orientations. these are the only pruning tables my program currently uses. note that they cannot "see the entire group". specifically, let H_0 = , H_1 = , H_2 = , and let T denote the intersection of these three subgroups. for a given position, the three distances to these subgroups depend only upon the corresponding coset in T \ G . thus T might be thought of as a "target subgroup". this target subgroup T is interesting. it consists of those positions that "look like" they're in the "square group" , i.e. F and B colors mix only with each other, and similarly for R and L , and also for U and D. however, this is strictly larger than the square group; it contains the square group as a subgroup of index 6. the searching is done in the way that korf describes as "IDA*" (or at least the "ID" part of that terminology). we traverse the tree of all sequences of length 1, hoping to find a solution. that generally fails, so we continue to sequences of length 2, and so forth, until a solution is found. the "A*" part of the algorithm is to use the pruning tables to avoid searching large parts of the tree that are guaranteed not to bear fruit. in his paper, korf uses the expected value of his heuristic functions to get an estimate of how effective they are at pruning the search tree. actually, he should subtract 1 from this expected value, since we must generate (at least partially) the top node of a subtree that gets pruned. this is only a rough estimate; getting a more precise figure is a delicate matter which i won't address here. korf reports an expected value of 8.878. i generated 10 million random cubes (i did not use the long sequence of random twists method) and got an expected value of 9.941. my program generates slightly more than 500000 nodes per second. korf generates them at 700000 per second, so i've got more overhead per node. however, it generates many fewer nodes, since it prunes the search tree more efficiently. i solved korf's ten random cubes, and found all minimal solutions, rather than stopping at the first. this entailed one complete search through length 16f, three through length 17f and six through length 18f. the position at distance 16f has a unique minimal solution, as do the three positions at distance 17f. of the six positions at distance 18f, one has a unique minimal solution, one has 3 minimal solutions, two have 4 minimal solutions and two have 6 minimal solutions. the total run time for these was just under 198 hours. korf estimates 4000 hours for the same search, so on these positions, my program is twenty times as fast. my computer has a 200 MHz pentium pro processor, and is configured with 128 megabytes of RAM. i'd expect a similar increase in performance for most positions, but not all. for example, positions inside the target subgroup T run very slowly, as do positions very close to it. hopefully, most of these are close enough to start, so that searches don't have to go very deep. i suspect that there are probably also positions that give korf's program difficulty. as you can see, i've made only minor modifications to korf's method. the only differences are: 1. use different pattern databases that allow more efficient pruning. 2. apply the same pattern database in multiple orientations. 3. allow a target subgroup larger than just the identity. it's clear that more experimentation is needed with different pattern databases. for any subgroup K of G , we could consider distances to that subgroup. it seems likely that we want small subgroups, so that the average distance is large. for this reason, using symmetry to reduce the size of the database is an important tool. i encourage others to experiment with different subgroups. more results to come ... mike From cube-lovers-errors@oolong.camellia.org Mon Jul 7 02:22:55 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id CAA04972; Mon, 7 Jul 1997 02:22:54 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707070459.AAA17039@life.ai.mit.edu> Date: Mon, 7 Jul 1997 01:04:35 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: symmetry reductions for superflip a simple counting argument shows that some cube positions are at least 18 face turns from start, and thus the diameter of the cube group is at least 18f. in january 1995, i showed, by exhaustive search, that the position "superflip" is exactly 20 face turns from start. therefore the diameter is at least 20f. this gave the first improvement to the lower bound obtained by the counting argument. the searching method i used at the time was my version of kociemba's algorithm. although my symmetry reductions fit together quite well with kociemba's algorithm, this might not be the most appropriate searching method to use for this purpose. (i guess i could have hacked it not to bother looking for solutions longer than 19f. i don't remember why i didn't do this.) my new optimal solving program can do an exhaustive search in much less time. the symmetry reductions are similar, but much simpler. i will try be more coherent this time with my explanation, hopefully without being overbearing. the first thing to note is that dik winter found a maneuver for superflip in 20f: F B U2 R F2 R2 B2 U' D F U2 R' L' U B2 D R2 U B2 U (20f) therefore our concern is with searching for maneuvers of length at most 19f. there are three ways to transform a maneuver for superflip to get another such maneuver, which do not change its length: 1. we may conjugate the maneuver by any symmetry of the cube. 2. we may cyclically shift the maneuver; i.e. replace sequence_1 sequence_2 by sequence_2 sequence_1 3. we may replace the maneuver by its inverse. (in fact, we won't use 3 here, but it might be helpful elsewhere.) our first result is proposition 1. any maneuver for superflip in 19f contains a 180 degree face turn. proof. if the proposition were false, then superflip would be an odd number of quarter turns from start, contradiction. qed. the relevance of this proposition is proposition 2. suppose that a maneuver for superflip contains a 180 degree face turn. then it can be transformed, using the above tranformations, into a maneuver that begins with U R2. proof. we first claim that the maneuver has two consecutive "syllables" such that the first contains a 90 degree face turn and the second contains a 180 degree face turn. a "syllable" is a sequence of one or two face turns along the same axis; e.g. U D2. by hypothesis, the maneuver has a syllable that contains a half turn. if the claim is false, then the preceding syllable contains no 90 degree turns, and therefore consists only of half turns. but then the syllable before that contains only half turns, by the same reasoning. continuing in this way, we see that every syllable consists only of half turns. therefore we have a maneuver for superflip consisting only of half turns. this is a contradiction, so the claim is true. now, since the individual face turns within a syllable commute, we may suppose that the maneuver has a 90 degree face turn followed by a 180 degree face turn, which are along different axes, and thus are adjacent faces. now we may conjugate by an appropriate symmetry of the cube to suppose that these turns are U R2. finally, we may cyclically shift the maneuver so that these are the first two turns. qed. proposition 3. suppose that superflip is exactly 19 face turns from start. then applying the sequence U R2 to it brings us 2 face turns closer to start, i.e. 17f from start. proof. apply proposition 1 and proposition 2. qed. we now know how to handle the case that superflip's distance from start is exactly 19f. if the distance is less than 19f, we use the following proposition 4. under any circumstances, applying the sequence U R2 to superflip brings us at least 1f closer to start. proof. a minimal maneuver for superflip must contain a 90 degree twist, and we may suppose that the next face turned is an adjacent one. by cyclically shifting the maneuver, we may bring these two turns to the beginning. furthermore, by symmetry, we may suppose that the first turn is U and the second is some twist of the R face. now by applying U to superflip, we've moved 1f closer to start, and applying R2 to this doesn't move us any further from start, since it either combines with, or cancels the next turn in the minimal maneuver. qed. putting this all together, we get our desired result. proposition 5. suppose that superflip is within 19f of start. then the position superflip U R2 is within 17f of start. proof. this is just combining props 3 and 4. qed. i don't claim that these are the best reductions possible. they suffice for our purposes. i tested the position superflip U R2 (i.e. the position obtained by first doing superflip, and then doing the sequence U R2) with my optimal solver. my program took 2 hours and 40 minutes to exhaustively search this position through 17 face turns (not including about 11 minutes to generate all the lookup tables). there were no solutions. thus superflip is exactly 20 face turns from start. when i did the search in january 1995, the run time was 6 days. so we see quite a bit of improvement. mike From cube-lovers-errors@oolong.camellia.org Tue Jul 8 00:16:07 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id AAA07853; Tue, 8 Jul 1997 00:16:06 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707080413.AAA00423@life.ai.mit.edu> Date: Tue, 8 Jul 1997 00:18:18 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: superfliptwist requires 20 face turns i can now show that the pattern "superfliptwist" is exactly 20 face turns from start. this position was proposed as a likely antipode of start by cubologist christoph bandelow. in the german edition of his book "einfuerung in die cubologie" he offered a prize for the shortest maneuver for this pattern. the prize was collected by rainer aus dem spring, who found a maneuver in 22 face turns. much later, a maneuver of length 20f was found by herbert kociemba: D F2 U' B2 R2 B2 R2 L B' D' F D2 F B2 U F' L R U2 F' (20f) as one of the first applications of his ingenious searching algorithm. i'll try not to be so verbose with my symmetry reductions this time. first note that "superfliptwist" does not describe a unique position of the cube; there are two possible orientations. in this context, i use the term "position" to refer to one of the 43252003274489856000 possible configurations, and the term "pattern" to refer to an equivalence class of positions under symmetries of the cube. (this concept has been discussed by dan hoey and jerry bryan as the "real size of cube space" i.e. the number of patterns.) the following two facts are easily verified: * superfliptwist commutes with the square of each face turn. * it does not commute with 90 degree slices (e.g. U D') or 90 degree antislices (e.g. U D), however, if A is a 90 degree slice or antislice, then A superfliptwist A^(-1) is also superfliptwist, but in the other orientation. these facts lead to the importance of the following proposition. superfliptwist is not in the subgroup generated by slices and antislices. (note that this group contains all squares of face turns.) proof. we may ignore the corners and just show that all edges cannot be flipped in this subgroup. to do this, we choose dominant facelets on the 12 edges as follows: choose the U or D facelet of the edges in the R-L slice, the R or L facelet of the edges in the F-B slice and the F or B facelet of the edges in the U-D slice. now we may define the flip of an edge that is not in its correct location. all edges start in the correct orientation. a 90 degree slice or antislice along the U-D axis changes the orientation of all eight edges in the F-B slices and R-L slices. similarly, a 90 degree slice or antislice along the F-B or R-L axis flips all edges in two different slices. within this subgroup, either all edges in a given slice are flipped, or none are flipped, and furthermore, the number of the three slices with flipped edges is even, i.e. 0 or 2. however, superfliptwist has all three slices with flipped edges, so it is not in this subgroup. qed. now consider the first syllable of a minimal maneuver for superfliptwist. ("syllable" was defined in my previous message.) if this is a single 180 degree turn, then we may cyclically shift this to the end of the maneuver. similarly, a slice squared may also be shifted to the end of the maneuver. furthermore, 90 degree slices and or antislices may also be shifted to the end of the maneuver, with only the mild effect of changing which orientation of superfliptwist we're doing. from the proposition, we eventually find a syllable which is not of these types, and is therefore of type U or D2 U. in the case of D2 U , we may shift the D2 to the back of the maneuver, so we may suppose that the first face turn is U . furthermore, by conjugating by the cube rotation C_U, if necessary, we may suppose that our maneuver solves our preferred orientation of superflip. the second face turn is in a different syllable, so it is an adjacent face. conjugating by C_U2, if necessary, brings this face to either R or F. therefore we may suppose that the first two face turns are one of the six sequences U R , U R2 , U R', U F , U F2 or U F' . to show that superfliptwist is not within 19f of start, i tested the six patterns obtained by applying these sequences to it. it took my program 7.5 hours to exhaustively search all of these through 17f. (these positions ran a bit faster than most of the others i've tested. this is partly because superfliptwist is 15 face turns from my "target" subgroup, so larger parts of the search tree are pruned.) no solutions were found, so superfliptwist requires 20 face turns. i also let the first situation run partially through depth 18f. in about 4 and a half hours, it found a solution which yields U R F' B U' D' F U' D F L F' L' U R D F U R L (20f, 20q) this is automatically minimal in the quarter turn metric! mike From cube-lovers-errors@oolong.camellia.org Tue Jul 8 17:33:24 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id RAA09883; Tue, 8 Jul 1997 17:33:24 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707082039.QAA04704@life.ai.mit.edu> Date: Tue, 8 Jul 1997 16:43:50 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: composition of superflip and pons asinorum the position which is the composition of superflip and pons asinorum is exactly 19 face turns from start. some time ago, jerry bryan found that this position is exactly 20 quarter turns from start, and he gave all minimal maneuvers, up to symmetry. one of these is 19 face turns long: B' D' L' F' D' F' B U F' B R2 L U D' F L U R D (19f, 20q) symmetry reductions for this position are much simpler (but not nearly as good) as for superflip and/or superfliptwist. if the first face turn is a 90 degree turn, then by symmetry, we may suppose it is U . if the first face turn is a 180 degree turn, then we may suppose it is U2 . i tested the two positions obtained by applying these possible initial turns. my program took about 6 and a half hours to exhaustively search these through 17 face turns. no solutions were found, and therefore the original position is more than 18 face turns from start. i realize that this is not nearly as satisfying as obtaining all minimal maneuvers. that will take about 13 times as long, but is feasible with my current program. mike From cube-lovers-errors@oolong.camellia.org Thu Jul 10 00:56:16 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id AAA12818; Thu, 10 Jul 1997 00:56:16 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707100450.AAA14097@life.ai.mit.edu> Date: Thu, 10 Jul 1997 00:54:58 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: all minimal maneuvers for superflip (face turn metric) i can now give all minimal maneuvers for superflip in the face turn metric. recall that there are three operations we may apply to any maneuver for superflip which give another maneuver of the same length: 1. we may conjugate the maneuver by any symmetry of the cube. 2. we may cyclically shift the maneuver; i.e. replace sequence_1 sequence_2 by sequence_2 sequence_1 3. we may replace the maneuver by its inverse. my original search (january 1995) for superflip in 19 face turns was divided into 16 cases. since i used my (unhacked) version of kociemba's algorithm, the search through each case produced maneuvers for superflip, and 8 of these cases found maneuvers of length 20f. i previously reported that these were each equivalent to dik winter's maneuver, using the three operations above. however, i was mistaken about this; there were two different maneuvers which differ only very slightly. to facilitate an exhaustive search through 20f, i'll use a result of a previous search. proposition. any maneuver for superflip in 20f contains a 180 degree face turn. proof. otherwise the maneuver would be 20 quarter turns long. however, i did an exhaustive search through 20q and found no maneuvers. qed. (in fact, this quarter turn result was later improved by jerry bryan, who showed that superflip is not within 22q of start, and therefore is exactly 24q from start.) now the symmetry reductions show that we may take the first two face turns to be U R2 . my program exhaustively searched the position superflip U R2 through 18f. it took 35 hours, and found 30 maneuvers, which came in two different types: U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f) U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f) note that these are the same except for the last 5 face turns. (this gives the relation R' L B2 U2 F2 R L' U2 B2 D2 = identity; alternatively, the same sequence produces (f++)(d++) in the supergroup.) from this, we can count the exact number of 20f sequences for superflip. both of the above may be cyclically shifted in 23 different ways. we get 23 different ways, instead of 20, because there are three separate pairs of consecutive twists of opposite faces. we'd consider sequence_1 U D sequence_2 and sequence_1 D U sequence_2 to be the same, but we wouldn't consider U sequence D and D sequence U to be the same. yet cyclic shifting of these last two produces the same maneuver. we can also conjugate by any of the 48 symmetries of the cube, and we can also invert any of the maneuvers. all these operations produce different maneuvers, so we get a total of 2 * 23 * 48 * 2 = 4416 different maneuvers. by counting, the number of different sequences of length <= 19f is about 82 times as many positions the cube has. thus a position has, on average, 82 maneuvers of length <= 19f, although superflip has 0. the number of different sequences of length 20f is about 1016 times the number of positions, so a position has, on average, 1016 different maneuvers of length 20f. superflip has more than 4 times that many. here are the 30 solutions my program found for superflip U R2. hopefully i haven't made any mistakes this time. they should all be equivalent to one of the two listed above. U R2 F U2 F2 D2 R' L U R2 F' B' R D2 L F2 R D2 R D (20f) U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f) U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f) U R2 F R' L F2 D2 B2 U R2 F' B' R D2 L F2 R D2 R D (20f) U R2 F2 L2 F D2 R L D R2 D F2 U R2 D F' B' D2 L D' (20f) U R2 F2 L2 F' U2 R' L' U' R2 U' F2 D' R2 U' F B U2 L' D' (20f) U R2 F2 L2 B D2 R' L' D F2 U R2 D F2 D F B D2 R D' (20f) U R2 F2 L2 B' U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' D' (20f) U R2 F' U2 B2 D2 R L' D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f) U R2 F' B' R D2 L F2 R D2 R U D R2 F U2 F2 D2 R' L (20f) U R2 F' B' R D2 L F2 R D2 R U D R2 F R' L F2 D2 B2 (20f) U R2 F' R L' B2 D2 F2 D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f) U R2 U B2 D R2 U F' B' U2 L F2 R2 B2 U' D B U2 R L (20f) U R2 U B2 D R2 U F' B' U2 L U' D R2 B2 L2 B U2 R L (20f) U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' U R2 D F2 (20f) U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 (20f) U R2 U2 L2 F' B R F2 U' D' F L2 B U2 F L2 F R L F2 (20f) U R2 B R' L B2 U2 F2 D R2 F' B' R U2 L B2 R U2 R D (20f) U R2 B D2 B2 U2 R' L D R2 F' B' R U2 L B2 R U2 R D (20f) U R2 B' R L' F2 U2 B2 U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f) U R2 B' D2 F2 U2 R L' U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f) U R2 D F2 U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' (20f) U R2 D F2 U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' (20f) U R2 D F2 U R' L' U2 B L2 F2 R2 U' D R U2 F B U F2 (20f) U R2 D F2 U R' L' U2 B U' D F2 R2 B2 R U2 F B U F2 (20f) U R2 D F2 D F B D2 R U D' R2 F2 L2 B D2 R' L' D F2 (20f) U R2 D F2 D F B D2 R B2 R2 F2 U D' B D2 R' L' D F2 (20f) U R2 D F' B' D2 L U D' R2 F2 L2 F D2 R L D R2 D F2 (20f) U R2 D F' B' D2 L B2 R2 F2 U D' F D2 R L D R2 D F2 (20f) U R2 D2 L2 F B' L B2 U D B D2 B L2 F D2 B R' L' B2 (20f) mike From cube-lovers-errors@oolong.camellia.org Sun Jul 13 19:35:33 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id TAA06922; Sun, 13 Jul 1997 19:35:33 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707132333.TAA15870@life.ai.mit.edu> Date: Sun, 13 Jul 1997 19:37:58 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for composition of superflip and pons asinorum i've finished calculating all minimal maneuvers (in the face turn metric) for the composition of superflip and pons asinorum. in my search for maneuvers of length <= 18f for this position, i used symmetry to show that we may suppose that the first face turn is either U or U2 . in fact, there is more symmetry available, and this time i will use it. in the case beginning with U , there are four symmetries, generated by the cube rotation C_U . using these, we may suppose that the second face turn is one of D , D2 , D' , R , R2 or R' . in the case beginning with U2 , there are eight symmetries. these are generated by the cube rotation C_U and reflection through the left-right plane. using these, we may assume that the second face turn is one of D , D2 , R or R2 . we can reduce these cases somewhat further. the cases beginning with U D2 and with U2 D are equivalent, so only one needs to be seacrhed. the cases beginning with U D' and with U2 D2 can also be eliminated. both U D' and U2 D2 commute with both pons asinorum and with superflip, so we may cyclically shift these turns to the end of the maneuver. our position cannot be achieved only using these "slice" turns, so we'll always be able to cyclically shift until we do not begin with a slice turn. (alternatively, note that any maneuver of length 19f , or any odd length cannot consist only of slice turns!) that leaves seven cases to search. my program took just less than one day to search all through 17f. it found 26 maneuvers, 16 for the case beginning with U D . however, this case has 8 symmetries, so there are just 2 different maneuvers, each in 8 different orientations. this leaves 12 different maneuvers, which come in 6 pairs of inverses. they are: U R F D R U' D L' U' D F' B2 R L' D' F' L' B' R' (19f) U D F R L' F B' L D2 R L F' B' U' L2 F B' U2 L' (19f) U D F' B' L' U2 F' B L2 U' R' L' F' U' D F' B D' L2 (19f) U2 R F U F B' L' D' F B' L B R L' U D2 B' R' U2 (19f) U2 R F U2 D' R' L F' L' F B' U L F B' D' B' R' U2 (19f) U2 R U2 D2 R U' L' U B R F2 U' D B' R' F' D B' L2 (19f) and their inverses. the first of these is the maneuver found by jerry bryan. it's also the only of these that is 20 quarter turns long, which is consistent with his findings. mike From cube-lovers-errors@oolong.camellia.org Mon Jul 14 12:37:17 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id MAA08816; Mon, 14 Jul 1997 12:37:16 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <3.0.1.32.19970714113921.006aa4a4@pop.tiac.net> X-Sender: kangelli@pop.tiac.net X-Mailer: Windows Eudora Light Version 3.0.1 (32) Date: Mon, 14 Jul 1997 11:39:21 -0400 To: cube-lovers@ai.mit.edu From: karen angelli Subject: hockey puck puzzle Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" I recently purchased a hockey puck puzzle from Whole Systems Design, who also brought us the Masterball. Since I received it, I've played with for only a couple of hours without beginning any systematic attempt at solving it. My initial impression is that, although it looks like is should be very easy, it may actually be pretty hare. Or, it's one of those puzzles that is so easy that I'm thinking too hard and can't see the answer. I've never heard anyone in the cube-lovers discuss the puzzle, and I wondered if any of you had any impressions of it. The puzzle has the shape, size and feel of a regulation hockey puck, and is divided into twelve wedges. It's basically a flattened masterball in which only the lines of longitude twist, and the lines of latitude do not. There are several designs, with various degrees of difficulty and redundancy. For example, on some, there are printed hockey players, Maple leafs, or American flags. On my version, the wedges are numbered consecutively from one through twelve. The pristine version is with the numbers lined up in consecutive order. I'm not really interested in learning a solution from anybody, but I would be interested in comments about whether you think the puzzle is harder or easier than it looks. YOu can see pictures of the hockey puck puzzle or order them at www.wsd.com/HockeyPuck/home. 'e-ya later, Pete. From cube-lovers-errors@oolong.camellia.org Mon Jul 14 14:02:22 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA08973; Mon, 14 Jul 1997 14:02:22 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Mon, 14 Jul 1997 13:59:16 -0400 (EDT) Message-Id: <199707141759.NAA00043@spork.bbn.com> From: Allan Wechsler To: karen angelli Cc: cube-lovers@ai.mit.edu Subject: hockey puck puzzle In-Reply-To: <3.0.1.32.19970714113921.006aa4a4@pop.tiac.net> References: <3.0.1.32.19970714113921.006aa4a4@pop.tiac.net> Please give a clearer description of the puck. The photo gives only a slight clue about how many pieces there are, and how they are arranged. Are there twelve pieces, or twenty-four, or more? -A From cube-lovers-errors@oolong.camellia.org Tue Jul 15 13:18:50 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA10963; Tue, 15 Jul 1997 13:18:49 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Tue, 15 Jul 1997 17:57:11 +0200 From: Rob Hegge Subject: Description of hockey puck puzzle To: cube-lovers@ai.mit.edu Message-id: <9707151557.AA06449@sumatra.mp.tudelft.nl> Content-transfer-encoding: 7BIT X-Sun-Charset: US-ASCII The hockey puck puzzle is a flat disk with a diameter of about 9 cm or 3.5 inches and a thickness of about 2.5 cm or 1 inch. It basically consists of a circle (the hart) and a "ring" surrounding the circle. The circle is cut into two equal halves like "(|)". The two halves are connected so that you can turn one half upside down, while holding the other half. The ring is cut (from front to back) into 12 equal wedges, each of which is attached to the circle by a dovetail so that the ring with the wedges can be moved around the circle. One can also flip six wedges including one half of the circle around so that afterwards those 6 wedges and the half circle face backwards. Thus the puzzle is similar to a puzzle called saturn (which has only 8 wedges ?). The type of moves reminds me of moves possible on square-1. In the puzzle I own the 12 wedges on the front are numbered from 1 to 12 and on the back with the letters of "hockeypuzzle", while the left half circle contains the letters "pu" and right half circle the letters "ck" as shown below. I do not have it here so this was straight from memory. front: back: 12 1 c k 11 2 o e 10 | 3 h | y | pu|ck 9 | 4 p | e 8 5 u l 7 6 z z The three "|" denote the cut through the circle. A flip as described above would give for instance 12 k c 1 11 e o 2 10 | y h | 3 |ck pu| 9 | e p | 4 8 l u 5 7 z z 6 while then a clockwise turn of the ring for one wedge would give: 11 12 1 2 10 k c 3 9 | e o | 4 |ck pu| 8 | y h | 5 7 e p 6 z l u z For a rotational puzzle it is not that difficult. Rob From cube-lovers-errors@oolong.camellia.org Tue Jul 15 13:18:23 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA10958; Tue, 15 Jul 1997 13:18:23 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Tue, 15 Jul 1997 09:41:28 -0400 (EDT) Message-Id: <199707151341.JAA00944@spork.bbn.com> From: Allan Wechsler To: jandr@xirion.nl Cc: Allan Wechsler , cube-lovers@ai.mit.edu Subject: Re: hockey puck puzzle In-Reply-To: <9707151143.AA27610@la1.apd.dec.com> References: <199707141759.NAA00043@spork.bbn.com> <9707151143.AA27610@la1.apd.dec.com> [Jan de Ruiter:] The puzzle contains edge pieces and center pieces, all with the same thickness as the puck. There are always two center pieces which together form the inner circle. In this case there are 12 edge pieces which together form the outer ring. Simpeler pucks might contain less than 12 edge pieces, but always an even number. The possible moves are: 1. rotate one center piece with half of the edge pieces 180 degrees relative to the other center piece and the other half of the edge pieces. 2. rotate the edge pieces around the center pieces, always multiples of 30 degrees, or 1/12 of a circle (or more if there are less edge pieces) I know the puck with 6 edge pieces is near trivial to solve. I haven't tried the other ones yet. Jan de Ruiter Thanks for the description -- Pete ("Karen Angelli") provided an identical one in a private reply. But this is still incomplete. Are the obverse and reverse of the individual pieces distinguishable? Suppose I manage to flip every other edge piece over in place (not sure this is possible). Does it then look solved? Or do the two sides have different colors or a distinguishing mark or something? I haven't tried it, but I can't imagine that this puzzle would be very difficult. -A From cube-lovers-errors@oolong.camellia.org Tue Jul 15 13:17:34 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id NAA10953; Tue, 15 Jul 1997 13:17:33 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <9707151143.AA27610@la1.apd.dec.com> Subject: Re: hockey puck puzzle To: Allan Wechsler Date: Tue, 15 Jul 1997 13:42:54 +0200 (CETS) From: Jan de Ruiter Cc: cube-lovers@ai.mit.edu In-Reply-To: <199707141759.NAA00043@spork.bbn.com> from "Allan Wechsler" at Jul 14, 97 01:59:16 pm Reply-To: jandr@xirion.nl X-Mailer: ELM [version 2.4 PL23] Content-Type: text > > Please give a clearer description of the puck. The photo gives only a > slight clue about how many pieces there are, and how they are > arranged. Are there twelve pieces, or twenty-four, or more? > > -A > The puzzle contains edge pieces and center pieces, all with the same thickness as the puck. There are always two center pieces which together form the inner circle. In this case there are 12 edge pieces which together form the outer ring. Simpeler pucks might contain less than 12 edge pieces, but always an even number. The possible moves are: 1. rotate one center piece with half of the edge pieces 180 degrees relative to the other center piece and the other half of the edge pieces. 2. rotate the edge pieces around the center pieces, always multiples of 30 degrees, or 1/12 of a circle (or more if there are less edge pieces) I know the puck with 6 edge pieces is near trivial to solve. I haven't tried the other ones yet. Jan de Ruiter From cube-lovers-errors@oolong.camellia.org Wed Jul 16 09:37:39 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id JAA12856; Wed, 16 Jul 1997 09:37:38 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: From: "joyner.david" To: "'cube-lovers@ai.mit.edu'" Subject: RE: hockey puck puzzle Date: Wed, 16 Jul 1997 07:53:21 -0400 X-Mailer: Microsoft Exchange Server Internet Mail Connector Version 4.0.994.63 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit >---------- >From: Jan de Ruiter[SMTP:jandr@apd.dec.com] >Sent: Tuesday, July 15, 1997 7:42 AM >To: Allan Wechsler >Cc: cube-lovers@ai.mit.edu >Subject: Re: hockey puck puzzle > >> >> Please give a clearer description of the puck. The photo gives only a >> slight clue about how many pieces there are, and how they are >> arranged. Are there twelve pieces, or twenty-four, or more? Spencer Robinson, a former student, and I have written a WWW page which sketches an easy but rather inefficient solution of the 12 piece hockey puck puzzle: http://www.nadn.navy.mil/MathDept/wdj/mball/puck.htm Have fun! - David Joyner >> >> -A >> > From cube-lovers-errors@oolong.camellia.org Wed Jul 16 10:18:34 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id KAA12915; Wed, 16 Jul 1997 10:18:33 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 16 Jul 1997 09:44:21 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: No Local Maxima 11q from Start To: cube-lovers@ai.mit.edu Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII X-X-Sender: jbryan@pstcc6.pstcc.cc.tn.us My new Shamir program has now generated the entire search tree for the standard cube group G up to 11q from Start. This search was accomplished once before using my old tape spinning programs, so there is limited new information. One good result is that all the numbers match between the two programs. The matching results were obtained using different programs, implementing different algorithms, in different programming languages, on different hardware platforms, and under a different operating systems. So I feel pretty good about the numbers. They have been posted before, so I won't post them again. With problems this big, it is always good to have some sort of independent verification because it is impossible to verify anything by hand. Another interesting result is in fact new. The old program was only able to determine local maxima up to 10q from Start while calculating the 11q tree. The new program is able to determine local maxima up to the same distance from Start it is searching. There are no local maxima 11q from Start. I find this result somewhat surprising, since there are four local maxima (unique up to M-conjugacy) 10q from Start. The new program did confirm the previously known 10q local maxima, but failed to find any 11q local maxima. In its search for local maxima, for each position x the program calculates the set E(x) of quarter turns with which a minimal process for the position can end. We call |E(x)| the maximality of x, and a position is a local maximum if its maximality is 12. At a distance of 11q from Start, there exist positions with maximality values for every number in 1..11. This is the first time we have found any positions with a maximality of 9 or 11. (See my note of 16 June 1995, "10q Local Maxima Search Matrix".) There seem to be more positions with even maximality values than odd, and a maximality of 11 is especially interesting because such a position is "almost" a local maximum. I am disappointed in the speed of my program. For this run, it identified about 1100 patterns (representative elements of M-conjugacy classes) per second. This corresponds to about 50,000 positions per second (about 48 times 1100). The program is running on a Pentium P166 with 16MB memory under Windows/95. My concern is that I have worked so hard to make the program run in small amounts of memory that it is running too slow. I am now going to take out a few of the memory saving techniques to see if I can speed it up a bit. The program is actually about 20MB, and runs successfully on a 16MB machine due to the good graces of virtual memory. In fact, I can calculate out to 11q from Start even on an 8MB machine. But trying to calculate out to 12q from Start fails on the 16MB machine (the program is the same size for 11q from Start and for 12q from Start because I am storing all positions up to 6q from Start. The program would only have to be made larger if I were to try calculating 13q from Start or 14q from Start.) When I say the program fails at 12q from Start, I mean that the virtual memory thrashes unmercifully, and therein lies an interesting tale. Why should the program be able to calculate 11q without thrashing, but thrash so badly at 12q? It has to do with the Shamir algorithm itself. Recall that we are producing products of the form ST in lexicographic order. To be specific, we are producing products of the from St in lexicographic order for all t in T and merging the results. S itself is already in lexicographic order. Think of processing a dictionary, and thing of processing S in lexicographic order. We essentially process all the A's, followed by all the B's, then all the C's, etc. There is very good locality of reference as far as the virtual memory goes. Moving up to St, we might first process all the N's, then all the E's, then all the Z's, etc, but there is still very good locality of reference. There is an occasional big jump in where we are referencing memory, but most of the time we reference elements of the set S which are very close together in memory. When we calculate 11q from Start, S is the set Q[6] of positions which are 6q from Start, and T is the set Q[5]. Because Q[5] is only about 1/9 as big as Q[6], the real memory working set to calculate Q[6]Q[5] is only about 10% of the total virtual memory of the program, maybe about 2MB. But when we move up to calculating 12q, we move up to Q[6]Q[6] and the real memory working set becomes the whole 20M program. This simply doesn't work on a 16MB machine. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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@oolong.camellia.org Thu Jul 17 14:17:15 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id OAA18327; Thu, 17 Jul 1997 14:17:14 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 16 Jul 1997 22:41:08 -0400 (EDT) From: Jerry Bryan Subject: Re: No Local Maxima 11q from Start In-reply-to: To: cube-lovers@ai.mit.edu Reply-to: Jerry Bryan Message-id: MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII On Wed, 16 Jul 1997, Jerry Bryan wrote: > (See my note of 16 June 1995, "10q Local > Maxima Search Matrix".) > There seem to be more positions with even > maximality values than odd, ... This statement is bogus, which is clear if you look at my chart from 1995. Most of the positions have a maximality of 1 this close to Start. I was thinking of another situation. That is, when I looked at local maxima in the corners only group, the inverses of the local maxima tended to have even maximality. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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@oolong.camellia.org Wed Jul 23 16:50:10 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id QAA04228; Wed, 23 Jul 1997 16:50:09 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Date: Wed, 23 Jul 1997 06:42:55 -0400 From: Edwin Saesen Subject: Where can I get...? To: CUBE Message-ID: <199707230643_MC2-1B6E-D8FD@compuserve.com> MIME-Version: 1.0 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Hi everyone, sorry if this is a boring question for most of you (and if it is, please reply by private mail instead of to the list), but I'm new here, and my most important question at the moment is where to get any of the followin= g puzzles: Rubik's Wahn (5x5x5) (maybe also called Professor's cube, Ultimate or Master Revenge) Rubik's Revenge (4x4x4) Rubik's Domino (3x3x2) Rubik's Pocket Cube (2x2x2) Pyraminx Pyraminx Octahedron Megaminx I'd prefer sources in germany, although I doubt it that I can find any of= them here... Michael From cube-lovers-errors@oolong.camellia.org Wed Jul 23 17:35:00 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id RAA04379; Wed, 23 Jul 1997 17:34:59 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-ID: <33D6768B.707E@ibm.net> Date: Wed, 23 Jul 1997 14:24:27 -0700 From: Jin "Time Traveler" Kim Reply-To: chrono@ibm.net Organization: The Fourth Dimension X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: CUBE Subject: Re: Where can I get...? References: <199707230643_MC2-1B6E-D8FD@compuserve.com> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Edwin Saesen wrote: > > Rubik's Wahn (5x5x5) (maybe also called Professor's cube, Ultimate or > Master Revenge) > Rubik's Revenge (4x4x4) > Rubik's Domino (3x3x2) > Rubik's Pocket Cube (2x2x2) > Pyraminx > Pyraminx Octahedron > Megaminx > > I'd prefer sources in germany, although I doubt it that I can find any of > them here... > > Michael You are quite in luck. There is a great source of puzzles in Germany. Christoph.Bandelow@rz.ruhr-uni-bochum.de bandecbv@rz.ruhr-uni-bochum.de I'm not sure which is correct, or maybe they both work, but Dr. Christoph Bandelow has a catalog available for people to buy various puzzles. He should have all of the above puzzles available for purchase, except the 4x4x4, 3x3x2, and the Megaminx. All of them are rather hard to locate. Especially tough is the 4x4x4 because of its high demand. I've been looking for one as well for a number of years. By many accounts, there are no more for sale anywhere except maybe private collections. And if you own one, you're not likely to part with it anyway. So good luck. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@oolong.camellia.org Wed Jul 23 22:49:31 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id WAA04820; Wed, 23 Jul 1997 22:49:30 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707240207.WAA24816@life.ai.mit.edu> Comments: Authenticated sender is From: Christoph Bandelow To: cube-lovers@ai.mit.edu Date: Thu, 24 Jul 1997 04:04:23 +0000 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: Where can I get...? Reply-to: Christoph.Bandelow@ruhr-uni-bochum.de Priority: normal X-mailer: Pegasus Mail for Windows (v2.42a) Edwin Saesen asked for a source of "Rubik's Wahn" (5x5x5 magic cube) and some other rotational puzzles. As our busy "Time Traveler" Jin Kim already kindly remarked, the 5x5x5 Magic Cube and many other rotational puzzles are still available from me. This refers also to the Megaminx, both the one made in Hong Kong - sold as Megaminx - and the slightly better one made in Hungary - sold as Super Nova or as Magic Dodecahedron. By the way, I do now sell the 5x5x5 cube under the name "Giant Magic Cube" (In Germany: "Riesen-Zauberwuerfel", in France: "Cube Magique Geant") , and I hope this sounds nice and doesn't create too much new confusion. At least the price is still the same: 40 DM or 24 USD. Just email me your postal address to receive your free copy of my mail order catalog. Cube-Lovers: Please notice my new slightly simplified email address. Christoph Christoph Bandelow mailto:Christoph.Bandelow@ruhr-uni-bochum.de From cube-lovers-errors@oolong.camellia.org Wed Jul 23 22:49:09 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id WAA04816; Wed, 23 Jul 1997 22:49:09 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org From: Tim Mirabile To: CUBE Subject: Re: Where can I get...? Date: Wed, 23 Jul 1997 23:16:49 GMT Organization: http://www.webcom.com/timm/ Message-ID: <33d68e99.988371@mail.htp.com> References: <199707230643_MC2-1B6E-D8FD@compuserve.com> <33D6768B.707E@ibm.net> In-Reply-To: <33D6768B.707E@ibm.net> On Wed, 23 Jul 1997 14:24:27 -0700, Jin "Time Traveler" Kim wrote: >He should have all of the above puzzles available for purchase, except >the 4x4x4, 3x3x2, and the Megaminx. All of them are rather hard to >locate. Especially tough is the 4x4x4 because of its high demand. I've >been looking for one as well for a number of years. By many accounts, >there are no more for sale anywhere except maybe private collections. >And if you own one, you're not likely to part with it anyway. So good >luck. Hmmm. I would imagine a large number of the 4x4x4 cubes sold in the U.S. as "Rubik's Revenge" were sold to non-cube-lovers and are now tucked away in attics and basements with all the other toys the kids have outgrown. For this reason I've considered checking out local garage and yard sales. (Before it became widely known what the value of old baseball cards could be, these sales were an excellent way for collectors to pick up large boxes of old cards for only a few dollars). Anyway, I hope Christoph still has the Megaminx because I just sent him a check for some items including this. :) There was no indication he was out of them in the catalog I just got. -- For USCF & FIDE rated chess on Long Island -> http://www.webcom.com/timm/ TimM on the Free Internet Chess Server - telnet://fics.onenet.net:5000/ Webmaster, tech support - ICD/Your Move Chess & Games: http://www.icdchess.com/ The opinions of my employers are not necessarily mine, and vice versa. From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 25 12:21:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA15510; Fri, 25 Jul 1997 12:21:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From R.F.Hegge@MP.TUDelft.NL Thu Jul 24 04:40:13 1997 Date: Thu, 24 Jul 1997 10:36:30 +0200 From: R.F.Hegge@MP.TUDelft.NL (Rob Hegge) Subject: Re: Where can I get...? To: CUBE-LOVERS@ai.mit.edu Message-Id: <9707240836.AA12191@sumatra.mp.tudelft.nl> Content-Transfer-Encoding: 7BIT X-Sun-Charset: US-ASCII > Rubik's Wahn (5x5x5) (maybe also called Professor's cube, Ultimate or > Master Revenge) > Rubik's Revenge (4x4x4) > Rubik's Domino (3x3x2) > Rubik's Pocket Cube (2x2x2) > Pyraminx > Pyraminx Octahedron > Megaminx > > I'd prefer sources in germany, although I doubt it that I can find any of > them here... Most of them can indeed be bought from C.Bandelow. The only source for a 4x4x4 that I know of is Puzzletts, where I bought mine a year ago for about 50 US $. You can see their online mailorder catalog at www.puzzletts.com. They are based in Seattle, USA. I myself am still looking for Rubik's Domino (3x3x2). Rob, r.f.hegge@ctg.tudelft.nl PS If you or anyone else is interested in puzzles like this. There still exist a club called "Nederlandse Kubus Club" (NKC) or Dutch Cubist Club. About half of its members are not Dutch, despite its name, but American, German, Japanese etc. They publish a magazine called Cubism For Fun (CFF) three times a year, which deals with all kinds of (mechanical) puzzles, like polyform puzzles, sliding puzzles, rotational puzzles, burrs etc. They also organise a Cube Day once a year in the Netherlands for members with talks about puzzles and where new and SECOND HAND puzzles can be traded, bought, admired etc. From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 25 12:39:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA15691; Fri, 25 Jul 1997 12:39:18 -0400 (EDT) Precedence: bulk Mail-from: From whuang@ugcs.caltech.edu Thu Jul 24 15:30:43 1997 To: mlist-cube-lovers@nntp-server.caltech.edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Where can I get...? Date: 24 Jul 1997 19:27:15 GMT Organization: California Institute of Technology, Pasadena Message-Id: <5r8aaj$335@gap.cco.caltech.edu> References: Nntp-Posting-Host: beat.ugcs.caltech.edu X-Newsreader: NN version 6.5.0 #2 (NOV) Tim Mirabile writes: >Hmmm. I would imagine a large number of the 4x4x4 cubes sold in the >U.S. as "Rubik's Revenge" were sold to non-cube-lovers and are now >tucked away in attics and basements with all the other toys the kids >have outgrown. For this reason I've considered checking out local >garage and yard sales. I suggest any cube-lover who is considering going to local garage and yard sales to not do so! This way, *I* can get all of the old cubes and other puzzles!! In fact, I've acquired TWO 4x4x4's by this method in the last five years, as well as one of Nob Yoshigahara's Pineapple Puzzles for only fifty cents! The old lady who sold it to me even cautioned me not to eat it! (Hmm... I'm not doing a very good job of convincing you guys to avoid these sales, am I? ;-) ) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- finger me for /etc/passwd From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 25 13:54:12 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA16199; Fri, 25 Jul 1997 13:54:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From pdwyer@ecn.net.au Fri Jul 25 05:16:15 1997 Message-Id: <199707250905.AA10662@ecn.net.au> From: "Peter Dwyer" To: "CUBE" Subject: Re: Where can I get...? Date: Fri, 25 Jul 1997 18:57:16 +1000 > Hmmm. I would imagine a large number of the 4x4x4 cubes sold in the > U.S. as "Rubik's Revenge" were sold to non-cube-lovers and are now > tucked away in attics and basements with all the other toys the kids > have outgrown. For this reason I've considered checking out local > garage and yard sales. (Before it became widely known what the > value of old baseball cards could be, these sales were an excellent > way for collectors to pick up large boxes of old cards for only a > few dollars). I got my 4x4x4 from a fleamarket for $1. I was so exicited when I saw it I told the lady I would pay anything and she said $1 :-) Donna From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 25 20:58:24 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA18386; Fri, 25 Jul 1997 20:58:24 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From chrono@ibm.net Fri Jul 25 17:00:49 1997 Message-Id: <33D8EF4F.69C5@ibm.net> Date: Fri, 25 Jul 1997 11:24:15 -0700 From: "Jin \"Time Traveler\" Kim" Reply-To: chrono@ibm.net Organization: The Fourth Dimension To: CUBE Subject: Re: Where can I get...? References: <199707250905.AA10662@ecn.net.au> Peter Dwyer wrote: > > > Hmmm. I would imagine a large number of the 4x4x4 cubes sold in the > > U.S. as "Rubik's Revenge" were sold to non-cube-lovers and are now > > tucked away in attics and basements with all the other toys the kids > > have outgrown. For this reason I've considered checking out local > > garage and yard sales. (Before it became widely known what the > > value of old baseball cards could be, these sales were an excellent > > way for collectors to pick up large boxes of old cards for only a > > few dollars). > > I got my 4x4x4 from a fleamarket for $1. I was so exicited when I saw > it I told the lady I would pay anything and she said $1 :-) > > Donna > Indeed, while we are sharing 4x4x4 stories, I got mine (on an "extended" borrow) while digging around in a friend's garage. (Just hunting for junk) I found the Rubik's Revenge in a plastic bucket behind some old paint cans with all teh stickers peeled off. The cube is in great shape, but the stickers are pretty trashed (had to Krazy Glue them back on). But at least it works well. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Fri Jul 25 21:02:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA18418; Fri, 25 Jul 1997 21:02:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From tim@mail.htp.com Fri Jul 25 19:32:56 1997 From: tim@mail.htp.com (Tim Mirabile) To: CUBE-LOVERS@ai.mit.edu Subject: Re: Where can I get...? Date: Fri, 25 Jul 1997 23:29:20 GMT Organization: http://www.webcom.com/timm/ Message-Id: <33dc3688.524425@mail.htp.com> References: <9707240836.AA12191@sumatra.mp.tudelft.nl> In-Reply-To: <9707240836.AA12191@sumatra.mp.tudelft.nl> On Thu, 24 Jul 1997 10:36:30 +0200, R.F.Hegge@MP.TUDelft.NL (Rob Hegge) wrote: >The only source for a 4x4x4 that I know of is Puzzletts, where I >bought mine a year ago for about 50 US $. You can see their online >mailorder catalog at www.puzzletts.com. They are based in Seattle, >USA. I tried to order from them first, in the beginning of June. There was no response to my order or the followup email I sent. -- TimM on the Free Internet Chess Server - telnet://fics.onenet.net:5000/ The opinions of my employers are not necessarily mine, and vice versa. From cube-lovers-errors@mc.lcs.mit.edu Sat Jul 26 12:33:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA21436; Sat, 26 Jul 1997 12:33:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ERCO@compuserve.com Sat Jul 26 05:11:01 1997 Date: Sat, 26 Jul 1997 05:06:24 -0400 From: Edwin Saesen Subject: Re: Re: Where Can I get... To: CUBE Message-Id: <199707260506_MC2-1B9B-F7B3@compuserve.com> Hi everyone, thanks for all of the responses for my original inquiry about those various cubes. Rob wrote: >There still exist >a club called "Nederlandse Kubus Club" (NKC) or Dutch Cubist Club. This sounds interesting. Can you tell me how to contact them? And, does anyone know if there's a similar organization in germany? Wei-Hwa Huang wrote: >I suggest any cube-lover who is considering going to local garage >and yard sales to not do so! This way, *I* can get all of the >old cubes and other puzzles!! Well, it seem s to me that all those people in the USA are much more lucky than I am in germany. I've been going to flea markets for years (not only for cubes, mainly for buying records), and I've never ever seen a 4x4x4 for sale there, only lots of 3x3x3s. Concerning Puzzletts: >The only source for a 4x4x4 that I know of is Puzzletts, >I tried to order from them first, in the beginning of June. There was >no response to my order or the followup email I sent. I asked a friend of mine in the USA to order one for me, so I simply *HOPE* that the above described situation isn't their regular way of doing business. What might help is the fact that one of their retailers is in the city where my friend lives, so he might have a chance of getting one there. Concerning my old 4x4x4: Is there any way to fix broken center pieces? That's the reason why I need a new one, I still have all the pieces but two of them are broken and I have no idea if there's a way to fix them (I doubt it, though). Furthermore, does anyone of you maybe have a broken 4x4x4 and is willing to sell that one? Maybe I can fix my old one with some of those pieces then. (But probably they wouldn't fit together if they were done by different companies). I also could do with someone willing to sell a working 4x4x4 by the way :-) Sorry for the length of this Michael From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 21:30:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29606; Sun, 27 Jul 1997 21:30:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Jul 27 11:09:16 1997 Date: Sun, 27 Jul 1997 11:05:58 -0400 (EDT) From: Nicholas Bodley To: Edwin Saesen Cc: CUBE Subject: Broken 4^3s; advice on repairs to plastic (medium length) In-Reply-To: <199707260506_MC2-1B9B-F7B3@compuserve.com> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Edwin's mention of broken center pieces reminds me of a period roughly 15 years ago when I was quite active disassembling various cubes and their relatives; I'm fairly sure that it was these center pieces that impressed me as being fragile. Even the wondrous innards of the 5^3 are not as delicate, although fascinating. I'm trying to remember without much success whether I actually broke a piece. I think I did. Some plastics, I'm just about sure, can't be dissolved by any readily-available liquids, and i think the 4^3s were made of such material. (Acrylics can be "solvent-welded" very successfully.) Plastic welding is possible with probably any thermoplastic (i.e., one that can be melted after being molded), but is a skill just as is metal welding, and needs expensive hot-air tools (others also?) designed for the purpose. I would not recommend it for something so precious as a 4^3, unless you can find an expert. Finally, adhesives are worth considering. After all, the colored stickers are retained by adhesive. Not at all sure, but I think I did have success with cyanoacrylate (CA) (famous in the USA for its tradename "Krazy Glue"). This is a strange substance that seems not to be well understood, and it might be a good idea to learn more about it (even if you already think you know) before trying a critical repair. At least, if your repair is unsuccessful it will come apart, with probably little damage (no promises!) to the surfaces. The adhesive can then be scraped off. (Of course, you'll let it cure before reassembly...) As with almost any adhesive, clean surfaces are quite important. 99% isopropyl alcohol (no longer costly; try a good drugstore) is a worthy cleaner; rather few, uncommon contemporary plastics are attacked by it (but the clear printhead drive rack in an Oki printer disintegrated in a very few minutes, maybe 8 years ago). I doubt that any plastics used for these sorts of puzzles would be sensitive to alcohol. Not sure of my information, but I believe that curing of CA is triggered by absence of oxygen combined with an imperceptibly thin film of water on the surfaces (don't try to wet them!). Very low humidity might inhibit curing. For those who are really serious, the model-builders' magazines have ads for different versions of CA adhesives; after all, models have been made of plastics since WW II. Look into (i.e., catalog pages (Web?)) the products of the Loctite Corp., which makes a variety of industrial adhesives. There are other companies like Loctite, but Loctite has the most-successful marketers IMHO. (Note that there's no "k" in "Loctite"!) By the way, the Pocket Cube (2^3) is a bear to disassemble and even worse to reassemble. If it weren't for the really-good-quality polymer chosen for it, it (more than likely) could not be manufactured. The difficulty is in that the cubies have to be distorted ("sprung") to disassemble it. Whether this plastic retains its ability over many years to be bent out of shape but not crack, I don't know! All of this is offered with the best intentions; if I'm wrong about some particular, and you're reasonably sure that I am, by all means please let me know! The brittle plastics used before WW II, in general, are a different matter; alcohol is probably a bad idea in general. The modest amount I do know is really off-topic. I hope that this isn't too lengthy; this List seems to be willing to carry long messages at times. |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 21:32:43 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29657; Sun, 27 Jul 1997 21:32:42 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Jul 27 11:32:09 1997 Date: Sun, 27 Jul 1997 11:28:40 -0400 (EDT) From: Nicholas Bodley Reply-To: Nicholas Bodley To: Edwin Saesen Cc: Cube Mailing List , "Dr. Christoph Bandelow" Subject: Fit of 4^3 pieces; 5^3 query In-Reply-To: <199707260506_MC2-1B9B-F7B3@compuserve.com> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII (Sorry to have forgotten to include this in the previous message about repairing plastic.) Others (such as Drs. Christoph Bandelow or David Singmaster) might know better than I, but I strongly suspect that there was only one specific maker, and probably one set of molding dies, for the 4^3. If so, all pieces regardless of by what path they reached the owner, should fit. A good, close look, perhaps under a magnifier, would give an initial judgment about whether a given piece should be trial-assembled. If it has about the same amount of friction as the others, it really ought to be OK. The 3^3 most definitely has been made by several different companies from their own molding dies; interchangeability is by no means assured! Could someone enlighten me about the 5^3? I got mine from Dr. Uwe Meffert; I wonder whether Dr. Bandelow's came from the same set of dies? |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 21:34:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29661; Sun, 27 Jul 1997 21:34:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Jul 27 12:09:36 1997 Date: Sun, 27 Jul 1997 12:06:18 -0400 (EDT) From: Nicholas Bodley To: Cube Mailing List Subject: 4^3 innards: There's a ball in there, but which way does it point? Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Uniquely to all cubes from the Pocket Cube to the 5^3, only the 4^3 (Rubik's Revenge) has a ball inside it. (I do hope that I don't cause curious people to break their fragile center pieces trying to open up theirs!) This is really a repetition of some old information, but with the current interest in 4^3s, it seems sufficiently interesting to repeat it. However, afaik, the questions about the ball's orientation are probably new. The ball (made of at least 9 pieces plus eight screws, as I recall) consists of a center piece, essentially a sphere (possibly two hemispheres) with "octants" fastened to it; these have the geometry of 90-90-90-degree right spherical triangles, although in practical detail they differ. Gaps between these octants define three circumferential grooves that correspond to the Earth's equator and two orthogonal meridians of longitude. (All 3 grooves are orthogonal.) (Sorry for the redundancies; trying to be clear to everybody). These grooves are "undercut" on one side, so they have an inverted L-shaped cross-section. The cubies (center ones only, as I recall) have feet that tuck under the extended edges of the octants' grooves; this keeps them in place. Machinists know well of the T-slots that work with clamps to hold work in place on machine tools; these are similar, but the cubies are free to slide in the L-slots. If I'm thinking clearly (not too sure!), the ball has a 120-degree rotational symmetry about an unique diameter. One octant has no undercuts; its opposite, I think, has all three edges undercut. Other octants have some edges undercut. Practical details dictate that when one half of the 4^3 is rotated, one half is definitely locked to the internal ball. However, you probably don't know which half it is! (The mathematical folk here might find it fun to predict which half is the locked half for any given configuration; this might even not be a trivial problem. (As well, does the locked part always end up in the same place when the Cube is solved? I suspect so, but am not sure.) (Anybody for a translucent-cubie 4^3?) Is it possible to maneuver the internal ball so that it has effectively revolved by a half turn (or quarter turn?) about any given axis, while preserving the exterior configuration? My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 21:38:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29673; Sun, 27 Jul 1997 21:38:20 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Jul 27 20:25:05 1997 Date: Sun, 27 Jul 1997 20:21:46 -0400 (EDT) From: Nicholas Bodley Reply-To: Nicholas Bodley To: Cube Mailing List Cc: Javier Susaeta , Mark Glusker Subject: Making parts for puzzles (somewhat off-topic) (medium length) Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII For reasons I'm not certain of, but probably having to do with a case of brain fade caused by not eating a proper breakfast (I live alone!), I posted a short article to the wrong mailing list, no less. It was to an unmoderated list that goes to people interested in mechanical calculators, so the few replies I have had so far found it interesting. I do hope this is not too far off-topic; I have included replies that contain material that would be helpful to anyone who has any ideas for making new or replacement parts, and has access to the technologies. The length is not hopeless... Subject: Rapid Prototyping and conceivable Cube reissues (Posted 09:53 AM 27/07/97 -0700 with follow-up (by gracious permission) from a couple of people who >do< know what they're talking about.) This is close to being peripheral to the topics of the List, but it might be worth pointing out that it is now possible by several methods to "print" solid objects in 3 dimensions from digital data in the proper format. I'm fairly sure it's even possible to make molds. These processes are sometimes known as Rapid Prototyping, and (betraying linguistic ignorance or lack of concern) something like 3-D Lithography. ("Lithography" has roots meaning writing on stone.) (I suggest Solid Object Synthesis...) The field is still evolving and improving. The data files for these processes are ordinarily created by CAD (Computer-Aided Design/Drafting) programs. The programs create the data for numerous thin slices of an object. The data for a slice directs the machine to make a solid counterpart, and the object is created progressively by creating a bonded stack of the requisite number of slices. (I shouldn't go into how, but one early method uses a pool of photopolymer and a flat "stage" that progressively moves deeper into the pool. A bright UV light source positioned by the data triggers polymerization (solidifies).) My point is that it is now considerably less expensive and much quicker than it used to be, to make a prototype of a puzzle design, although the materials used for this process (afaik) don't have the requisite durability yet. With copyrights taken care of, it might be possible to make limited numbers of given designs. It would (at present) be too costly to make individual puzzles by these processes, but the economics could change in a decade. Whether it is possible to pour melted plastic into molds to make the pieces of a Cube-like puzzle is at present very doubtful, but it might be, in the future. (Rattlebacks, also known as celts, look just like solidified poured plastic.) It's unfortunate that the economics that make possible the affordable production of large numbers of such puzzles as the 4^3 also mean that starting another production run has to be economically justified. Let's hope that the tooling for manufacturing such miracles as the 5^3 and the Magic Dodecahedron is preserved! Failing that, at least any CAD files for doing numerical machining of the molding dies really ought to be kept and backed up. Hope I'm not too far off-topic. {"afaik" = "as far as I know", pointed out as a courtesy to non-native users of English} {END of the text I posted to the wrong List -nb} * * * {The following replies are posted with the gracious permission of their respective authors. } Here's a reply from Javier Susaeta: Yes, I have read a bit about rapid prototyping and believe that it has a tremendous potential. I cannot tell you where I read it, but I remember a case where a complete intake manifold was designed by CAD/CAM and then a single copy was (slowly) built, layer by layer, with one of the just-born rapid-prototyping machines. The material used was aluminium powder plus a plastic agglomerant. Once the manifold was "finished" it was baked in an oven in order to eliminate the agglomerant and weld together the metal particles. The result was not so solid as a cast aluminium manifold, but nevertheless perfectly usable. (Referring to more details, Javier said:) I remember having read it in internet, a few months ago. The intake manifold was for a specially-built big diesel of a bulldozer or a similar machine. I am not sure, but perhaps it was from Caterpillar or a similar US company. They resorted to rapid prototyping because of costs. It was a very special engine, and they needed 1 (yes, one) intake manifold. The fixed costs for a single casting were enormous, and a manifold is so convoluted that machining it out of a block of metal was impossible. So they resorted to this new technique. Regards Javier Susaeta Here's some more on the topic, from Mark Glusker: Date: Sun, 27 Jul 1997 14:33:36 -0700 (PDT) From: Mark Glusker This process has improved dramatically over the past few years. The materials are now quite durable. It is not necessarily cheaper to use stereo lithography: it is best for small parts with lots of detail-per- cubic-inch. Simple parts or large parts are best fabricated using conventional machining methods. If you are making a part to be used as a master for a mold, don't forget to enlarge the original by several percent (depending on the final material) to account for molding shrinkage. Despite the improvements in this field, the best stereo lithography part is still not nearly as good as a well machined part. It's very much like the difference between a well printed photograph and a scanned image printed on a good laserwriter. Used appropriately, it is a great tool but Bridgeport is not about to go the way of Friden or Monroe. {Bridgeport is a very-famous maker of milling machines; Friden and Monroe were once-vital makers of mechanical desktop calculators; they are now history. -nb} [Mark also wrote:] -- Mark Glusker, glusk@sgi.com >From glusk@mechcad3.engr.sgi.com Sun Jul 27 19:13:05 1997 Date: Sun, 27 Jul 1997 16:04:04 -0700 (PDT) From: Mark Glusker I just received some stereo lithography parts last week that were made of polycarbonate. It is finally possible to use that process for more than just verifying the shape of objects. However, if you wanted to replicate a cube (I assume you mean a Rubik-like mechanism) you would need to do lots of post-finishing to remove the "raster" ridges from the parts so they will move smoothly against one another. This post-finishing is done by hand and will certainly affect your final tolerances. I use ProEngineer (on a Silicon Graphics workstation, naturally) which can automatically generate an STL file, the standard data format used by the stereo lithography vendors. There are lots of vendors around here, and several have relationships with prototype die casters for limited production runs of parts (5 to 100 pieces, typically). {At this point, Mark offered help; however, since the help was a personal offer to me, I don't want to post his comments directly. -nb} {snip} ... I could go on for quite some time on this subject! There are similar digital processes for replicating flat metal parts from a CAD file, with similar economic tradeoffs, in this case related to perimeter-per-area of part. That would be a great way to replicate a missing piece of a mechanical calculator,... {Here, Mark's helpful comments were welcome, but off-topic for this List. -nb} Regards, Mark |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 22:51:19 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA00608; Sun, 27 Jul 1997 22:51:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Sun Jul 27 22:45:48 1997 Date: Sun, 27 Jul 1997 22:45:37 -0400 Message-Id: <199707280245.WAA12048@sun30.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Cc: Nicholas Bodley Subject: Administrata and a reference The administrata is that Alan Bawden has asked me to take over cube-lovers-request for a few weeks, while he is recovering from surgery. After seventeen years of running cube-lovers, he deserves a vacation, but I would rather he find someplace more pleasant to spend it than a hospital. I'm sure we all wish him a speedy recovery. I will be adding and removing addresses and filtering out abuse, but I will not be updating the archives or the collection of reader contributions. As always, send to cube-lovers-request@ai.mit.edu for administrative services. The reference is for Nicholas Bodley, who in one of his very informative messages on Rubik's Revenge raised questions of the possible orientations achievable by the internal sphere without changing the exterior. I answered that question in my message on "Invisible Revenge" on 9 August 1982. The sphere can be placed in any of 24 orientations, and I showed how to do so. If we consider the sphere modulo its functional symmetry (fixing one corner of the cube) we will distinguish only 8 of these orientations. I also mentioned how to determine which of these 8 orientations the interior sphere is in on a physical cube, without disassembly. See ftp://ftp.ai.mit.edu/pub/cube-lovers/cube-mail-4 for details. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 28 12:11:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA03296; Mon, 28 Jul 1997 12:11:59 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From CFolkerts@compuserve.com Mon Jul 28 05:22:17 1997 Date: Mon, 28 Jul 1997 05:18:01 -0400 From: Corey Folkerts Subject: 2^3 Reassembly To: Message-Id: <199707280518_MC2-1BB6-48C8@compuserve.com> My 2^3 burst into pieces while I was playing around with it a while back. I was amazed and intrigued by the number of internal pieces it contained; many more than the 3^3. Anyway, after a couple minutes I got it all put back together, and started playing with it again. One problem: when I attempted to rotate the cube on one of the axes, it gave me a lot of resistance. If I continued to force it, the whole thing burst and was reduced once again to a pile of little black plastic pieces. After a few more random tests, I examined the pieces and noticed, as I'm sure many have, that some of the small internal pieces are slightly different than the others. This fact leads me to believe that the 'special' pieces need to be oriented correctly with respect to each other in order for the cube to work correctly. I would be most appreciative if someone could please inform the manner in which they need to be placed. Thanks in advance, Corey Folkerts From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 28 12:14:45 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA03315; Mon, 28 Jul 1997 12:14:44 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From R.F.Hegge@MP.TUDelft.NL Mon Jul 28 05:30:18 1997 Date: Mon, 28 Jul 1997 11:26:33 +0200 From: R.F.Hegge@MP.TUDelft.NL (Rob Hegge) Subject: Re: Where Can I get... To: CUBE-LOVERS@ai.mit.edu Message-Id: <9707280926.AA23236@sumatra.mp.tudelft.nl> Sorry, but I have just been informed that Puzzletts has been out of 4x4x4's for several years now. I probably received one of their last. >I tried to order from them first, in the beginning of June. There was >no response to my order or the followup email I sent. Maybe they put your order on their 'wish list' and will contact you after they obtained one. But given the fact the 4x4x4 are quite rare now it does not seem likely. > >There still exist > >a club called "Nederlandse Kubus Club" (NKC) or Dutch Cubist Club. > This sounds interesting. Can you tell me how to contact them? And, does > anyone know if there's a similar organization in germany? I will look up the address etc and send the info to the list asap. For germany I would not know. Rob From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 28 12:56:42 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA03556; Mon, 28 Jul 1997 12:56:41 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From hart@netprofitlc.com Mon Jul 28 11:16:15 1997 Date: Mon, 28 Jul 1997 09:01:14 -0600 (MDT) From: Paul Hart To: "Jin \"Time Traveler\" Kim" Cc: CUBE Subject: Re: Where can I get...? In-Reply-To: <33D8EF4F.69C5@ibm.net> Message-Id: On Fri, 25 Jul 1997, Jin "Time Traveler" Kim wrote: > Indeed, while we are sharing 4x4x4 stories, I got mine (on an "extended" > borrow) while digging around in a friend's garage. I've got another interesting story, too, about my two 4x4x4 cubes. Earlier this year, by sheer coincidence, I happened across two Rubik's Revenge cubes at a local hobby store. Each of the cubes was authentic, and still in the (unopened) original box. I easily slapped down my $12.95 (USD) for each cube, needlessly to say. :-) Paul Hart From cube-lovers-errors@mc.lcs.mit.edu Mon Jul 28 15:20:07 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA04297; Mon, 28 Jul 1997 15:20:06 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From chrono@ibm.net Mon Jul 28 14:58:51 1997 Message-Id: <33DCE54F.5463@ibm.net> Date: Mon, 28 Jul 1997 11:30:39 -0700 From: "Jin \"Time Traveler\" Kim" Reply-To: chrono@ibm.net Organization: The Fourth Dimension To: Cube-Lovers@ai.mit.edu Subject: Re: 2^3 Reassembly References: <199707280518_MC2-1BB6-48C8@compuserve.com> Corey Folkerts wrote: > > My 2^3 burst into pieces while I was playing around with it a while > back. I was amazed and intrigued by the number of internal pieces it > contained; many more than the 3^3. Anyway, after a couple minutes I > got it all put back together, and started playing with it again. One > problem: when I attempted to rotate the cube on one of the axes, it > gave me a lot of resistance. If I continued to force it, the whole > thing burst and was reduced once again to a pile of little black > plastic pieces. After a few more random tests, I examined the pieces > and noticed, as I'm sure many have, that some of the small internal > pieces are slightly different than the others. This fact leads me to > believe that the 'special' pieces need to be oriented correctly with > respect to each other in order for the cube to work correctly. I > would be most appreciative if someone could please inform the manner > in which they need to be placed. > > Thanks in advance, > Corey Folkerts You have experienced a problem which has led me to purchase a total of FOUR 2x2x2 cubes. Not even of my own undoing either. In two cases, friends attempted to play with the cube and disassembled them, and were unable to properly reassemble them. I have pieces of each in separate boxes, minus several pieces each. In one case I dropped it, it flew open, and I DID manage to reassemble it properly. But it ALSO fell victim to a careless reassembly by a friend who also carelessly disassembled it. That one I reassembled AGAIN and gave it to someone. They later told me they "broke" it, which means it's in pieces and since they live 450 miles away, I can't exactly help them. I have a fourth, still in its bag, untouched by human hands. Oh yes, and I bought a fifth one (actually, it was the fourth, so the untouched one is technically the fifth) but I had to return it and get another one (the fifth) because it had been disassembled before and reassembled incorrectly (the pieces only turned on one axis). -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 29 10:14:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA09785; Tue, 29 Jul 1997 10:14:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ERCO@compuserve.com Tue Jul 29 02:39:59 1997 Date: Tue, 29 Jul 1997 02:22:43 -0400 From: Edwin Saesen Subject: Re: Where can I get...? To: CUBE Message-Id: <199707290223_MC2-1BD0-779D@compuserve.com> Paul wrote: >I've got another interesting story, too, about my two 4x4x4 cubes. >I easily slapped down my $12.95 >(USD) for each cube, needlessly to say. :-) >From all of these stories of almost everyone in the USA either finding two copies or a single one for around US$1, I take it that they were MUCH more common in the USA than they were in germany :-( Rob wrote: >but I have just been informed that Puzzletts has >been out of 4x4x4's for several years now. I probably >received one of their last. :-((( Ok then, I'm desperate now. Is there ANYONE willing to sell a spare 4x4x4 one? I'll be willing to pay the $50 puzzletts were charging for them (as I tried to order one from them...). (Hey Paul, this is *THE* chance for you to get your money back, and have your own copy of the 4x4x4 for free plus having about $24 leftover to buy 24 copies of the 4x4x4 on your local flea markets...). But honest, if anyone ever sees one of those for sale somewhere, please get it for me. I think my chances of ever finding one here in germany are virtually zero, as I haven't been able to replace my own copy for six years or so... Michael From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 29 10:52:34 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA09946; Tue, 29 Jul 1997 10:52:33 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Tue Jul 29 03:22:14 1997 Date: Tue, 29 Jul 1997 03:10:32 -0400 (EDT) From: Nicholas Bodley To: Corey Folkerts Cc: Cube-Lovers@ai.mit.edu Subject: Re: 2^3 Reassembly In-Reply-To: <199707280518_MC2-1BB6-48C8@compuserve.com> Message-Id: My, it's been a while since I opened up mine. I hope I remember. What perplexes me (I was about to say, "puzzles me") is that I don't recall how it could be assembled wrong; surely there aren't two internal mechanisms that differ in some details? Perhaps it would help if I describe the internal structure (from memory). The basic structure that I remember is built on a "jack"; it's basically mutually-orthogonal extensions from a center, so to speak; like a physical embodiment of the axes of 3-D Cartesian coordinates. What I recall is that three of these, all adjacent, have a square cross-section; the other three have a circular cross-section, with a diameter significantly smaller than a side of the square. Each of three round projections fits into a hole through a rotating long, thin square prism. (In the vernacular, square sticks with round holes through their centers.) In mine, I am just about certain that all of these had the same length. Each cubie (all were identical internally) is hollow, but cut away with concave arcs that allow them to turn with respect to their neighbors. The cubies are kept together by 12 "clips". These fit into the cutout arcs; when you assemble the Cube, you put two cubies next to each other (they touch) and fit this "clip" so that it keep s them together. To install it, you move the clip away from the imaginary geometrical center of the whole puzzle. As I recall them, the "clips" are essentially quadrants (1/4 circles). They consist of two parallel planes with a gap between them; the sides of the cubies fit into this gap. The parallel planes are joined at the inner edges. When the whole Cube is assembled, the square extensions of the "jack", as well as the square sticks that turn on the other ends of the jack serve to keep the clips from moving toward the center of the whole Cube. This is a structure that could not be either assembled or disassembled if it were made of rigid materials. It's only because the cubies (at least!) are made of a strong plastic that has good mechanical spring properties and can be harmlessly deformed (within limits), that it is possible to make this structure. >>> It's conceivable that the cube was misassembled so that one or more "clips" didn't actually straddle both of its cubies. When assembled, there should not be any gaps between the cubies, and all movement should be reasonably free of friction. I'd love to know how these were assembled in the first place; did the mfr. have special tools to temporarily deform the parts? Did the assemblers develop very strong hand muscles? Btw, it's a challenge to describe the innards in words. I hope this helps! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 29 14:33:28 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA11052; Tue, 29 Jul 1997 14:33:27 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From chrono@ibm.net Tue Jul 29 13:43:28 1997 Message-Id: <33DE2A54.62F2@ibm.net> Date: Tue, 29 Jul 1997 10:37:24 -0700 From: "Jin \"Time Traveler\" Kim" Reply-To: chrono@ibm.net Organization: The Fourth Dimension To: Cube-Lovers@ai.mit.edu Subject: Re: 2^3 Reassembly References: Nicholas Bodley wrote: > The cubies are kept together by 12 "clips". These fit into the cutout > arcs; when you assemble the Cube, you put two cubies next to each other > (they touch) and fit this "clip" so that it keep s them together. To > install it, you move the clip away from the imaginary geometrical center > of the whole puzzle. > > As I recall them, the "clips" are essentially quadrants (1/4 circles). > They consist of two parallel planes with a gap between them; the sides of > the cubies fit into this gap. The parallel planes are joined at the inner > edges. Part of the problem is that the clips weren't all identically shaped. If all of the clips were shaped the same, then reassembly wouldn't be a problem. But because they ARE shape differently, there is a question of whether the position of one is important relative to the position of the others. -- Jin "Time Traveler" Kim chrono@ibm.net VGL Costa Mesa http://www.geocities.com/timessquare/alley/9895 http://www.slamsite.com From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 29 15:10:58 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA11199; Tue, 29 Jul 1997 15:10:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From CFolkerts@compuserve.com Tue Jul 29 14:09:01 1997 Date: Tue, 29 Jul 1997 13:58:09 -0400 From: Corey Folkerts Subject: Re: 2^3 Reassembly To: Cube-Lovers Message-Id: <199707291358_MC2-1BDB-4183@compuserve.com> Nicholas Bodley writes: > The cubies are kept together by 12 "clips". These fit into the cutout >arcs; when you assemble the Cube, you put two cubies next to each other >(they touch) and fit this "clip" so that it keep s them together. To >install it, you move the clip away from the imaginary geometrical center >of the whole puzzle. My 2^3 has these 12 clips as I discovered when it first burst. However, I would like to confirm something. Nine of my clips are identical, the 1/4 circle shape. However, its the other three that are causing me trouble. One of them is identical to the other nine except that on one of the two planes it has a very small notch cut out of it. The notch is an arc and I'm guessing it is probably about 1 mm deep. The other two have one of the 1/4 planes identical to the first nine, but the second plane extends far beyond, doubling the "height" of the clip. If viewed from the side which has the extended plane it is a diamond instead of a 1/4 circle. All of the other internal pieces are identical to your description I would like to know if everyone else has these altered clips in their 2^3s. Corey Folkerts From cube-lovers-errors@mc.lcs.mit.edu Tue Jul 29 16:24:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA11498; Tue, 29 Jul 1997 16:24:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From whuang@ugcs.caltech.edu Tue Jul 29 14:32:02 1997 To: Cube-Lovers@AI.MIT.Edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: Broken 4^3s; advice on repairs to plastic (medium length) Date: 29 Jul 1997 18:28:24 GMT Organization: California Institute of Technology, Pasadena Message-Id: <5rlco8$em3@gap.cco.caltech.edu> References: Nicholas Bodley writes: > By the way, the Pocket Cube (2^3) is a bear to disassemble and even worse >to reassemble. If it weren't for the really-good-quality polymer chosen >for it, it (more than likely) could not be manufactured. The difficulty is >in that the cubies have to be distorted ("sprung") to disassemble it. >Whether this plastic retains its ability over many years to be bent out of >shape but not crack, I don't know! Does anyone know whether the mechanisms for the old 2^3's are the same as the recent new 2^3 releases? -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- I hate formication. It should be abolished entirely. From cube-lovers-errors@mc.lcs.mit.edu Wed Jul 30 10:35:07 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA14669; Wed, 30 Jul 1997 10:35:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Wed Jul 30 08:15:13 1997 Date: Wed, 30 Jul 1997 08:11:44 -0400 (EDT) From: Nicholas Bodley To: "Jin \"Time Traveler\" Kim" Cc: Cube-Lovers@ai.mit.edu Subject: Re: 2^3 Reassembly In-Reply-To: <33DE2A54.62F2@ibm.net> Message-Id: On Tue, 29 Jul 1997, Jin "Time Traveler" Kim wrote: }Nicholas Bodley wrote: } }> The cubies are kept together by 12 "clips". These fit into the cutout {Snips} }Part of the problem is that the clips weren't all identically shaped. }If all of the clips were shaped the same, then reassembly wouldn't be a }problem. But because they ARE shaped differently, there is a question of }whether the position of one is important relative to the position of the }others. } }-- }Jin "Time Traveler" Kim }chrono@ibm.net It's not likely that there are two or more designs of the 2^3; sorry if I misled anyone. I might have been lucky; however, I don't recall needing to sort the clips. This >is< something to look out for if you disassemble a 2^3. It's likely that the clips for the swiveling long blocks would be different from those for the rigid extensions of the "jack". Less likely is that the different cavities used to mold several clips at once had different shapes where the differences had no effect on operation, but anyone familiar with such mechanisms would be able to tell. The mechanical engineer on this project has the answers! Sure hope this isn't memory fade; I'm going on 62... My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Jul 30 12:26:38 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA15194; Wed, 30 Jul 1997 12:26:37 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Wed Jul 30 08:21:49 1997 Date: Wed, 30 Jul 1997 08:18:30 -0400 (EDT) From: Nicholas Bodley To: Corey Folkerts Cc: Cube-Lovers Subject: Re: 2^3 Reassembly In-Reply-To: <199707291358_MC2-1BDB-4183@compuserve.com> Message-Id: As to the notches Corey mentions, they might not matter, but all three pieces I would guess might of a later design that would simplify assembly; they are probably the last three pieces to be assembled. It sounds as though one needs to very careful and make sketches when disassembling a 2^3! All of mine were bought probably around 1985 or so, or earlier; they are old. Please don't think that simply because I'm partly informed that I'm an expert! I'm not. Good luck to all, NB On Tue, 29 Jul 1997, Corey Folkerts wrote: }Nicholas Bodley writes: } }> The cubies are kept together by 12 "clips". These fit into the cutout }>arcs; when you assemble the Cube, you put two cubies next to each other }>(they touch) and fit this "clip" so that it keep s them together. To }>install it, you move the clip away from the imaginary geometrical center }>of the whole puzzle. } }My 2^3 has these 12 clips as I discovered when it first burst. However, I }would like to confirm something. Nine of my clips are identical, the 1/4 }circle shape. However, its the other three that are causing me trouble. One }of them is identical to the other nine except that on one of the two planes }it has a very small notch cut out of it. The notch is an arc and I'm }guessing it is probably about 1 mm deep. The other two have one of the 1/4 }planes identical to the first nine, but the second plane extends far }beyond, doubling the "height" of the clip. If viewed from the side which }has the extended plane it is a diamond instead of a 1/4 circle. All of the }other internal pieces are identical to your description } } I would like to know if everyone else has these altered clips in }their 2^3s. } } Corey Folkerts } } |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Jul 30 15:23:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA15891; Wed, 30 Jul 1997 15:23:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From whuang@ugcs.caltech.edu Wed Jul 30 14:13:18 1997 To: Cube-Lovers@AI.MIT.Edu From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Subject: Re: 2^3 Reassembly Date: 30 Jul 1997 18:09:49 GMT Organization: California Institute of Technology, Pasadena Message-Id: <5ro01d$as6@gap.cco.caltech.edu> References: Corey Folkerts writes: >Nicholas Bodley writes: >> The cubies are kept together by 12 "clips". These fit into the cutout >>arcs; when you assemble the Cube, you put two cubies next to each other >>(they touch) and fit this "clip" so that it keep s them together. To >>install it, you move the clip away from the imaginary geometrical center >>of the whole puzzle. >My 2^3 has these 12 clips as I discovered when it first burst. However, I >would like to confirm something. Nine of my clips are identical, the 1/4 >circle shape. However, its the other three that are causing me trouble. One >of them is identical to the other nine except that on one of the two planes >it has a very small notch cut out of it. The notch is an arc and I'm >guessing it is probably about 1 mm deep. The other two have one of the 1/4 >planes identical to the first nine, but the second plane extends far >beyond, doubling the "height" of the clip. If viewed from the side which >has the extended plane it is a diamond instead of a 1/4 circle. All of the >other internal pieces are identical to your description > I would like to know if everyone else has these altered clips in >their 2^3s. I believe so. Those "special" serve the same purpose as the protruding octants on the 4^3 internal ball -- to anchor one of the blocks in each plane. Otherwise, one of the planes of "clips" may be offset 45 degrees (not obvious from the outside), and the other planes become unturnable. Make sure that one weird clip is in each plane before assembly. -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- I hate formication. It should be abolished entirely. From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 31 17:38:54 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA22750; Thu, 31 Jul 1997 17:38:53 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ponder@austin.ibm.com Thu Jul 31 16:55:58 1997 Date: Thu, 31 Jul 1997 15:52:23 -0500 From: ponder@austin.ibm.com (Ponder) Message-Id: <9707312052.AA16660@roosevelt.austin.ibm.com> To: Cube-Lovers@ai.mit.edu Subject: Rubik's octahedron's etc. Cc: ponder@austin.ibm.com I have a Rubik's Cube and a Megaminx Dodecahedron. There are some tetrahedral puzzles available but they do not correspond precisely to the Rubik's cube, in that they do not have well-defined center pieces and the corners are freely rotating. As far as I can tell, nobody has an octa- hedron or an icosahedron that works on these principles either. Its hard to expect the puzzle companies to come out with anything like these since they're in it for a profit. I heard that Meffert's company closed down before they could produce most of the puzzles they intended. Does anyone have designs for puzzles like these that could be built in a machine-shop? (Preferably that you've already patented, to eliminate any legal concerns!!). I imagine I could try to hack something together, but it would take an awful lot of trial-and-error especially since some internal designs would hold together better than others. I'm publishing a paper in the Journal of Recreational Mathematics on solving these other puzzles, but it would be real nice to have demo models, even if it takes some work. The Octahedron is particularly interesting because it forbids edge-flips and it would be more convincing if I do more than show it on paper. Thanks, Carl Ponder ponder@austin.ibm.com From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 31 18:52:22 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA22953; Thu, 31 Jul 1997 18:52:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ponder@austin.ibm.com Thu Jul 31 17:16:55 1997 Date: Thu, 31 Jul 1997 16:13:34 -0500 From: ponder@austin.ibm.com (Ponder) Message-Id: <9707312113.AA33486@roosevelt.austin.ibm.com> To: Cube-Lovers@ai.mit.edu Subject: puzzle to be simulated Cc: ponder@austin.ibm.com I've seen a number of Rubik's Cube simulations on the web, and was wondering if any of you would be interested in implementing the following puzzle that I call "the hell-hole". It has 16 faces that each work like the faces of a rubik's cube, but: 1] The faces are layed out on a 4x4 grid. Each "corner" joins four surrounding faces instead of three. 2] The grid is rolled into a cylinder and then joined at both ends to forma torus. However, the torus is given a "twist" when you join the two ends together, as follows: 1 2 3 4 _ _ _ _ a|_|_|_|_|b b|_|_|_|_|c c|_|_|_|_|d d|_|_|_|_|a 1 2 3 4 First join the 1-2-3-4 sides together to form the cylinder, then the a-b-c-d ends together to get the torus. Each of the squares is a 3x3 face like a Rubik's Cube. The combinatorics get a *lot* messier because of the twist. Without it, you can't "flip" the edge pieces. With it, you can, but only by moving the edge-piece in a full-circle around the torus. No way to build it, either, since the pieces would need to flex between convex and concave. It could be simulated on a computer, however. I have a paper coming out in the Journal of Recreational Mathematics on how to solve these kinds of things, and it is pretty messy. Thanks, Carl Ponder ponder@austin.ibm.com From cube-lovers-errors@mc.lcs.mit.edu Thu Jul 31 21:36:12 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA24419; Thu, 31 Jul 1997 21:36:12 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Thu, 31 Jul 1997 21:38:18 -0400 Message-Id: <199708010138.VAA24279@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Cube-Lovers@ai.mit.edu, ponder@austin.ibm.com In-Reply-To: <9707312113.AA33486@roosevelt.austin.ibm.com> Subject: Re: puzzle to be simulated Carl Ponder describes his "hell-hole" puzzle that (if I undestand correctly) has 36 facets laid out like m : n n n : p p p : r r r : s s s : a ..:.......:.......:.......:.......:.. s : a a a : b b b : c c c : d d d : e s : a a a : b b b : c c c : d d d : e s : a a a : b b b : c c c : d d d : e ..:.......:.......:.......:.......:.. d : e e e : f f f : g g g : h h h : j d : e e e : f f f : g g g : h h h : j d : e e e : f f f : g g g : h h h : j ..:.......:.......:.......:.......:.. h : j j j : k k k : l l l : m m m : n h : j j j : k k k : l l l : m m m : n h : j j j : k k k : l l l : m m m : n ..:.......:.......:.......:.......:.. m : n n n : p p p : r r r : s s s : a m : n n n : p p p : r r r : s s s : a m : n n n : p p p : r r r : s s s : a ..:.......:.......:.......:.......:.. s : a a a : b b b : c c c : d d d : e with opposite boundaries identified so that the letters match up. A turn rotates 25 facets--one of the 3x3 "faces" marked with dots--and the sixteen neighboring facets from the neighboring faces. > The combinatorics get a *lot* messier because of the twist. > Without it, you can't "flip" the edge pieces. With it, you > can, but only by moving the edge-piece in a full-circle around > the torus. If I've got the puzzle right, you could get edge flippability just by using a 3x3 array of faces instead of 4x4. Or perhaps 3x5. Another nice idea that uses a "square" torus is k : l l l : m m m : n n n : d ..:.......:.......:.......:.. h : a a a : b b b : c c c : j h : a a a : b b b : c c c : j h : a a a : b b b : c c c : j j j : k k k : l ..:.......:.......:.......:.......:.......:.. n : d d d : e e e : f f f : g g g : h h h : a n : d d d : e e e : f f f : g g g : h h h : a n : d d d : e e e : f f f : g g g : h h h : a ..:.......:.......:.......:.......:.......:.. c : j j j : k k k : l l l : m m m : n n n : d c : j j j : k k k : l l l : m m m : n n n : d c : j j j : k k k : l l l : m m m : n n n : d ..:.......:.......:.......:.......:.......:.. f : g g g : h h h : a a a : b b b : c c c : j This corresponds to a square of side sqrt(13) with opposite edges identified, cut on the bias into 13 square faces. ............................ :.' a .' `.. m .' `.: : `.. .' `..' n .': : `..' b .' `.. .' : : d .' `.. .' `..' d: : .' `..' c .' `..: :`..' e .' `.. .' : :.' `.. .' `..' j : :' `..' f .' `.. .: : k .' `.. .' `..': :.. .' `..' g .' `: : `..' l .' `.. .' : :h .' `.. .' `..' h: : .' a `..' m .'`.. : :.'.......'b`.......'.n..`.: after which each face is cut up into nine facets. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 1 09:34:41 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA27030; Fri, 1 Aug 1997 09:34:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Thu Jul 31 22:03:44 1997 Date: Thu, 31 Jul 1997 22:00:20 -0400 (EDT) From: Nicholas Bodley To: Ponder Cc: Cube-Lovers@ai.mit.edu Subject: Re: Rubik's octahedrons etc. In-Reply-To: <9707312052.AA16660@roosevelt.austin.ibm.com> Message-Id: If you were lucky, you could use a good CAD program to define the shapes, and NC machine tools to produce them; also possible is Rapid Prototyping. A graphic computer simulation would also be a substitute for a physical puzzle, although holding one in your hand beats just about any graphics. Good luck! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 1 10:47:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA27313; Fri, 1 Aug 1997 10:47:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Fri Aug 1 09:14:33 1997 Message-Id: <199708011310.OAA00278@mail.iol.ie> From: "David Byrden" To: Subject: Re: Rubik's octahedrons, etc. Date: Fri, 1 Aug 1997 14:09:19 +0100 > From: Ponder > I'm publishing a paper in the Journal of Recreational > Mathematics on solving these other puzzles, but it would > be real nice to have demo models, even if it takes some > work. The Octahedron is particularly interesting because > it forbids edge-flips Just for you, I have put up a new Java Octahedron at the Rubik Gallery http://www.iol.ie/~goyra/Rubik.html The new one is in the Cousteau Collection and has corner-centred slices.There is another one in the Plato Collection with face-centred slices. If the one you are thinking about is deeper or it twists in a different way, drop me a line and I can probably brew it up for you. David From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 1 20:34:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA01174; Fri, 1 Aug 1997 20:34:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Aug 1 20:32:48 1997 Date: Fri, 1 Aug 1997 20:31:56 -0400 Message-Id: <199708020031.UAA28142@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Cube-Lovers@ai.mit.edu, ponder@austin.ibm.com In-Reply-To: <199708010138.VAA24279@sun30.aic.nrl.navy.mil> (message from Dan Hoey on Thu, 31 Jul 1997 21:38:18 -0400) Subject: Re: puzzle to be simulated Continuing on my quest for Ponderesque cube-planes made of a square torus, possibly cut on the bias, I notice that any square torus will have k=a^2+b^2 faces for some a>b; if b=0 there is no bias. Carl Ponder suggests (if I understand it correctly) cutting each face into nine facets, and permuting the puzzle by turning a face together with the 16 neighboring facets. As with Rubik's cube, the edge facets move in pairs (called edge cubies). The corner facets move in quadruplets (called corner cubies). There are k corner cubies that can apparently achieve any of four orientations (twist) each, and 2k edge cubies that can apparently achieve any of two orientations (flip) each. The face center cubies can achieve any of four orientations (twist) each but never move. So the "constructible" group size--the size of the group before we consider parity and orientation constraints--is k! (2k)! 16^k, or k! (2k)! 64^k for the supergroup. But if a+b is even, we can shade the faces in a checkerboard, and the shades never change when we turn the faces. So in this case, the edges never flip, and the corners have only two orientations. The checkerboard-constructible group size is then k! (2k)! 2^k, or k! (2k)! 8^k for the supergroup. Everyone who knows Rubik's cube will suspect (and everyone who understands orientation theory will know!) that the corner orientations must sum to zero (mod 4) and the edge orientations must sum to zero (mod 2). If a+b is even, there is only a corner orientation constraint (mod 2). [See my article of 23 September 1982 for a sketch of orientation theory. Essentially, if the orientation group of a kind of piece is Abelian then there is an orientation constraint of the order of that orientation group.] The permutation parity constraint is also familiar to anyone who knows the cube. The edge permutation parity must equal the corner permutation parity, and in the supergroup the parity of the face center twist must also be equal (mod 180 degrees). So we should find groups of size k! (2k)! 2^f(k), where f(k)=4k - 4 a+b odd, or f(k)= k - 2 a+b even for the permutation group, and f(k)=6k - 5 a+b odd, or f(k)=3k - 3 a+b even for the supergroup. I've used GAP to find the group sizes for (a,b) = (2,0), (2,1), (3,0), (3,1), (3,2), (4,0), (4,1), and the group sizes agree except for (2,0) and (2,1). The group is smaller than expected by a factor of 5040 for the (2,0) permutation group, 20160 for the (2,0) supergroup, 6 for the (2,1) permutation group, and 12 for the (2,1) supergroup. I'm not too surprised about the (2,0) groups (for instance, all four corner cubies move cyclically on every turn!) but I don't see why (2,1) is pathological. Maybe it's one of those special group things that happen for just one permutation group. By the way, I suggest that a simulation of these should not try to map them to a curved torus, but to a toroidal tesselation of the plane. Then when you turn one piece, you see a lattice of other pieces turning in synchrony. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 4 10:22:51 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA16653; Mon, 4 Aug 1997 10:22:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From R.F.Hegge@MP.TUDelft.NL Mon Aug 4 09:38:48 1997 Date: Mon, 04 Aug 1997 15:34:47 +0200 From: R.F.Hegge@MP.TUDelft.NL (Rob Hegge) Subject: CFF was Re: Where Can I get... To: Cube-Lovers@ai.mit.edu Message-Id: <9708041334.AA09170@sumatra.mp.tudelft.nl> Edwin wrote: > Rob wrote: > >There still exist > >a club called "Nederlandse Kubus Club" (NKC) or Dutch Cubist Club. > This sounds interesting. Can you tell me how to contact them? And, does > anyone know if there's a similar organization in germany? Sorry I do not know of anything similar to NKC in Germany. I put most of the information about CFF/NKC including some tables of contents on http://wwwtg.mp.tudelft.nl/~rob/cff.html Please don't hesitate to email me if you still have questions. After finally finding an original Rubik's Cube I have some 3x3x3's left. What are the most interesting bandaged cubes or other puzzles one can make of them ? Rob From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 4 19:43:16 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA19964; Mon, 4 Aug 1997 19:43:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Mon Aug 4 19:41:34 1997 Date: Mon, 4 Aug 1997 19:41:24 -0400 Message-Id: <199708042341.TAA09295@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Cube-Lovers@ai.mit.edu, ponder@austin.ibm.com In-Reply-To: <199708020031.UAA28142@sun30.aic.nrl.navy.mil> (message from Dan Hoey on Fri, 1 Aug 1997 20:31:56 -0400) Subject: Re: puzzle to be simulated I've found out why the cube-plane groups related to the 1^2+2^2 square torus are 1/6 the size we would expect. It's the corners. The group has five corners {1,2,3,4,5} and five generators {A,B,C,D,E} that operate on corners as 5..CC/DDD`5 A: (1,2,4,3) EEE`1..DD/E B: (2,3,5,4) .EE/AAA`2.. C: (3,4,1,5) B`3..AA/BBB D: (4,5,2,1) B/CCC`4..BB E: (5,1,3,2) 5..CC/DDD`5 These generators do not generate the 120-element group S5, rather they generate a 20-element subgroup known to GAP as 5:4 = A split extension of C5 by C4 or equivalently H(2^2,5) = . Neither of these tells me a lot, except that the fact that this group has index 6 in S5 means that there are six "orbits" of corner permutations. Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 4 21:26:36 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA20319; Mon, 4 Aug 1997 21:26:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Mon Aug 4 20:14:53 1997 Message-Id: <33E6709C.4277@idirect.com> Date: Mon, 04 Aug 1997 20:15:25 -0400 From: Mark Longridge To: cube lovers Subject: Megaminx a.k.a. Supernova Has anyone ever written a simulation of the Megaminx for the PC? I'm thinking about Megaminx moves and I'm on the verge of writing a simulation from scratch (starting with my file for GAP), to help myself to compose sequences for patterns. I am particularly interested in processes for the 10-spot and 12-spot. Ultimately I should write one, something with coarse face movement, algebraic move entries like ( F+ B- )^12 with whole megaminx rotations. This is interesting to me as it lies outside of recorded cube literature. Mark Web Page At: http://web.idirect.com/~cubeman I've also managed to compile Mike Reid's ANSI C cube solver for MS DOS using DJGPP. It makes good use of DPMI. From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 10:16:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA23477; Tue, 5 Aug 1997 10:16:48 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Tue Aug 5 06:34:57 1997 Message-Id: <199708051031.LAA16445@mail.iol.ie> From: "David Byrden" To: " From: Mark Longridge > Has anyone ever written a simulation of the Megaminx for the PC? Mark: There is a virtual Megaminx at my Rubik Gallery http://www.iol.ie/~goyra/Rubik.html It's not written for the PC: it's in Java, so it works on alll the major kinds of computer. In fact, you don't need to download or install anything, just use a Java browser and you can play with the puzzles immediately. Control is via the mouse. The puzzles are not self-solving but if anyone wants to write a solver and make use of my graphical representation of the puzzles, get in touch. > I am particularly interested in processes for the 10-spot and > 12-spot. What exactly are these, I may want to put them in the Gallery. David From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 11:42:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA23888; Tue, 5 Aug 1997 11:42:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Tue Aug 5 06:34:57 1997 Message-Id: From: "David Byrden" To: Cube-Lovers@ai.mit.edu Subject: Re: Megaminx a.k.a. Supernova Date: Tue, 5 Aug 1997 11:30:44 +0100 > From: Mark Longridge > Has anyone ever written a simulation of the Megaminx for the PC? Mark: There is a virtual Megaminx at my Rubik Gallery http://www.iol.ie/~goyra/Rubik.html It's not written for the PC: it's in Java, so it works on alll the major kinds of computer. In fact, you don't need to download or install anything, just use a Java browser and you can play with the puzzles immediately. Control is via the mouse. The puzzles are not self-solving but if anyone wants to write a solver and make use of my graphical representation of the puzzles, get in touch. > I am particularly interested in processes for the 10-spot and > 12-spot. What exactly are these, I may want to put them in the Gallery. David [ Moderator's note: Sorry for those of you who get this Cube-Lovers message twice. I accidentally sent a version with mangled headers, which several mailer daemons refused to process. -- Dan] From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 12:36:40 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA24116; Tue, 5 Aug 1997 12:36:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Tue Aug 5 12:04:25 1997 Date: Tue, 5 Aug 1997 12:04:08 -0400 Message-Id: <199708051604.MAA13056@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Goyra@iol.ie Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708051031.LAA16445@mail.iol.ie> (Goyra@iol.ie) Subject: Re: Megaminx a.k.a. Supernova Mark Longridge wrote: > I am particularly interested in processes for the 10-spot and > 12-spot. "David Byrden" asked for clarification. Spot patterns are those in which all the corner and edge cubies agree with each other, but not necessarily with all the face centers. They are so named because the non-matching face centers show up as contrasting spots. Mark reported some analysis on them on 31 Oct 95, apparently from GAP. As he reported, there are five conjugacy classes: 0. The identity, 1. The 72-degree twelve-spot, 2. The 144-degree twelve-spot, 3. The 120-degree ten-spot, 4. The 180-degree ten-spot. The angle given is the displacement of the corners-and-edges ensemble from the face-centers ensemble. In cases 1 and 2, the rotation is about an axis through two opposite face centers; in case 3, through opposite corners; in case 4, through opposite edges. Of course, there's no reason to expect optimal processes for these patterns to the same length. Interestingly, while the square of the 72-degree is the 144-degree, it is also the case that the square of the 144-degree is the 72-degree (up to conjugacy). It's also interesting to consider star patterns, in which the edges agree with the face centers, and the corners agree with each other. These come in the same classes as the spots. A third type of pattern is a distorted checkerboard, in which the corners and face centers agree with each other, and the edges agree with each other. These come in the same classes as well. I had hoped to find some in which the edges were apparently reflected with respect to the face centers (as in the Pons Asinorum and the order 6 6-X patterns on the cube) but they seem to be in the wrong orbit for the Megaminx. Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 15:30:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA24768; Tue, 5 Aug 1997 15:30:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Tue Aug 5 14:31:16 1997 Message-Id: <199708051827.OAA07000@life.ai.mit.edu> Date: Tue, 5 Aug 1997 14:33:36 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: 20f maneuvers for superfliptwist because of the historical interest in the pattern superfliptwist, i decided to find all 20f maneuvers for it. this took about 114 hours of searching, using the symmetry reductions i described in an earlier message. by "all" maneuvers, i mean that any 20 face turn sequence for superfliptwist can be transformed in to one on my list by conjugating by one of the 24 symmetries that fix superfliptwist, or by inverting the sequence and conjugating by the cube rotation C_U , and perhaps again by one of the 24 symmetries. some extra processing by hand was required to eliminate inverse pairs. hopefully i haven't made any errors here. the list of maneuvers is U R F' B U' D' F U' D F L F' L' U R D F U R L (20f, 20q) U F D L U R' F' R F U' D F U' D' F' B L U R L (20f, 20q) U R' F' U' F' R' B' D2 R' D R L' B R F B2 R' U' B D' (20f, 22q) U F L D L F R U2 F U' F' B R' F' R2 L' F D R' D' (20f, 22q) U R' U' D F' U' F2 B' U L F' R L' U B L B U F R2 (20f, 22q) U R' U' D F' U' B' R' F' R U F2 R U B L B U F R2 (20f, 22q) U F L D L F R U2 F U' F' B R' F' L' U' R' U F R2 (20f, 22q) U F R' F' L' U' R' U' D F' U F2 R U L B L U F R2 (20f, 22q) U F D L D F R L' U' L F U2 D' F' U' F' B R' F R2 (20f, 22q) U F D L D F R U2 F R U' R' D' F' U' F' B R' F R2 (20f, 22q) U R B D2 L B R' D' R' B L' D2 L B' D2 R' F B2 D' R (20f, 24q) U R' B L2 U' L2 U' B U' L2 D R B D F U2 R' L' B' R' (20f, 24q) U F B' R' U2 L U' R2 B' L' F2 U' R' D' L2 U D B D' B (20f, 24q) U R L2 F U F U F L D2 L' D' L U' D F2 B L' F R2 (20f, 24q) U R L' B' R' F R' F' B' D2 F U B L2 D R U2 B D' B2 (20f, 24q) U R' U2 D F B' R F' R' F2 R U B U B U R2 L B L2 (20f, 24q) U R' B R U' L' U2 B' R2 L' D B2 L U' B R F U B L2 (20f, 24q) U R' D2 B' U' F2 R' D' L' U2 R L B L' B R F B' D' R2 (20f, 24q) U F D L U F' R U2 B R' L2 U' F2 R' F' L U L' F R2 (20f, 24q) U F2 R' U' F' R' L2 U B U L' F B' U2 D L U' D' B R2 (20f, 24q) U F' B' L F B2 U' D L' B U B R' L2 D' B' R' D2 B R2 (20f, 24q) U R2 B L' U2 B' R' L F2 D F L2 D R' F2 D' R L' U2 B (20f, 26q) U F R' L D B R2 U2 L2 D' R' D' R L2 U' F L D2 B R2 (20f, 26q) U F2 L D B' R L2 F' R' F' L2 B2 R2 U F R' L D B R2 (20f, 26q) U F2 R' L' U F' U' D2 B2 U' B D R' L2 D2 L2 D2 F U2 B' (20f, 28q) the maneuver that herbert kociemba found is equivalent to the 28q maneuver. the two maneuvers that are 20q long can be obtained from one another by inverting the first, then cyclically shifting the antislice to the end of the maneuver, and then reorienting. the first of the 26q maneuvers is quite interesting. it can also be written as (U R2 B L' U2 B' R' L F2 D C_UF)^2 where C_UF is a cube rotation about the UF - DB edge axis (as in bandelow's book). mike From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 19:12:26 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA27472; Tue, 5 Aug 1997 19:12:26 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Tue Aug 5 19:11:22 1997 Date: Tue, 5 Aug 1997 19:11:12 -0400 Message-Id: <199708052311.TAA13218@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Cube-lovers@ai.mit.edu Subject: Glyph patterns Years ago I thought for a while on a taxonomy of some of the pretty patterns. Mark's bringing some of them up on the Megaminx has reminded me of them. My favorite class of pretty patterns is the "glyph" patterns. These are the patterns on which each face of the cube has facets of at most two colors. They include most of the pretty patterns we've discussed on the list. The "glyphs" here are the partition of the nine facets into two colors, where we aren't concerned with which colors but with the partition. I call the part of the glyph that includes the face center the "figure" and its complement the "ground". There are only 51 glyphs up to the symmetry of the square, or 70 if we distinguish chiral pairs. Some of the common ones we have discussed are blank, X, plus, dot, bar, T, slash, and H. I recall seeing a cubing book that assigns 26 of the glyphs to letters of the alphabet, where you try to place all the letters of your favorite six-letter word on the cube, or something like that. Classification and analysis of glyph patterns is often simplified by separating out the corner-glyph from the edge-glyph. There are only 6 each of these subglyphs (up to symmetry), mostly determined by how many "figure" facets of each type there are. Name 0 1 2 D 3 4 +-----+-----+-----+-----+-----+-----+ |. .|X .|X X|X .|X X|X X| Corner | . | . | . | . | . | . | |. .|. .|. .|. X|. X|X X| +-----+-----+-----+-----+-----+-----+ | . | X | X | X | X | X | Edge |. . .|. . .|X . .|. . .|X . X|X . X| | . | . | . | X | . | X | +-----+-----+-----+-----+-----+-----+ So a type-2D glyph would have the corner-glyph 2 and the edge-glyph D. There are two type-2D glyphs, called T and U. An important subclass of the glyph patterns are the "isoglyphs", which have the same glyph on all six faces. We've talked about the 6-T, 6-plus, 6-X, 6-H, and 6-spot patterns. Recall that you can twist just two opposite corners of the cube--I think Hofstadter called this a boson or something. I was amused to find that there is just one other 6-corner isoglyph of the cube. Another subclass are the "continuous" glyph patterns, in which the glyphs on neighboring faces match along the edge. That is to say, a facet of an edge cubie and an adjacent facet of a corner cubie have the same color if and only if the other facet of the edge cubie and the adjacent facet of the corner cubie have the same color. This matching condition gives the 6-plus patterns much of their charm. When every cubie of a continuous glyph pattern has either all "figure" facets or all "ground" facets, we call the pattern a "reassembled" glyph pattern. In this case, we can envision the cube having been cut apart into figure and ground cubies and put back together in a different orientation. Note that the reorientation may include a reflection, as we see in the Pons Asinorum. Some of the prettiest reassembled glyph patterns have corner type 4 on all faces--I call them "path patterns", because you can consider them a road map going around the cube. In 1981 Dave Ackley found one he called the "four-way street", which is the unique continuous type-41 isoglyph. If you can find it, you know what I'm talking about. I've been considering writing a program (or perhaps sparking someone else's interest in writing a program) to count and classify all the glyph patterns, possibly by using corner-edge reduction. It might be interesting to see if there is a set of nine cubes that exhibits all 51 glyphs, or if not what the smallest panglyphic set is. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 6 11:16:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA01366; Wed, 6 Aug 1997 11:16:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Tue Aug 5 22:42:50 1997 Message-Id: <199708060239.WAA23888@life.ai.mit.edu> Date: Tue, 5 Aug 1997 22:45:08 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: more patterns at distance 20f i can also give three more patterns that are exactly 20 face turns from start. all three are very symmetric; they have 24 symmetries. the symmetry group is "H" in dan hoey's taxonomy. in particular, all are local maxima (in the quarter turn metric). see hoey and saxe's "symmetry and local maxima" (december 14, 1980) for more info about this. the first pattern is the composition of superfliptwist with pons asinorum. you may recall that i suggested this pattern to dik winter when he was looking for positions that couldn't be solved in 20 face turns or less. he did succeed in solving it in 20f, using kociemba's algorithm, but it took much longer than most other positions did. the other two are inverses of one another. they can both be described as the composition of superfliptwist with 6 H's. however, the patterns "6 H's" and "superfliptwist" each come in two orientations. therefore, fix your favorite orientation of 6 H's; now there are two different orientations of superfliptwist which may be composed. this gives two distinct patterns, and the positions are inverses. by symmetry, we may assume that the first face turn of a maneuver for any of these positions is either U or U2. to confirm that the pattern superfliptwist . pons asinorum is not within 19f of start, we need to search the positions superfliptwist . pons asinorum . U and superfliptwist . pons asinorum . U2 completely through depth 18f. similarly, for the second pattern, two complete searches through depth 18f were required. the third pattern is the same distance from start as is its inverse, so this one doesn't require further testing. my optimal solver did not find the minimal maneuvers for these, although it certainly would have, if i'd let it continue searching long enough. however, one can find 20f maneuvers using kociemba's algorithm: superfliptwist . pons asinorum: D' R' B' L2 U' L B' D' R' D' B2 D2 B' U D2 R2 F2 D' L' B' (20f) superfliptwist . 6 H's: B' L2 D B2 R' D2 F' L2 U' L' F' B U' R D' R2 F2 R2 U' D2 (20f) it would be nice to find a position that was not within 20f of start. of course, we don't know if any such positions exist. my guess is that they do, but that's only a hunch. dik winter examined 9000 random positions and found that they were all within 20f of start. therefore the positions we're looking for are extremely scarce. i think that looking at positions with a lot of symmetry seems to be the right way to approach this. i've tested some of the most symmetric positions, but each that i examined was solved in 20f or less. mike From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 6 12:37:59 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA01892; Wed, 6 Aug 1997 12:37:58 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From saxonia@imn.htwk-leipzig.de Wed Aug 6 11:44:11 1997 Date: Wed, 6 Aug 1997 17:40:38 +0200 From: saxonia@imn.htwk-leipzig.de (Ralf Laue) Message-Id: <9708061540.AA24642@imn.htwk-leipzig.de> To: Cube-Lovers@ai.mit.edu Subject: Rubik's Cube World Records RUBIK'S CUBE WORLD RECORD LIST ------------------------------ I have created a list of Rubik's Cube world records in the WWW with the URL: http://www.imn.htwk-leipzig.de/~saxonia/list/rubik.html It is about speed cubing world records and other funny stuff (cube marathon record etc.) I would be glad about comments and corrections to this list. (New record categories are very welcome: Particularly I am interested in the 4x4x4 cube speed solving world record!) If you do not have access to the WWW, just send me an e-mail, and I will send you the list by e-mail. If you have your own WWW site about Rubik's Cube, I would be glad if you would create a link to my URL. (The list is a part of my WWW information about unusual world records at http://www.imn.htwk-leipzig.de/~saxonia/homepage.html ) Very Best Wishes, Ralf Laue ___________________________________________________________________________ Please excuse me for a delay in replying to your e-mail. I cannot read my incoming mail daily. ----------------------------------------------------------------------------- Ralf Laue e-mail: saxonia@imn.htwk-leipzig.de P. O. Box 80 Read my Homepage about unusual world records: 04181 Leipzig http://www.imn.htwk-leipzig.de/~saxonia/homepage.html Germany ----------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 6 19:25:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA04684; Wed, 6 Aug 1997 19:25:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Wed Aug 6 19:23:53 1997 Date: Wed, 6 Aug 1997 19:10:30 -0400 Message-Id: <199708062310.TAA17135@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Cube-lovers@ai.mit.edu Subject: Reassembled patterns (was Glyph patterns) I wrote: > When every cubie of a continuous glyph pattern has either all "figure" > facets or all "ground" facets, we call the pattern a "reassembled" > glyph pattern. In this case, we can envision the cube having been cut > apart into figure and ground cubies and put back together in a > different orientation.... On second thought, I prefer the definition that a (2-part) reassembled pattern is one that can be partitioned into two sets of cubies, where the cubies of each set are in agreement with each other. This definition differs from the previous in two ways. Reassembled patterns need not be continuous--"laughter" is a noncontinuous glyph pattern. And not all continous glyph patterns with figure/ground cubies meet this definition--e.g. flip the LF and RD edge cubies. We may also speak of 3-part reassembled patterns, though they are not necessarily glyph patterns. Are there any particularly nice ones? Cube-in-a-cube-in-a-cube comes to mind. Call an "N-part" pattern one that requires cutting into at least N parts for reassembly. Surely every position can be reassembled from at most 21 parts, since that's all the pieces there are. Is this achievable? We could restrict the reorientation of the parts to C, but in some cases (e.g. pons asinorum) we can manage with fewer parts if we allow reorienting some of the edges by M. Is there a 20-part pattern that would require 21 parts if the orientations were restricted to C? In the supergroup, can we manage a 24-part position? A 23-part position that requires 24 parts for C-reorientation? Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 6 22:13:35 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA05412; Wed, 6 Aug 1997 22:13:34 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Wed Aug 6 22:12:07 1997 Message-Id: <199708070208.WAA08375@life.ai.mit.edu> Date: Wed, 6 Aug 1997 22:14:21 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: optimal solver for quarter turns i have my optimal cube solver working for quarter turns. it seems to be as effective as the face turn version. some minimal maneuvers it has found are cube in a cube in a cube U' L' U' F' R2 B' R F U B2 U B' L U' F U R F' (20q) six X's, order 6 F U' L2 F' L' D R U' D L' B U2 F' L' D' F D R (20q) ron's cube within the cube F D' F' R D F' R' D R D L' F L D R' F D' (17q) and it has also confirmed minimality of known maneuvers for several other patterns. mike From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 7 10:59:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA07805; Thu, 7 Aug 1997 10:59:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Wed Aug 6 23:51:22 1997 Message-Id: <199708070348.XAA10924@life.ai.mit.edu> Date: Wed, 6 Aug 1997 23:53:38 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: superflip requires 24 quarter turns with my optimal solver, i can show that superflip is exactly 24 quarter turns from start. this was already shown by jerry bryan, so this confirms his result. first some history. david plummer gave a 28q maneuver for superflip on december 10, 1980. apparently there was no improvement to this until january 1995, when i implemented kociemba's algorithm for quarter turns. after a lot of searching, where i specified the initial sequence R' U2 , it found R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q) mark longridge notes that this sequence has a remarkable symmetry, namely that it may be written as (R' U2 B L' F U' B D F U D' C_-1)^2 , where C_-1 denotes central reflection. later in january 1995, i completed an exhaustive search for superflip in 20 quarter turns, without finding any maneuvers. i used my quarter turn version of kociemba's algorithm, which took 29 cpu hours. this improved the lower bound of the diameter of the cube group to 22q. the previous lower bound was 21q, obtained by a counting argument. in february 1995, jerry bryan improved this result to show that superflip is not within 22 quarter turns, and thus is exactly 24 quarter turns from start. this also improved the lower bound for the diameter to 24q. we'll use symmetry to reduce the size of the search space dramatically. consider three cases for a minimal maneuver for superflip. 1) it contains a half turn (i.e. two consecutive quarter turns of the same face). 2) it does not contain a half turn, but contains two consecutive turns of opposite faces. 3) otherwise. in case 1, as in the face turn situation, we may suppose that the first three quarter turns are U R2 . in case 2, by cyclically shifting, we may suppose these two turns are the first two. if they form a slice (U D') then we may take the first three quarter turns to be U D' R . if they form an antislice, then we may take the first three quarter turns to be either U D R or U D R' . in case 3, i claim that we may find three consecutive turns of mutually adjacent faces. otherwise, if the first two faces turned were U and R, then we'd only be turning U , R , D and L . however, edges cannot change orientation when only these faces are turned. thus the claim holds, and by cyclically shifting, we may suppose that these three faces are U , R and F . by symmetry, we may suppose that they're turned in that order. now we have eight cases: U R F U R F' U R' F U R' F' U' R F U' R F' U' R' F U' R' F' we can eliminate two of these by using inversion. inverting the case U' R F gives F' R' U . conjugating this by the appropriate cube reflection gives U R F' , and these three turns can be cyclically shifted to the beginning of the maneuver. similarly, the case U' R' F can be transformed to the case U' R F . thus ten cases remain. to show that superflip is not within 22q of start, these cases must be searched through 19q. my program took 22 hours to searched these completely, and no maneuvers were found. iw would be nice to know all the minimal maneuvers for superflip. the branching factor is about 9.37, so an exhaustive search would take about 22 * (9.37)^2 hours, which is about 80 days. this is feasible, but is definitely a long term project. i've already searched the first case, (beginning with U R2) which would seem to be the most likely, through 21q. this took about 147 hours. i expected it to find a lot of maneuvers, but it only found 4, in two inverse pairs. the first is equivalent to the maneuver above, and the new one is U R2 F' R D' L B' R U' R U' D F' U F' U' D' B L' F' B' D' L' (24q) mike From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 7 15:46:48 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA09383; Thu, 7 Aug 1997 15:46:47 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 7 15:46:42 1997 Message-Id: <199708071943.PAA07912@life.ai.mit.edu> Date: Thu, 7 Aug 1997 15:48:49 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: composition of superflip and pons asinorum my optimal cube solver has also found all minimal maneuvers for the composition of superflip and pons asinorum. this was previously done by jerry bryan, so the purpose here is to confirm his results. up to symmetry, there are 10 maneuvers of length 20q, which occur in 5 inverse pairs. they are U R F D R U' D L' U' D F' B2 R L' D' F' L' B' R' (20q) U R U F U F B' L' F B' R L' B' R L' U F' U' R' U' (20q) U R U F D R L' B' R L' F B' L' F B' D F' U' R' U' (20q) U R D B U R L' F' R L' F' B L' F' B U B' D' R' U' (20q) U R D B D F' B L' F' B R L' F' R L' D B' D' R' U' (20q) this agrees exactly with jerry bryan's results. mike From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 8 11:22:01 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA02825; Fri, 8 Aug 1997 11:22:00 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Fri Aug 8 09:15:32 1997 Message-Id: <33EB1ABE.1DC8@hrz1.hrz.th-darmstadt.de> Date: Fri, 08 Aug 1997 15:10:22 +0200 From: Herbert Kociemba To: Dan Hoey , cube-lovers@ai.mit.edu Subject: Continuous isoglyph patterns References: <199708052311.TAA13218@sun30.aic.nrl.navy.mil> Dan Hoey made some proposals concerning 2-colored cube patterns. The "coninuous"-condition i find especially interesting. I added this feature to my Cube Explorer program and found exactly 34 continuous isoglyphs (plus the trivial solution). I don't know if there are any among them, which have not been described somewhere else before. Here are generators for the patterns (I searched only about 1 minute for most generators of the patterns, so there is no claim for the maneuvers to be optimal): D F2 D' . R B2 R' D F2 D' R B2 R' (12) U . L' D' U B2 D U' L' U' (9) U2 L' B2 . F R' D' R2 D B2 F' U L (12) D' U . L' R B' F D' U (8) R2 D L2 U' B2 D' U2 . R' F' U R B' L' D' F L2 B2 R U' (19) F2 L2 U L2 U' F2 D . B L R' D' U' F' D2 F' R2 F R (18) B2 U' B2 D' B2 D . L' B2 F' U R D2 R' D' U' F U (17) L2 D2 B D2 B' D2 B L2 . D B R D2 F L2 D F' R' (17) B2 U' B2 L2 B2 U2 B2 U' D2 . R U R' D' L U F U' D' L (19) U R2 D . F' L D2 U2 R' D2 U2 F D' R2 U' (14) U B2 . L B F' L2 R' B' F D U2 L' B2 U' (14) D' U . F' U L' R B' U F D' U R' (12) U' B2 F2 L2 U B2 U' L2 F2 . B' U R' F D' R2 D2 R' F' (18) F2 R2 D R2 D U F2 D' . R' D' F L2 F' D R U' (16) D U2 R2 D' U' . R D B2 R2 B2 R2 D B2 D2 R U' (16) D U2 L2 U R2 U' L2 U . R' B2 L2 F' L2 B' R' F' L D U' (19) D2 U F2 D' L2 U R2 B2 . R B2 R2 U2 B' L2 D2 R2 D R' U' (19) D2 R B2 R . F L B' F U' R L' U' F' D2 F' L2 (16) D' B2 F2 D' U L2 . F' L R' F' D U' R D B2 R (16) U' R2 F2 U2 . L' D2 B' L2 U' L2 D2 L U2 F' U2 (15) L2 U2 R' . B' D U' B2 D' R' D L D2 F D U2 L2 (16) U' F2 U . R U2 R2 U2 R' F' R2 F U' F2 U (14) U2 R2 F2 U B2 D' . L' F L' F L' F D B2 U (15) D2 R F2 L' D2 R . B D2 F' L2 U' R' D L F D L' D L' (19) D' L2 F2 L2 B2 R2 U F2 U2 . L' F R B D R U' L F' U2 F (20) B2 L2 R2 U B2 R2 D F2 U' . B F U2 R' B2 L2 D U' B' L' R' (20) U B2 U2 L2 U F2 R2 B2 U' L2 D2 F2 U' . B L2 R2 D2 U2 F' (19) L2 . R' B2 F2 D2 B2 F2 L2 R2 U2 R' (11) D U L2 B2 D U' . F' U F' R F2 R' F D' B2 L2 D' U' (18) L2 U' B2 F2 D . R D F' U' R2 B2 U' B D2 B' F' L U' (18) D F2 R2 F2 R2 U F2 . R F2 R D2 U' F L' F' L D (17) B2 R2 F' U2 D2 L2 R2 B . U' L R B' F U D B2 F2 R' F2 (19) D' L2 R2 D2 B2 F2 U' . R' B' F D' U L R' F2 D2 U2 F' (18) B2 F2 L2 R2 D2 U2 (6) If you copy and paste the maneuvers from this message to a text file, you can load them into Cube Explorer and directly watch the results. The response to my Cube Explorer 1.0 program showed me, that the userinterface and the terminology of the program are confusing (if not to say bad) and some features are missing which should be there. I almost completed Version 1.5 now. When I put it to my homepage in a few days, I will send another message. --Herbert From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 8 12:47:55 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA03262; Fri, 8 Aug 1997 12:47:55 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From ERCO@compuserve.com Fri Aug 8 03:45:46 1997 Date: Fri, 8 Aug 1997 00:52:32 -0400 From: Edwin Saesen Subject: Alexander's star etc. To: CUBE Message-Id: <199708080052_MC2-1CA0-3F67@compuserve.com> Hi everyone, can anyone tell me a way to disassemble an Alexander's Star without breaking it? I've had this one for YEARS (about 10 or so), and never really found a way to do it. It's very difficult to use, and I think a little lubricationg will do wonders :-) On a different note, I visited Christoph Bandelow on this wednesday, and it was absolutely incredible to see the range of puzzles he has (although he said he's not collectiong anymore). Anyway, I got quite a few nice things from him, including a 5x5x5 and two replacement center pieces for my 4x4x4 (YES!!! YES!!!!!!) :-) Michael PS: This doesn't mean I'm not interested in a second 4^3 one anymore - I would feel more comfortable with a second 4^3 to have a chance to replace broken pieces myself, but at least now it's working again. From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 9 15:13:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA08470; Sat, 9 Aug 1997 15:13:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Sat Aug 9 15:09:29 1997 Date: Sat, 9 Aug 1997 15:09:01 -0400 Message-Id: <199708091909.PAA07964@sun30.aic.nrl.navy.mil> From: Dan Hoey To: kociemba@hrz1.hrz.th-darmstadt.de Cc: cube-lovers@ai.mit.edu Subject: Re: Continuous isoglyph patterns Bravo, Herbert! A very nice list. It's surprising how many of them are reassembled patterns, too. Only the second and tenth are not reassembled, and both fail by using a reassembled pattern to camouflage a small distortion. Pattern #2 is pattern #3 composed with a two-flip, and pattern #10 is pattern #9 composed with a three-cycle of edges. There are four elements of M used to perform the reorientation of the reassembled patterns. Over half of them use the order-3 major-diagonal rotation, of Plummer's cross: patterns 1, 3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 30, 31, and 32. Several use the order-2 major-diagonal rotation, of Christman's cross: patterns 5, 8, 14, 20, 26, 28, and 29. The only pattern reassembled by the major diagonal reflected rotation is the order-6 6-X, pattern 33, And the only pattern reassembled by central reflection is Pons Asinorum, pattern 34. I've also classified these patterns by the glyph that appears on the faces (modulo my clerical errors). Patterns I know traditional names for are given with an asterisk; I've made up temporary descriptive names otherwise. Glyph Type Patterns X . X 2. Girdle 3-cycle, distorted X X X 01 3. Girdle 3-cycle X X X X . X 8. Christman's girdle . X X 02 9. Off-girdle 3-cycles X X X 10. Off-girdle 3-cycles, distorted 11. Girdle 3-cycles X . X . X X 03 21. Plummer's C's X . X X . X 32. Plummer's X . X . 04 33. Order-6 X X . X 34. * Pons Asinorum . X X X X X 10 1. * Meson X X X . X X X X . 11 7. Meson & girdle 3-cycle X X X . . X 13. Plummer's cluster . X X 12 14. Christman's cluster X X X . X X X X . 12 16. Meson & girdle 3-cycles X . X X . X X X . 13 24. Plummer's Y's X . . . . X 25. Plummer's cluster & girdle 3-cycles . X . 14 26. Christman's cluster & girdle X . X . X . X X X 30 15. Plummer's rabbits . X X . X X . X X 31 22. Plummer's P's . X . X X . X X . 32 23. * Cube in a cube . . . . X . 29. Christman's arrow X X . 32 30. Plummers's arrow . . X X . . . X X 33 16. Plummer's bend . . . X . . 5. Christman's comma . X . 34 6. Plummer's comma . . . . X . 27. * Plummer's Cross X X X 40 28. * Christman's Cross . X . . X . X X X 41 31. * Four-way street . . . . X . 18. Plummer's cube out of cube in a cube X X . 42 19. * Worm . . . 20. Christman's cube out of cube in a cube . X . . X . 43 12. Plummer's U's . . . . . . . X . 44 4. * Six-spot . . . Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 14 18:29:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA12833; Thu, 14 Aug 1997 18:29:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 14 18:19:32 1997 Message-Id: <199708142216.SAA16395@life.ai.mit.edu> Date: Thu, 14 Aug 1997 18:21:24 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: patterns with 24-fold symmetry i've finished computing minimal maneuvers for those positions with 24-fold symmetry. these positions were classified by dan hoey and jim saxe in their note "symmetry and local maxima." there are 24 such positions; they form an abelian subgroup of type 6, 2, 2. we may take as generators superfliptwist, pons asinorum, and 6 H's. of these 24 positions, 4 have 48-fold symmetry; i'll include these here as well. the other 20 positions occur in 10 pairs which differ only in orientation; i.e. there are 10 "patterns". some of these maneuvers were found earlier by others; i'll acknowledge this to the extent that i'm aware of it. in addition, most maneuvers were found by kociemba's algorithm, and a few by my optimal solver, which is based on the same ideas. positions with 48-fold symmetry start (0q, 0f) no turns needed superflip R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q) U R2 F' R D' L B' R U' R U' D F' U F' U' D' B L' F' B' D' L' (24q) U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f) U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f) dik winter was the first to find a 20f maneuver. pons asinorum U2 D2 F2 B2 R2 L2 (12q, 6f) F B' U2 D2 R2 L2 F B' (12q) F B' U' D F B' U' D F B' U' D (12q) this last maneuver is due to dan hoey. pons asinorum composed with superflip B' D' L' F' D' F' B U F' B R2 L U D' F L U R D (20q, 19f) F' U' B' R' F R L' D' R L' U D' L' U D' F R B U F (20q) B' R' F' U' F R L' D' R L' U D' L' U D' F U F R B (20q) B' R' B' D' F U' D L' U' D R L' U' R L' F D B R B (20q) R U R B R' U' D F U' D F B' D F B' R' B' R' U' R' (20q) U D F R L' F B' L D2 R L F' B' U' L2 F B' U2 L' (19f) U D F' B' L' U2 F' B L2 U' R' L' F' U' D F' B D' L2 (19f) U2 R F U F B' L' D' F B' L B R L' U D2 B' R' U2 (19f) U2 R F U2 D' R' L F' L' F B' U L F B' D' B' R' U2 (19f) U2 R U2 D2 R U' L' U B R F2 U' D B' R' F' D B' L2 (19f) jerry bryan found the 20q maneuvers. positions with 24-fold symmetry superfliptwist U R F' B U' D' F U' D F L F' L' U R D F U R L (20q, 20f) herbert kociemba was the first to find a 20f maneuver. supertwist U R' B D B U L D B' D2 R U' F L F R D L F' L2 (22q) B' L2 U D R2 B' D2 F2 D' R2 F B L2 D' B2 U2 (16f) dik winter first found the 16f maneuver. 6 H's D2 L2 B2 U2 D2 B2 R2 D2 (16q, 8f) jim saxe found this maneuver. superflip composed with 6 H's U F' L' F' B U R F' B U' B' U D' R2 L' B U' (18q, 17f) superfliptwist composed with pons asinorum U F B D R L U' F2 B2 R L D' F B D R L D F' B' (22q, 20f) dik winter was the first to find a 20f maneuver. supertwist composed with pons asinorum F L D F U' B2 R F R' F' R F L2 U' R D B R (20q) B2 L U2 F' B' U2 R' F2 L2 F' U2 R' L' U2 B R2 (16f) superfliptwist composed with 6 H's (type 1) U F B U' R L U F B R2 L2 D' F B U' R L D' R' L' (22q, 20f) superfliptwist composed with 6 H's (type 2) inverse of type 1 supertwist composed with 6 H's (type 1) U2 L U B D L U B' R' L' F' D R U F D L' U2 (20q) L' B2 U' D' B2 R' U2 L2 U B2 R L F2 U R2 U2 (16f) supertwist composed with 6 H's (type 2) inverse of type 1 some of these maneuvers have some symmetry. i find the maneuver for superfliptwist composed with pons asinorum especially interesting. it is composed of: twists of the U or D face, and antislices along the R-L and F-B axes: U (FB) D (RL) U' (F2B2) (RL) D' (FB) D (RL) D (F'B') therefore, when we conjugate this maneuver by the cube rotation C_U2, we get the same maneuver! the maneuver for superfliptwist composed with 6 H's has the same type of symmetry. mike From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 15 11:12:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA16020; Fri, 15 Aug 1997 11:12:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 14 21:48:30 1997 Message-Id: <199708150145.VAA22661@life.ai.mit.edu> Date: Thu, 14 Aug 1997 21:50:27 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: isoglyphs dan's idea of glyph patterns, especially isoglyphs is quite interesting. herbert has given all the "continuous" isoglyphs. i spent some time looking for other isoglyphs, and was surprised at how many exist. herbert, how did you find those? is that part of your pattern generator? if your program can also find all "discontinuous" isoglyphs, then i guess there's not much point in trying to do it by hand. mike From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 15 13:38:15 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA16787; Fri, 15 Aug 1997 13:38:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From goyra@iol.ie Fri Aug 15 12:42:42 1997 Message-Id: <199708151639.RAA24296@GPO.iol.ie> From: "Goyra" To: Subject: Got a new shape for a Rubik puzzle? Date: Fri, 15 Aug 1997 17:38:13 +0100 The Rubik Gallery is a website with Java Rubik puzzles. There are dodecahedrons, icosahedrons, cubes, etc etc. The software can support any shape at all, twisting in any way imaginable, provided the axes all meet at one point. I'd like to add some more intricate and strange shapes of puzzle, and before I sit down to bust my brain over the geometry, I want to ask the list members for ideas. You guys are mathematical geniuses and I'm sure some of you already have ideas for wierd puzzles that will never see the light of day in physical form; perhaps ones with Penrose tiles as the facelets, perhaps ones that twist in non-intuitive ways to create surprising shapes.If you'd like us all to see, twist and solve your puzzle, please tell me about it. David http://www.iol.ie/~goyra/Rubik.html From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 15 16:56:00 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA17814; Fri, 15 Aug 1997 16:56:00 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Aug 15 15:27:45 1997 Date: Fri, 15 Aug 1997 15:27:33 -0400 Message-Id: <199708151927.PAA03768@sun30.aic.nrl.navy.mil> From: Dan Hoey To: reid@math.brown.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708150145.VAA22661@life.ai.mit.edu> (message from michael reid on Thu, 14 Aug 1997 21:50:27 -0400) Subject: Re: isoglyphs Mike, I'm glad you like glyphs, and I'd also like to know about the other isoglyphs (by which I don't want to minimize my interest in the wealth of optimal processes you've been producing!). In particular, we've seen isoglyphs with all corner types except D (which we know is impossible) and with all edge types. So we might wonder if all the glyph types not involving corner type D are achievable. But I know there is no isoglyph of type 4D (stripe). Are there others? One superset of the isoglyphs that might be worth looking is the partial isoglyphs, in which all faces are either the same glyph or blank (type 00). This allows corner type D (laughter is 4 type D4 + 2 type 00). These even come in continuous varieties (slice is 4 type 4D + 2 type 00). Is there a partial isoglyph pattern for every glyph? And what about chiral partial isoglyphs, for which isoglyphicity is redefined to require the same handedness for orientable patterns? I'm pretty sure all the isoglyphs we've seen so far are chiral, but are there isoglyphs achievable only non-chirally? Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 15 18:58:41 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA18159; Fri, 15 Aug 1997 18:58:41 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Aug 15 19:00:35 1997 Date: Fri, 15 Aug 1997 19:00:26 -0400 Message-Id: <199708152300.TAA04077@sun30.aic.nrl.navy.mil> From: Dan Hoey To: reid@math.brown.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708142216.SAA16395@life.ai.mit.edu> (message from michael reid on Thu, 14 Aug 1997 18:21:24 -0400) Subject: Re: patterns with 24-fold symmetry Mike Reid writes: > i've finished computing minimal maneuvers for those positions with > 24-fold symmetry. these positions were classified by dan hoey and > jim saxe in their note "symmetry and local maxima." there are 24 > such positions; they form an abelian subgroup of type 6, 2, 2. It took me a while to understand that. For the benefit of other cube-lovers, since any finite Abelian group can be decomposed into a direct product of cyclic groups, it can be typified by listing the orders of its factors. > we may take as generators superfliptwist, pons asinorum, and 6 H's. > of these 24 positions, 4 have 48-fold symmetry; i'll include these > here as well. the other 20 positions occur in 10 pairs which differ > only in orientation; i.e. there are 10 "patterns". It may be better to take the order-6 generator to be one of the 6-H-supertwists. Then you can tell the M-symmetric positions because they project to the identity of the 6-factor. Writing p, f, t, h for pons, superflip, supertwist, and 6-H, I get the following table of positions (suffixed with optimal qtw:ftw). i p f fp ............................................... i : i 0:0 p 12:6 f 24:20 fp 20:19 : th : th 20:16 th' 20:16 fth 22:20 fth' 22:20 : t : t 22:16 pt 20:16 ft 20:20 fpt 22:20 : h : h 16:8 h 16:8 fh 18:17 fh 18:17 : t : t 22:16 pt 20:16 ft 20:20 fpt 22:20 : th': th' 20:16 th 20:16 fth' 22:20 fth 22:20 The last two rows could be omitted, just as the last column could be with your decomposition: i h p h ............................................... i : i 0:0 h 16:8 p 12:6 h 16:8 : ft : ft 20:20 fth 22:20 ftp 22:20 fth' 22:20 : t : t 22:16 th 20:16 tp 20:16 th' 20:16 : f : f 24:20 fh 18:17 fp 20:19 fh 18:17 : t : t 22:16 th' 20:16 tp 20:16 th 20:16 : ft : ft 20:20 fth' 22:20 ftp 22:20 fth 22:20 This has the advantage of having patterns on each row nearer each other. By the way, this isn't a complete list of optimal maneuvers, is it? Are you looking to find such a list? Or would it be too difficult (or too voluminous)? And I'm looking forward to seeing optimal maneuvers for the T-symmetric positions (if I'm not being too presumptuous). Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 15 20:32:05 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA18522; Fri, 15 Aug 1997 20:32:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Fri Aug 15 19:48:02 1997 Message-Id: <199708152344.TAA05912@life.ai.mit.edu> Date: Fri, 15 Aug 1997 19:49:54 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs perhaps i am missing something, but doesn't D2 R2 U2 D2 R2 U D (12q, 7f) produce an isoglyph of type 4D ? are there any isoglyphs of type 21 or 23 ? i haven't found any. each has three possible patterns: ... .*. .*. *** .** *** *** , *** and *.* , .*. ... ... .*. **. .*. *.* , *.* and *** . i hadn't even considered chiral versus achiral isoglyphs. indeed, all the "continuous" isoglyphs given by herbert are chiral. achiral isoglyphs certainly exist, for example D2 R2 U' B' L B U B L F2 R D' L2 U2 B2 D (22q, 16f) of type 11; pattern *.. *** *** and others can be derived from this. i suspect that there is no chiral form of this isoglyph, but i'm not absolutely certain. another interesting note is that the inverses of the "continuous" isoglyphs are also isoglyphs; in fact the same pattern, perhaps in a different orientation. however, there is at least one (probably more) isoglyph whose inverse is not an isoglyph. instead of giving it right here, i'll challenge other readers to find it/them. mike From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 16 20:08:01 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA22718; Sat, 16 Aug 1997 20:08:01 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sat Aug 16 05:28:59 1997 Message-Id: <33F571EE.764B@hrz1.hrz.th-darmstadt.de> Date: Sat, 16 Aug 1997 11:25:02 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs References: <199708152344.TAA05912@life.ai.mit.edu> michael reid wrote: > perhaps i am missing something, but doesn't > > D2 R2 U2 D2 R2 U D (12q, 7f) > > produce an isoglyph of type 4D ? That's right, and it is the only one which exists of this type. But it is no continuous isoglyph. > > are there any isoglyphs of type 21 or 23 ? i haven't found any. > each has three possible patterns: > > ... .*. .*. > *** .** *** > *** , *** and *.* , > > .*. ... ... > .*. **. .*. > *.* , *.* and *** . Here are generators for all isoglyphs of your second pattern: D2 R2 B2 L2 U2 F2 U2 B2 R2 U2 R2 U2 (12) U2 R2 F2 R2 U2 B2 D2 B2 L2 U2 R2 U2 (12) U2 L2 F2 R2 U2 B2 D2 B2 R2 U2 R2 U2 (12) D2 L2 B2 L2 U2 F2 U2 B2 L2 U2 R2 U2 (12) and here for the fifth: D2 R2 F2 L2 U2 B2 D2 F2 L2 U2 L2 U2 (12) U2 R2 B2 R2 U2 F2 U2 F2 R2 U2 L2 U2 (12) U2 L2 B2 R2 U2 F2 U2 F2 L2 U2 L2 U2 (12) D2 L2 F2 L2 U2 B2 D2 F2 R2 U2 L2 U2 (12) For the others, no isoglyphs exist. --Herbert From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 16 20:49:30 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA22827; Sat, 16 Aug 1997 20:49:29 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sat Aug 16 05:29:55 1997 Message-Id: <33F571F4.2C84@hrz1.hrz.th-darmstadt.de> Date: Sat, 16 Aug 1997 11:25:08 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs References: <199708150145.VAA22661@life.ai.mit.edu> michael reid wrote: > > dan's idea of glyph patterns, especially isoglyphs is quite interesting. > herbert has given all the "continuous" isoglyphs. i spent some time > looking for other isoglyphs, and was surprised at how many exist. > herbert, how did you find those? is that part of your pattern generator? > if your program can also find all "discontinuous" isoglyphs, then i guess > there's not much point in trying to do it by hand. With the pattern generator it's indeed very easy to find the isoglyphs. I restricted myself to continuous isoglyphs, because I had the most interest in them an because the number is quite limited. There are many, if you do not use the "continuous" condition. By the way, Mike, it would be nice to complete the chapter "continous isoglyphs" by computing the shortest generators for them with your program. This hopefully should not take too long, because most of the generators seem to be rather short. Here are for example the (not necessarily continuous) isoglyphs, which built the "snake patterns". There are 13 of them: R2 B' U2 B' . D' F' U D' L B' F L D R2 D F2 (16) D' B2 F2 D' U L2 . F' L R' F' D U' R D B2 R (16) R L2 B2 R' . D' L' D B2 F L2 U' L U' F' (14) B2 U2 F2 D2 F2 U . R' F' L2 U2 L R U' L2 F2 L' F (17) D B2 L2 D2 . F' D2 L B2 D F2 U2 F U2 R' D2 (15) D2 R D2 F2 L U2 . B R' D R2 D' R B D2 L U2 L' (17) U' B2 R2 U2 . F' D2 L' F2 U' F2 D2 F U2 R' U2 (15) D2 B2 L B2 F2 L' . U B F' L F2 L' B' F D2 U' (16) D2 U' B2 D2 U2 L2 U2 . B' U2 L F2 U B2 D2 B D2 R' U2 (18) U2 R' F2 R' D2 F2 . B U' L' U' B D L' F2 D' (15) U' F2 D R2 U L2 U2 . B' U B' U R D' L F' R D2 (17) U2 F2 D F2 L2 U2 . F' U2 L F2 D B2 D2 F D2 R' F2 (17) D2 F2 D2 U2 R D2 U2 R' . U R L' B D2 B' R' L D2 U' (18) P.S.: I could not hold what I promised, the Cube Explorer 1.5 version (which for example has the "continuous" feature) still is not ready.... --Herbert From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 16 21:30:20 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA22864; Sat, 16 Aug 1997 21:30:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Sat Aug 16 17:12:29 1997 Message-Id: <199708162109.RAA29706@life.ai.mit.edu> Date: Sat, 16 Aug 1997 17:14:10 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: patterns with 24-fold symmetry dan asks > By the way, this isn't a complete list of optimal maneuvers, is it? > Are you looking to find such a list? Or would it be too difficult (or > too voluminous)? for most, it isn't a complete list. the main reason is that it's too time consuming to calculate all of them. (perhaps it is also too voluminous, but i'm not sure.) on an individual basis, here's what i gave: superflip: all 20f maneuvers, up to inversion, cyclic shifting, and conjugation by cube symmetries. all _known_ 24f maneuvers, up to the same transformations. pons asinorum: all 12q maneuvers, up to conjugation by cube symmetries. all 6f maneuvers, up to conjugation by cube symmetries. (the inverse of each maneuver is the same as some conjugate by a cube symmetry.) superflip composed with pons asinorum: all 20q and 19f maneuvers, up to inversion and conjugation by cube symmetries. for the H-symmetric patterns (24-fold symmetry), i was less ambitious. for each, i gave a single minimal maneuver in each metric. also, in the cases where there is a maneuver that is minimal in both metrics, i gave such a maneuver. > And I'm looking forward to seeing optimal maneuvers for the > T-symmetric positions (if I'm not being too presumptuous). how did you know what i'm working on next? ;-) i also plan to examine AC-symmetric positions and X-symmetric positions (if i understand your terminology correctly). however, there are so many of these (124) in the X-symmetric case, that i probably will have to settle for sub-optimal maneuvers. mike From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 16 22:34:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA23066; Sat, 16 Aug 1997 22:34:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Sat Aug 16 22:35:27 1997 Date: Sat, 16 Aug 1997 22:35:13 -0400 Message-Id: <199708170235.WAA07984@sun30.aic.nrl.navy.mil> From: Dan Hoey To: reid@math.brown.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708152344.TAA05912@life.ai.mit.edu> (message from michael reid on Fri, 15 Aug 1997 19:49:54 -0400) Subject: Re: isoglyphs > perhaps i am missing something, but doesn't > D2 R2 U2 D2 R2 U D (12q, 7f) > produce an isoglyph of type 4D ? Oops, you're right. I goofed because I half-remembered a different result, that there are no 6-bar patterns of the nice symmetric sort, with three mutually perpendicular pairs of parallel bars. > ... > i hadn't even considered chiral versus achiral isoglyphs. indeed, > all the "continuous" isoglyphs given by herbert are chiral. > achiral isoglyphs certainly exist, for example > D2 R2 U' B' L B U B L F2 R D' L2 U2 B2 D (22q, 16f) > of type 11; pattern > *.. > *** > *** > and others can be derived from this. i suspect that there is no > chiral form of this isoglyph, but i'm not absolutely certain. Modulo some oversight, I think this is true, and not hard to demonstrate. Recall that a "ground" facet is one that is not on its home face. First note that a corner cubie will have 0, 2, or 3 ground facets. So on any isoglyph of corner type 1, there are a total of 6 ground corner facets, and these ground facets must appear on two corner cubies (three ground facets each) or three corner cubies (two ground facets each). If two corner cubies, those cubies must be antipodes, and they are either rotated (forming a meson, FTR+ BLD- or equivalent) or exchanged ((FTR,BLD) or equivalent, implying odd edge permutation parity). If ground facets appear on three corner cubies, the cubies must be a three-cycle of cubies on nonadjacent corners ((FTR,FDL,BTL) or equivalent). I've done some analysis by facets on these three cases, which is too messy to describe, but which leads me to the conclusion that the above position is the only isoglyph of its pattern, implying the conclusion that there is no chiral form. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 13:05:48 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA02085; Mon, 18 Aug 1997 13:05:43 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Sun Aug 17 00:42:09 1997 Date: Sun, 17 Aug 1997 00:38:39 -0400 (EDT) From: Jerry Bryan Subject: Calculating Local Maxima in Face Turn Metric To: Cube-Lovers Reply-To: Jerry Bryan Message-Id: While everybody else has been doing isoglyphs and minimal maneuvers for highly "symmetric" positions, I have been trying to figure out how to calculate local maxima for the face turn metric with my Shamir program. It's a bit trickier than with the quarter turn metric, but I think I have it. If anybody sees any holes in the following, please let me know. Recall that I have been creating Start-rooted search trees by forming products of the form z=xy for |x|=n and |y|=m, for 1<=m<=n. Prior to my Shamir program, I would fix n (it gets larger iteratively), and just let m=1 to advance one level of the tree at a time. But the Shamir method lends itself to jumping forward several levels at one fell swoop. If E(w) is the set of all moves with which a minimal maneuver for w can end, then E(z) is the union of E(y) over all the y which can be used to create z. We now introduce an alternative interpretation for E(w). E(w) is the set of all moves whose inverses carry w one move closer to Start. The alternative interpretation works for both quarter turns and face turns. My program is all set up to calculate E(w) for either the quarter turn metric or the face turn metric. The only difference is that the representation of E(w) for the quarter turn metric is a bit string of 12 bits and for the face turn metric is a bit string of 18 bits. But using E(w) to calculate local maxima for the face turn metric will yield only what we have agreed to call strong local maxima, namely those local maxima where every face turn moves one move closer to Start. We desire also to calculate weak local maxima, where one or more face turns may leave the distance from Start unchanged. To this end, we define E2(w) to be the set of all moves whose inverses leave w the same distance from Start. For quarter turns, the required initializations are E(q)={q} for all q in Q, the set of twelve quarter turns. E2(q) is of course null in all cases. For face turns, the required initializations are: E(q) = {q} for all q in Q E(q2) = {q2} for all q2 in H, the set of six half turns E2{q} = {q',q2} for all q in Q E2{q2} = {q,q'} for all q2 in H To be pedantically complete, we could define E3(w) to be the set of all moves whose inverse leaves w one move further from Start. Note that E(w), E2(w), and E3(w) are disjoint, and their union is Q for quarter turns and Q+H for face turns. For quarter turns, we have defined the maximality of w to be |E(w)|, wherein we have a local maximum if |E(w)|=12. The corresponding definition of maximality for face turns is an ordered pair (|E(w)|,|E2(w)|), where w is a local maximum if |E(w)|+|E2(w)|=18 and where w is a strong local maximum if |E(w)|=18. (A local maximum which is not a strong local maximum is a weak local maximum.) The only thing I am worried about is the following. Given the proposed initializations and calculations for E(w) and E2(w) for face turns, will E(w) and E2(w) be disjoint automagically, or is their disjointedness something which will have to be tested? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 13:49:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA02319; Mon, 18 Aug 1997 13:49:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sun Aug 17 03:43:15 1997 Message-Id: <33F6AA41.3C98@hrz1.hrz.th-darmstadt.de> Date: Sun, 17 Aug 1997 09:37:37 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs References: <199708170235.WAA07984@sun30.aic.nrl.navy.mil> [Mike Reid wrote:] > > ... > > i hadn't even considered chiral versus achiral isoglyphs. indeed, > > all the "continuous" isoglyphs given by herbert are chiral. > > achiral isoglyphs certainly exist, for example > > > D2 R2 U' B' L B U B L F2 R D' L2 U2 B2 D (22q, 16f) > > > of type 11; pattern > > > *.. > > *** > > *** > > > and others can be derived from this. i suspect that there is no > > chiral form of this isoglyph, but i'm not absolutely certain. > Dan Hoey wrote: >... > I've done some analysis by facets on these three cases, which is too > messy to describe, but which leads me to the conclusion that the above > position is the only isoglyph of its pattern, implying the conclusion > that there is no chiral form. There are two more isoglyphs of this pattern, B2 D . R D' B2 F' R2 U' B' R' U F R' (13) D2 R D2 L B2 R' B2 . D' R F D' L' F' D2 L' (15) Could someone tell me, what chiral and achiral exactly mean? --Herbert From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 14:10:15 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA02395; Mon, 18 Aug 1997 14:10:14 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Sun Aug 17 14:08:40 1997 Date: Sun, 17 Aug 1997 14:05:08 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Calculating Local Maxima in Face Turn Metric In-Reply-To: To: Cube-Lovers Message-Id: On Sun, 17 Aug 1997, Jerry Bryan wrote: > The only thing I am worried about is the following. Given the proposed > initializations and calculations for E(w) and E2(w) for face turns, will > E(w) and E2(w) be disjoint automagically, or is their disjointedness > something which will have to be tested? > Well, I see that I never actually gave the calculation of E2(w), just the initialization. But it works the same as for E(w), namely if z=xy, then E2(z) is the union of E2(y) over all the y which can be used to make minimal maneuvers for z. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990 From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 16:16:56 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA02836; Mon, 18 Aug 1997 16:16:56 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Aug 17 14:33:24 1997 Date: Sun, 17 Aug 1997 09:23:04 -0400 (EDT) From: Nicholas Bodley To: Dan Hoey Cc: Cube Mailing List Subject: Patterns on larger cubes (Was Re: isoglyphs) In-Reply-To: <199708170235.WAA07984@sun30.aic.nrl.navy.mil> Message-Id: Perhaps many List subscribers have realized that (unless I'm very confused!) while the current highly-evolved discussion of patterns (isoglyphs, etc.) pertains only to 3^3s, there must be enormous "worlds to conquer" when one considers [maneuvers] to create patterns on the 4^3 (Rubik's Revenge) and the 5^3. I understand little of the current discussion about isoglyphs (even if the term itself makes sense); nevertheless, it's delightful to see such discussions going on, and I have great respect for those who do understand and can contribute. (I wouldn't have it any other way!) It's perhaps of interest to consider a Theory of Mechanisms, in which it would be possible (eventually) to design an optimum set of innards for, say, a 5^3, or to rigorously prove that what exists is optimal. Connections with topology and kinematics would be not at all unexpected. Close connections with CAD would also make sense. My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. -------------------------------------------------------------------------- From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 16:49:39 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA02974; Mon, 18 Aug 1997 16:49:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sun Aug 17 22:56:57 1997 Message-Id: <33F7B8DA.4C23@hrz1.hrz.th-darmstadt.de> Date: Mon, 18 Aug 1997 04:52:11 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Cube Explorer 1.5 now available! I finally succeeded in updating my Cube Explorer program. You can download it from http://home.t-online.de/home/kociemba/cube.htm The current discussion in this mailing list concerning isoglyphs shows that is pretty error prone to classify these patterns by hand. The program is a good help here. --Herbert From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 17:45:09 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA03210; Mon, 18 Aug 1997 17:45:09 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From scotth@ichips.intel.com Mon Aug 18 17:36:26 1997 Message-Id: <199708182132.OAA19914@ichips.intel.com> To: Cube Mailing List Subject: d-dimensional cube mechanisms Date: Mon, 18 Aug 1997 14:32:59 -0700 From: Scott Huddleston Several years ago I worked out a solution to the d-cube 3^d, for d>3. This is most interesting combinatorially if you assume you're restricted to only rotating entire (d-1)-faces at a time, so that's what I assumed in my solution. But when I thought about building a mechanism for the d-cube, I came to the surprising (to me) conclusion that any natural extension of the 3^3 mechanism to d dimensions would allow you to rotate any 2-face. I concluded that any mechanism that would restrict you to only rotating entire (d-1)-faces would require some sort of complex interlocking mechanism that would have to engage and disengage whenever a (d-1)-face was to be rotated. Has anyone else thought about this problem (d-cube mechanisms) enough to confirm or refute my conclusions? Best, - Scott Huddleston scotth@ichips.intel.com From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 18 18:37:28 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA03449; Mon, 18 Aug 1997 18:37:28 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Mon Aug 18 18:16:31 1997 Date: Mon, 18 Aug 1997 18:16:18 -0400 Message-Id: <199708182216.SAA00604@sun30.aic.nrl.navy.mil> From: Dan Hoey To: kociemba@hrz1.hrz.th-darmstadt.de Cc: cube-lovers@ai.mit.edu In-Reply-To: <33F6AA41.3C98@hrz1.hrz.th-darmstadt.de> (message from Herbert Kociemba on Sun, 17 Aug 1997 09:37:37 +0200) Subject: Re: isoglyphs Herbert Kociemba asks: > Could someone tell me, what chiral and achiral exactly mean? "Chirality" is Lord Kelvin's word for "handedness" as in "appearing in two mirror-image varieties." A "chiral isoglyph" is one in which the handedness of the glyph is taken into account in testing for isoglyphy,* so that the glyph appears only in one variety. Neither Mike's original isoglyph nor the *.. ..* two you found are chiral isoglyphs--they all have both *** and *** . *** *** Mike used "achiral" for an isoglyph that fails to be a chiral isoglyph, though I would tend to use "non-chiral". I would rather use "achiral" for a situation that lacked chirality, as in an isoglyph of a mirror-symmetric glyph. [* Thanks to Allan Wechsler for inventing the word "isoglyphy". His alternate term, "isoglyphism", is still looking for a good use. ] Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 19 10:53:09 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA06719; Tue, 19 Aug 1997 10:53:09 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Tue Aug 19 01:54:06 1997 Message-Id: <199708190550.BAA21896@life.ai.mit.edu> Date: Tue, 19 Aug 1997 01:55:51 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs herbert writes > With the pattern generator it's indeed very easy to find the isoglyphs. i'm still unclear about what your pattern generator does. could you describe what it does, for the benefit of those who haven't seen your program? > By the way, Mike, it would > be nice to complete the chapter "continous isoglyphs" by computing the > shortest generators for them with your program. i will do this soon. right now the program is busy with T-symmetric positions. after that (or if there's a break) i'll give it the continuous isoglyphs to think about. there's one last pattern for which i could not find any isoglyph. it's the 32 pattern of type ..* *.. *** all others, except those previously mentioned as impossible (patterns of corner type D, and the 21 and 23 types which we previously discussed) have isoglyphs. can your program find isoglyphs of this type, or show that none exist? mike From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 19 13:29:02 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA07342; Tue, 19 Aug 1997 13:29:01 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Tue Aug 19 13:17:24 1997 Date: Tue, 19 Aug 1997 13:13:34 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: isoglyphs In-Reply-To: <199708190550.BAA21896@life.ai.mit.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Tue, 19 Aug 1997, michael reid wrote: > > > With the pattern generator it's indeed very easy to find the isoglyphs. > > i'm still unclear about what your pattern generator does. could you > describe what it does, for the benefit of those who haven't seen your > program? > I'll take a crack at this one. (The program is great, by the way.) The basic mode of the pattern generator allows you to specify a pattern for one of the 3x3 faces of the cube, and the program finds all the positions (unique up to symmetry) where each of the six 3x3 faces has this same pattern. It doesn't really matter which colors you specify in your one face, since you are really only specifying a pattern. For example, I have played with corner facelets and center facelet all one color and edge facelets all another color, or center facelet one color and all the edge and corner facelets another color (yields the 6-spot), etc. The patterns I have played with have very few (or sometimes, no) solutions. I don't know what happens if you choose a pattern with many, many solutions (maybe there really aren't all that many such positions, given that all six 3x3 faces have to have the same pattern). There is an expanded mode which I haven't played with much yet where you can give up to four 3x3 patterns. Each of the six faces on the cube then has to have a pattern that matches any one of the (up to) four which you specified. The so-called pattern editor I have described seems to operate essentially instantaneously. But having generated the position, you can then ask the program to find a near-optimal solution using the Kociemba algorithm. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 Wed Aug 20 08:33:01 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id IAA11354; Wed, 20 Aug 1997 08:33:00 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Wed Aug 20 00:53:03 1997 Message-Id: <33FA7810.3446@idirect.com> Date: Wed, 20 Aug 1997 00:52:32 -0400 From: Mark Longridge To: cube lovers Subject: Mike Reid's Cube Program I've updated my web page to include Mike Reid's cube program. I've also updated my own cube program to save arrangements the way Mike's program requires. The MS-DOS source and executables are available at: http://web.idirect.com/~cubeman/rubik.zip http://web.idirect.com/~cubeman/miker.zip I guess cubing is back in fashion. P.S. To Herbert Kociemba, your 1.5 version is great! From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 09:33:07 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id JAA11544; Wed, 20 Aug 1997 09:33:07 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From CFolkerts@compuserve.com Wed Aug 20 03:55:06 1997 Date: Wed, 20 Aug 1997 03:50:41 -0400 From: Corey Folkerts Subject: 5x5x5 Solution To: Cube-Lovers Message-Id: <199708200350_MC2-1DA6-9307@compuserve.com> I recently got my hands on a 5x5x5 from Dr. Christoph Bandelow, however, I'm am at an almost complete loss as to how to solve it. I think that if I ignore the center row in all axes then it is pretty much a 4x4x4, but if I then treat it as a 4x4x4 the center axes get all screwed up. I would really like to know if there is a site that has a description of the solution to a 5x5x5 or if someone could describe it to me in a message. Thanks in advance. Corey Folkerts From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 13:02:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id NAA12498; Wed, 20 Aug 1997 13:02:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Wed Aug 20 12:36:22 1997 Message-Id: <199708201632.RAA08938@mail.iol.ie> From: "David Byrden" To: " From: Corey Folkerts > I recently got my hands on a 5x5x5 from Dr. Christoph Bandelow, > however, I'm am at an almost complete loss as to how to solve it. I just extended the technique that had worked for me on the smaller cubes. Solve the corners, then solve the inner edges, then eventually the faces in the interior of the sides. By choosing these 5 subsets of the faces, which of course do not exchange faces with each other, you break the cube into a sequence of 5 smaller problems. Working inwards from the outermost faces is best because you can easily find operators (combinations of moves) that affect the inner faces in some way but preserve the outer ones that you have already solved. David From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 14:23:23 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA12867; Wed, 20 Aug 1997 14:23:22 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Goyra@iol.ie Wed Aug 20 12:36:22 1997 Message-Id: From: "David Byrden" To: Cube-Lovers@ai.mit.edu Subject: Re: 5x5x5 Solution Date: Wed, 20 Aug 1997 17:28:52 +0100 > From: Corey Folkerts > I recently got my hands on a 5x5x5 from Dr. Christoph Bandelow, > however, I'm am at an almost complete loss as to how to solve it. I just extended the technique that had worked for me on the smaller cubes. Solve the corners, then solve the inner edges, then eventually the faces in the interior of the sides. By choosing these 5 subsets of the faces, which of course do not exchange faces with each other, you break the cube into a sequence of 5 smaller problems. Working inwards from the outermost faces is best because you can easily find operators (combinations of moves) that affect the inner faces in some way but preserve the outer ones that you have already solved. David [ Moderator's note-- The previous copy of this message had bad headers. Sorry. --Dan] From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 15:06:25 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA13572; Wed, 20 Aug 1997 15:06:24 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Wed Aug 20 14:21:03 1997 Message-Id: <33FB355F.3A5D@idirect.com> Date: Wed, 20 Aug 1997 14:20:15 -0400 From: Mark Longridge To: cube lovers Subject: Corrections Oops.. those were the wrong URLs Here are the correct ones: http://web.idirect.com/~cubeman/rubik/rubik.zip http://web.idirect.com/~cubeman/rubik/miker.zip Or just go to http://web.idirect.com/~cubeman and click on the appropriate link. -> Mark <- From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 16:12:54 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA13881; Wed, 20 Aug 1997 16:12:53 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Wed Aug 20 14:55:47 1997 Message-Id: <199708201852.OAA00834@life.ai.mit.edu> Date: Wed, 20 Aug 1997 14:57:28 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: Mike Reid's Cube Program mark longridge writes > I've updated my web page to include Mike Reid's cube program. this is my sub-optimal solver, i.e. kociemba's algorithm, which can handle either quarter turns or face turns. (it's not yet in its final form, but that may take a while.) to those who are interested in my optimal solver, it will be available soon. it requires about 85 megabytes, so your computer will need at least that much RAM. mike From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 17:27:39 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA14857; Wed, 20 Aug 1997 17:27:39 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Wed Aug 20 17:22:17 1997 Date: Wed, 20 Aug 1997 17:22:07 -0400 Message-Id: <199708202122.RAA08061@sun30.aic.nrl.navy.mil> From: Dan Hoey To: scotth@ichips.intel.com Cc: Cube-Lovers@ai.mit.edu In-Reply-To: <199708182132.OAA19914@ichips.intel.com> Subject: Re: d-dimensional cube mechanisms Scott Huddleston writes: > Several years ago I worked out a solution to the d-cube 3^d, for d>3. > This is most interesting combinatorially if you assume you're > restricted to only rotating entire (d-1)-faces at a time, so that's > what I assumed in my solution. I thought that was most natural, but there are certainly at least two others. In my article of 22 Dec 1993 in ftp://ftp.ai.mit.edu/pub/cube-lovers/cube-mail-11 I mentioned how to show they are different, at least in the 3^4 hypercube. I didn't go into enough detail to determine "most interesting"--can you expand on what you mean? By the way, do you know if this is the Kamack and Keane model? I've never seen their paper, and I never figured out how to read Charlie Dickman's document.) > But when I thought about building a mechanism for the d-cube, I > came to the surprising (to me) conclusion that any natural extension > of the 3^3 mechanism to d dimensions would allow you to rotate any > 2-face. Wow! I can verify it for the most obvious natural extension I can describe. We could form the 3^d Rubik's ball by taking the unit ball B_d={z in R^d : |z|<=1} and some sufficiently small constant c, and cutting the ball with the hyperplanes P_i,s = {z in B_d : z_i = s c} for each index i in {1,...,d}, for each sign s in {-1,1}. We arrange that all rotations are allowed that keep these pieces inside the unit ball (perhaps enforcing this by attaching extensions in the shape of a cube). It's then easy to see that a representative 2-face {z in B_d : z_3 > c, z_4 > c, ..., z_d > c} can be rotated by mapping (z_1,z_2,z_3,...,zn) -> (z_1 cos th + z_2 sin th, z_1 sin th + z_2 cos th, z_3, z_4, ..., z_d). That is a very surprising observation to me, too! Hands up anyone who _isn't_ surprised!! In fact, I think it may be worse (geometrically, if not combinatorically). For instance, I expected that a cubical hyperface of the 3^4 could only be turned by rotating the cube about an orthogonal axis. But it looks to me like you could rotate the cube around any axis you like. Maybe you have to eventually rotate it so it occupies it's original space before you rotate a perpendicular hyperface, but I'm still somewhat annoyed that you can put it in weird orientations in between. > I concluded that any mechanism that would restrict you to > only rotating entire (d-1)-faces would require some sort of complex > interlocking mechanism that would have to engage and disengage > whenever a (d-1)-face was to be rotated. > Has anyone else thought about this problem (d-cube mechanisms) > enough to confirm or refute my conclusions? What I haven't proven is that say, any decomposition of the ball into 3^d pieces that admit (d-1)-face rotations will also admit 2-face rotations. If you've got that, it would pretty much support your conclusion, but I don't know how to verify it off-hand. Dan Hoey Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 19:26:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id TAA15689; Wed, 20 Aug 1997 19:26:36 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Wed Aug 20 19:12:10 1997 Message-Id: <33FB67F0.3D1C@hrz1.hrz.th-darmstadt.de> Date: Wed, 20 Aug 1997 23:56:00 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs References: <199708190550.BAA21896@life.ai.mit.edu> michael reid wrote: > i'm still unclear about what your pattern generator does. could you > describe what it does, for the benefit of those who haven't seen your > program? Though I used the word "pattern generator" myself I would like to ban it now, because the word generator is already uses for maneuvers (solvers versus generators). Let's talk about "pattern search". The pattern search is implemented in principle in the same way as you could try to built a pattern manually: First you "remove" all cubies, then again you add one after the other to the next free position and check if there is any contradiction with the pattern(s) in the Pattern Editor. If yes, the cubie is removed and added again in a different orientation or location. This is done recursivly, until al positions are filled. If there is no bug in the code, the pattern search should find *all* cubes which are possible with the patterns given in the Pattern Editor. > > there's one last pattern for which i could not find any isoglyph. > it's the 32 pattern of type > > ..* > *.. > *** > > all others, except those previously mentioned as impossible (patterns of > corner type D, and the 21 and 23 types which we previously discussed) > have isoglyphs. can your program find isoglyphs of this type, or show > that none exist? My program finds no solution, so there also should not exist any. --Herbert From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 20 21:46:19 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA16469; Wed, 20 Aug 1997 21:46:18 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From tim@mail.htp.com Wed Aug 20 21:13:21 1997 From: tim@mail.htp.com (Tim Mirabile) To: Cube-Lovers@ai.mit.edu Subject: Re: 5x5x5 Solution Date: Thu, 21 Aug 1997 01:09:43 GMT Organization: http://www.webcom.com/timm/ Message-Id: <33fc8852.2330703@mail.htp.com> References: In-Reply-To: The method I use is inefficient I'm sure, but I was able to solve the 5x5x5 right away only learning one specialized move. First I solve the 3x3x3 centers on all six sides using mostly intuitive methods, with an occasional 3x3x3 move (turning the two outer slices together on each twist to simulate a 3x3x3) thrown in. I usually work on the "corner centers" first, then the "edge centers", just like I would on a 3x3x3. Then I solve the "outer" corners (holding the three center slices together), followed by the "center" edge pieces (holding the two outer slices together) also using purely 3x3x3 methods. Then I work on the "off center" edge pieces using moves like r1 U2 r3 U2 or l3 U2 l1 U2. This messes up a row of center pieces but if you do it three times total they are restored, and 5 of the off center edges are permuted. I usually improvise by making a twist or two to get the edges I want to permute in the right place, followed by reversing these afterward. I also work with the slightly messed up centers at times making sure that I restore them later as I permute other sets of edges. Finally I use one 4x4x4 specific move from Mark Longridge's page at http://web.idirect.com/~cubeman/revenge.txt which also explains the extended notation. (http://web.idirect.com/~cubeman/ is his main page of course). The move I use is this: p3: Flip UF edge pair: r2 (D2 l1)^2 D1 l3 r3 d2 l1 r1 D3 l3 r3 d2 B2 r1 B2 l3 B2 l1 B2 r2 On the 5x5x5, this not only flips (and swaps) the edges, but swaps two center pieces. But if you hold the edges you want to flip at UB instead of UF, and do (p3) U2 (p3) U2 (p3), the edges will be flipped and the centers restored. You can also improvise here if you need to flip more than one pair - messing up centers with the first flip and restoring them with the second. Since these edges here are swapped as well as flipped, you can also use these moves to swap a single pair of edges. -- Webmaster, tech support - ICD/Your Move Chess & Games: http://www.icdchess.com/ The opinions of my employers are not necessarily mine, and vice versa. From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 21 10:17:18 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA18772; Thu, 21 Aug 1997 10:17:17 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 21 00:10:16 1997 Message-Id: <199708210406.AAA21524@life.ai.mit.edu> Date: Thu, 21 Aug 1997 00:11:54 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs thanks for the description of your program, herbert. i was not so much concerned with the algorithm the pattern finder uses (although this is nice to know) as i was with how the user may specify patterns, etc. jerry bryan gave a good description of this. here are several other applications that your pattern finder should be able to handle easily: 1. classify all "snakes". you've already done the part where they consist only of faces of type 42; there's also face type 4D to consider. 2. confirm my results on the "czech check" problem, i.e. classify all patterns that have exactly eight squares correct on each face. 3. find all "partial isoglyphs". and they are certainly many other applications. these three come to mind immediately. it looks like i'll have to find a windows machine to try out your program; it sure sounds excellent. mike From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 21 12:54:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA19523; Thu, 21 Aug 1997 12:54:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 21 00:45:19 1997 Message-Id: <199708210441.AAA22489@life.ai.mit.edu> Date: Thu, 21 Aug 1997 00:47:02 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: partial isoglyphs 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 6 solid faces: start 5 solid faces: no possibilities 4 solid faces: other two faces opposite: types 02, 0D and 04 are possible other two faces adjacent: type 0D is possible 3 solid faces: mutually adjacent: type 02 is possible not mutually adjacent: types 01 and 0D are possible 2 solid faces: adjacent: types 01, 02, 0D and 03 are possible opposite: many possible types 1 solid face: types 01, 02 and 0D are possible but it seems like herbert's cube explorer program can settle this matter completely. maybe he, or someone else wants to investigate this. mike From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 21 14:21:46 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id OAA20968; Thu, 21 Aug 1997 14:21:45 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Thu Aug 21 12:51:15 1997 Date: Thu, 21 Aug 1997 12:20:58 -0400 Message-Id: <199708211620.MAA00539@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Goyra@iol.ie Cc: cube-lovers@ai.mit.edu Subject: Re: Megaminx a.k.a. Supernova In-Reply-To: <199708051604.MAA13056@sun30.aic.nrl.navy.mil> Glyph-lovers may recall I was led to that discussion by analyzing the five conjucacy classes of spot patterns on the Megaminx. I called them 0. The identity, 1. The 72-degree twelve-spot, 2. The 144-degree twelve-spot, 3. The 120-degree ten-spot, 4. The 180-degree ten-spot. Thanks to David Singmaster for noticing this is wrong. It should have been 0. The identity, 1. The 72-degree ten-spot, 2. The 144-degree ten-spot, 3. The 120-degree twelve-spot, 4. The 180-degree twelve-spot. I'm glad to see someone's paying attention around here. Dan Hoey@AIC.NRL.Navy.Mil From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 21 16:12:37 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA21460; Thu, 21 Aug 1997 16:12:35 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Thu Aug 21 16:11:04 1997 Date: Thu, 21 Aug 1997 16:07:30 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: isoglyphs In-Reply-To: <199708210406.AAA21524@life.ai.mit.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Thu, 21 Aug 1997, michael reid wrote: > here are several other applications that your pattern finder should > be able to handle easily: > > 2. confirm my results on the "czech check" problem, i.e. classify all > patterns that have exactly eight squares correct on each face. If I understand your question correctly, and if I am using Herbert's program correctly, there are 54 such isoglyphs unique up to M-conjugacy. 3 of them involve only corners as the incorrect facelet, and 51 of them involve only edges as the incorrect facelet. (I am assuming that by definition the center facelet is always correct, thus eliminating the 6-spot from consideration. If you count the 6-spot, then there are of course 2 such isoglyphs unique up to M-conjugacy.) If you tell Herbert's program to consider only continuous isoglyphs with exactly eight squares correct on each face, there are 3 such isoglyphs unique up to M-conjugacy. 1 of them involves only corners as the incorrect facelet, and 2 of them involve only edges as the incorrect facelet. (I suppose you would say that the other 51 are discontinuous.) Herbert's program only lists positions which are unique up to M-conjugacy. Here's a modest suggestion. It might be nice for the program to list the size of the M-conjugacy class for each such position. That way, you could count both "positions" and "patterns", where I am using "position" to mean any element of G and "pattern" to mean an M-conjugacy class (or a representative thereof). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 Aug 21 18:46:52 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA22876; Thu, 21 Aug 1997 18:46:51 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Thu Aug 21 18:12:32 1997 Message-Id: <33FCB0F1.3E08@hrz1.hrz.th-darmstadt.de> Date: Thu, 21 Aug 1997 23:19:45 +0200 From: Herbert Kociemba To: cube-lovers@ai.mit.edu Subject: Re: isoglyphs michael reid wrote: > here are several other applications that your pattern finder should > be able to handle easily: > > 1. classify all "snakes". you've already done the part where they > consist only of faces of type 42; there's also face type 4D to > consider. There are 57 different patterns, which have face type 42 or 4D. I don't think I should list generators for them here. The pattern-computation with my program took only about 3 minutes on a PC with 486 processor, so anybody who wants could repeat the computation and then create generators for the patterns he/she(?) is interested in. > 2. confirm my results on the "czech check" problem, i.e. classify all > patterns that have exactly eight squares correct on each face. Cube Explorer finds 56 patterns, which confirms your result. > and they are certainly many other applications. these three come > to mind immediately. it looks like i'll have to find a windows machine > to try out your program; it sure sounds excellent. And I thought, window machines are very popular.... --Herbert From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 22 21:50:40 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA04121; Fri, 22 Aug 1997 21:50:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Fri Aug 22 21:15:50 1997 Message-Id: <199708222345.TAA12112@life.ai.mit.edu> Date: Fri, 22 Aug 1997 19:51:01 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for T-symmetric positions dan hoey denotes by T a subgroup of size 12 of the symmetries of the cube that preserves one of the long diagonals. there are 4 conjugates of T , each preserves a different long diagonal. i'll choose the UFR - DLB diagonal. (note that "preserving" allows the diagonal to be reversed in direction.) in "symmetry and local maxima," hoey and saxe classify all cube positions with T-symmetry. for each choice of a T subgroup, there are 16 such positions. they form a commutative subgroup of the cube group of type 2, 2, 2, 2. (this means that it is isomorphic to a product C_2 x C_2 x C_2 x C_2 of cyclic groups of order 2.) we may take generators of these cyclic factors to be superflip, pons asinorum, and the two positions (UB+) (UL+) (FL+) (FD+) (RD+) (RB+) (number 1 below) and (DRF, UBL) (FLD, BUR) (LUF, RBD) (UB, DF) (UL, DR) (FL, BR) (number 5 below). 4 of these positions have more symmetry, namely the subgroup generated by superflip and pons asinorum. for the other 12 positions, minimal maneuvers are given below. i've also given a maneuver that is minimal in both metrics, whenever such a maneuver exists. 1. B U L' F' U R U2 D2 F' L U' B' L D R2 L2 B2 (22q, 17f) 2. F U D' R2 U2 R' B' U' F R' D R L' F U' F U' R' (20q, 18f) 3. U R U' F D R L' B' L' F R F B' U' L' D B' D' (18q, 18f) 4. D' R' U B' D' R' L F L B' R' F B' U L D' F U' D2 (20q, 19f) 5. D' B' D' R' B L B U' B U R D R L D R' L2 D (19q) B2 L U' L D R' L' D2 R U L' B2 U R2 U2 F2 U (17f) 6. U L U D F B' U' D L2 F U D B' R L B' U' D' F U (21q) D' L F' B' L F2 B2 U R L' U D' L F' R2 L2 F2 U' D2 (19f) 7. F U D L2 F2 L2 U' F B D F' B' D' F' (17q, 14f) 8. U F B' L U F B' L D F D' R2 L F U' B' L F2 R (21q) U B R' F2 U' D' L2 D2 R2 B2 L' F' B R2 F2 R' D F (18f) 9. U B U2 L F' B2 U' F' B L U' B' D' F R U B R L' (21q) U F2 D B' U' B2 R B2 D' F2 U' D2 B2 L' U2 B D2 (17f) 10. U F B D' L' U' B' L' F R' L' D L U F U B D B R' D' (21q) U F U D' B' D2 R U D R2 D2 B R L2 F2 B' D R2 L' (19f) 11. U R F' B D B' U' F B' D' F R L' D2 R D' B' U' D F' (21q) D' F2 U2 B2 R F' L U' F2 B R' F' D L2 D R2 F2 U' F2 (19f) 12. U B' D' R F' R' F' R2 B' D F' B2 D B' L' U F B R' (21q) U F2 B R2 F2 B' U L U B' U2 R2 L' U B U' L' U' (18f) mike From cube-lovers-errors@mc.lcs.mit.edu Mon Aug 25 11:47:21 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA18038; Mon, 25 Aug 1997 11:47:21 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Wed Aug 20 12:41:14 1997 Date: Wed, 20 Aug 1997 12:37:38 -0400 (EDT) From: Jerry Bryan Subject: Open and Closed Subgroups of G To: Cube-Lovers Message-Id: I am going to use the terms open and closed with a certain trepidation. These terms may already have a conventional meaning in group theory of which I am not aware. If so, my apologies in advance. Anyway,... The Hofstadter articles about the cube in Scientific American many years ago made one point very strongly. Namely, that in solving the cube by hand you often temporarily have to give up hard won progress. For example, suppose you solve the cube bottom layer first, middle layer (the middle between the top and bottom) second, and the top layer last. (This isn't the way I do it, and I doubt that anybody does it in quite this way, but it makes for a good example.) Using this method, it is almost certain that you will have to disturb the bottom layer to solve the middle layer, and it is almost certain that you will have disturb the bottom and middle layers to solve the top layer. (It's an interesting exercise to characterize those few positions where you wouldn't have to disturb previously solved layers to complete your task.) The set of positions where the bottom layer is fixed constitute a group, as do those positions where the two bottom layers are fixed. Hence, the series of plateaus involved in this particular solution define a sequence of nested subgroups. The Thistlethwaite algorithm reverses the roles of the solution algorithm and the sequence of nested subgroups. Instead of a solution algorithm defining a sequence of nested subgroups, a sequence of nested subgroups defines a solution algorithm. In some ways, I think this is a distinction without a difference; it is more like two sides of the same coin. However, I think there is one really fundamental difference with the Thistlethwaite algorithm. Namely, you never have to give up any of your "hard won progress". That is, after you make it to a particular subgroup in the nested sequence, no subsequent move takes you out of that subgroup. So let's define any subgroup of G having this property as closed, and any subgroup which is not closed as open. To be a little more specific, a closed subgroup is a subgroup such that for every position in the subgroup, there is a maneuver back to Start which never leaves the subgroup. I am a quarter-turner, but I think that perhaps closed subgroups are an argument which has not yet been specifically articulated for counting half-turns as one move. That is, many of the subgroups which are used in Thistlethwaite and Thistlethwaite-like algorithms forbid quarter-turns along one or more axes at some point in the process. Restricting quarter-turns assists your sequence of nested subgroups all to be closed subgroups. It seems to me that such an approach leads very naturally to counting half-turns as one move. Finally, when I first read about Thistlethwaite's algorithm, I naively assumed that the algorithm's maneuvers from the next to last subgroup in the sequence to the last subgroup in the sequence (namely to Start itself) were minimal in G. This is clearly not true in general. The first example I remember where this began becoming clear to me was the group. We could solve a cube as G -> -> I, although the jump from G to is quite a big jump. But a minimal maneuver in is certainly not necessarily minimal in G. Let's call a subgroup H of G a closed minimal subgroup if every minimal maneuver in H is also minimal in G. is closed, but it is not closed minimal. I have tried to think about which subgroups of G are clearly closed minimal. I can't think of many. In fact the only examples I can think of are subgroups generated as , where s is a syllable or syllables along the same axis. So , , , , and and their conjugates are closed minimal subgroups. (Well, I and G are closed minimal, but it hardly seems fair to count them.) This whole area was discussed rather thoroughly in the recent spate of messages about Korf's paper and Kociemba's algorithm, but without using the terms closed or closed minimal. I do not believe any of the subgroups which were discussed were claimed to be closed minimal, although I think essentially all of them were closed. The most interesting subgroup which was discussed was Mike Reid's co-called T, which is the intersection of with its conjugates. I wonder if Mike's T subgroup is closed minimal? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 [Moderator's note: This message has been delayed by a clerical error on my part.--Dan] From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 26 12:05:11 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA24505; Tue, 26 Aug 1997 12:05:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cubeman@idirect.com Tue Aug 26 01:20:45 1997 Message-Id: <3402678A.7E62@idirect.com> Date: Tue, 26 Aug 1997 01:20:10 -0400 From: Mark Longridge To: cube lovers Subject: Old URL no longer works! Sorry to trouble everyone with this... I just discovered my defaults on my page are different. That is, the URL http://web.idirect.com/~cubeman no longer works! Use http://web.idirect.com/~cubeman/index.html instead! More interesting stuff to follow... From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 26 21:10:39 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA26778; Tue, 26 Aug 1997 21:10:38 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Tue Aug 26 16:33:22 1997 Date: Tue, 26 Aug 1997 16:29:44 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Open and Closed Subgroups of G In-Reply-To: <199708252135.RAA25741@cyber1.servtech.com> To: cube-lovers@ai.mit.edu Reply-To: Jerry Bryan Message-Id: On Mon, 25 Aug 1997, christopher f. chiesa wrote: > > Jerry, judging from the fact that you were ABLE to go into a lot of > details, it strikes (humble, little ol', un-group-theory-educated) me that > perhaps you "read too much into" Hofstadter's remark about "giving up > hard-earned progress" -- either that, or _I_ have for years been reading > TOO LITTLE into the remark! > > To wit, Jerry, although I can't really follow all the stuff about groups > and subgroups and closure and all that, I think it suffices at the > layman's level to say that "you certainly make it SOUND" as though there > are a multitude of non-trivial cases in which one DOESN'T "have to give up > hard-earned progress" en route to a solution of the (3^3) Cube. > > Me, I don't get it. Seems to me that Hofstadter was speaking from > the simple perspective of the uneducated layman who, by hook or by > crook, manages after hours/days/weeks of effort to solve, say, the > TOP LAYER of the cube. To the layman, at this point, ANY turn of ANY face > other than U itself, or D which does not directly interact with it, is > going to constitute "giving up hard-earned progress!" Now, maybe, > from the perspective of all that math I can't follow, a simple "quarter > turn of any face other than U or D" at this point does NOT constitute > "giving up progress," but I assure you, the poor fellow who's just solved > the top layer for the first time is going to have a HEART ATTACK if you so > much as walk up and give the R face a quarter turn. And I'm pretty sure > it was from THAT simplistic perspective that Hofstadter was speaking. > > Does this conflict with YOUR interpretation, Jerry? I'd hate to think I > blew all this hot air into the mailinglist over NOTHING. :-) > > Is there a layman-comprehensible description of "the Thistlethwaite > algorithm" available anywhere (preferably for free, preferably online)? > I've been hearing about it for years but have never seen any details. > I'll take a crack at a number of your questions. There are probably a lot of people on the list who are not conversant with group theory, but I'll bet essentially everybody knows at least a little bit about basic set theory. I'm old enough that when I was taking algebra in high school in the early 60's, "new math" was all the rage. It turns out that "new math" was really just set theory. Many traditionalists were aghast that this wierd "new math" stuff was being taught. Never mind that it was hundreds of years old. And never mind that set theory is the foundation of nearly all modern mathematics. For example, most formal treatments of the concept of a function view a function as a set of ordered pairs (which has to satisfy certain rules). Anyway, sets are just collections of objects where the basic rule is that for any object you can unequivically determine that the object either is or is not in the set. We might write a set as something like {a,b,c} where the braces denote the set and the elements a, b, and c are listed within the braces. We often give names to sets, as in A={a,b,c}. And finally, we have subsets, where for example {a,b}, {a,c}, {a}, etc. are subsets of {a,b,c}. Subsets can have names as well. For example, we might say B={a,b} and then we would say that B is a subset of A. If you know what sets are, then it's easy to talk about groups. Oversimplifying slightly, a group is just a set, an operation on that set, and a short list of rules. As a simple example, the real numbers and addition form a group. The real numbers are the set and addition is the operation. Exactly what the short list of rules is does not matter for now, but be assured that real numbers and addition do comply with the required rules for a group. Just as there are subsets of sets, there are subgroups of groups. For example, the rational numbers form a subset of the real numbers. Similarly, the rational numbers and addition form a subgroup of real numbers and addition. The integers form a subset of the rational numbers. The integers and addition form a subgroup of the rational numbers and addition. It is very common to be a little sloppy and simply identify a group as being the set if the operation is well understood. So we might say that the integers form a group if it is well understood that we are talking about addition as being the group operation. So I will be a little sloppy myself to make my sentences a little shorter. Not every subset is a subgroup. For example, the set of even integers is a subgroup of the integers, but the set of odd integers is not a subgroup of the integers. If you add two even integers together, the result is an even integer. But if you add two odd integers together, the result is not an odd integer. One of the group rules is that if you combine two elements from the set together, then the result must also be in the set or else you don't have a group. In the case of the cube, the set is the collection of all positions which can be reached by scrambling the cube in all possible ways, and the operation is "followed by". For real numbers and addition, we might write x+y to indicate adding x and y together. For cube positions, we might write XY to mean "X followed by Y". Even though it is not multiplication in the every day sense of real numbers, XY is often called a product and the "followed by" operation is often called multiplication. Basic operations on the cube consist of twisting one face. These operations are called F, B, U, D, L, and R if you twist the Front, Back, Up, Down, Left, and Right faces clockwise by 90 degrees. The respective counterclockwise twists are called F', B', U', D', L', and R'. The respective 180 twists are called F2, B2, U2, D2, L2, and R2. For 180 degree twists, it doesn't matter whether your twist is clockwise or counterclockwise. Notice, for example, that FF=F2. That is "F followed by F" is the same thing as turning the Front face by 180 degrees. Also, F'F'=FF=F2, etc., and FFF=F' (90+90+90 degrees clockwise is the same thing as 90 degrees counterclockwise), etc. There are many ways to define a group or a subgroup. One of the more common ways is in terms of generators. The generator notation is , where S is some set or list of elements. For example, with the integers and addition, we might define a subgroup as <3>. This means {3, 3+3, 3+3+3, ...} so <3> is the group of all integers which are divisible by 3. One of the rules for groups is that every element in the set must have an opposite, usually called an inverse. For example, with integers and addition the opposite of 3 is -3, and the opposite of -3 is 3. The generator notation automatically includes inverses. So if we write <3> for integers and addition, it is the same as writing <3,-3>. So we could write <3> or <-3> or <3,-3> and it would all mean the same thing, namely {..., -6, -3, 0, 3, 6, ...}. (To simplify things, I lied slightly in the previous paragraph when I left out the negative numbers. Groups require inverses, and with addition the way you get inverses is to include the negative numbers.) With the cube, the way you get inverses is that F' is the inverse of F and F is the inverse of F', etc. F2 is its own inverse, so we would write (F2)'=F2. Given all that, the way we would write generators for the cube group would be as . Remember that we do not have to include F', B', etc. because they are included automatically. On the other hand, if you are left handed you might want to write the generators as and you would not have to include F, B, etc. because they would be included automatically. Also, you do not have to include F2, B2, etc. because we can get F2 as FF, we can get B2 as BB, etc. The notation essentially says the following. Beginning with a cube which is solved (which is at Start), we get the cube group by combining together the F, B, U, D, L, and R operations in all possible ways. This is just another way of saying that we would scramble the cube in all possible ways by turning all six of the faces in all possible ways. We now have enough definitions and notations in place to start talking about Thistlethwaite's algorithm. Consider what it means to say . This means that you can twist the Up face any way you want, but you can't twist any of the other faces. This also means that the group is {U,UU,UUU,UUUU}. If you are new at this, you ought to have a few questions. For example, where is U'? Well, U' is the same thing as UUU, so U' is included (270 clockwise is the same thing as 90 degrees counterclockwise). Where is U2? Well, U2 is the same thing as UU, so U2 is included. What about UUUUU and UUUUUU etc.? Well, UUUU is the same thing as not twisting at all (360 degrees clockwise is the same thing as not twisting), so UUUUU=U, UUUUUU=UU, etc. No matter how you twist, as long as you confine yourself to the Up face, there are only four possible positions. UUUU is normally written as I (for the identity). Every group must have an identity. For addition, the identity is zero. For the cube, we normally just write I. It should be obvious, for example, that UU'=I and that U'U=I. That is, if you twist the Up face 90 degrees clockwise and immediately twist the Up face 90 degrees counterclockwise, you are back where you started. Now, let's go back to the idea of solving the bottom two layers of the cube first, then solving the Up layer. It is very likely that after solving the bottom two layers, the Up layer would look very scrambled. But we might get very lucky and discover that Up layer was already in . That is, it might already be in one of the four positions, U, UU, U'=UUU, or I. If the Up layer were already at I by accident, then the whole cube would already be solved. If it were in one of the other three positions, then we could finish solving the cube by simply twisting the Up face and there would be no need to disturb either of the bottom two layers. The Thistlethwaite algorithm accomplishes the same sort of thing, except that it is by design rather than by luck. We have already considered the group where you scramble the cube in any way you want using any twist of any face. Now, consider the group . What this means is that starting with a solved cube, we scramble it any way we want by making any twists we want of the U, D, L, and R faces, but for the F and B faces we can only make 180 degree twists. It turns out that by so doing, we cannot reach as many positions as we can if we allow 90 degree twists of all six faces. Hence, we would say that is a subgroup of . A key point of the subgroup is that if we can create a position in it by using only the indicated moves, than we can also reverse the process and solve any position in it by using only the indicated moves. A position in the subgroup is "somewhat solved" in much the same sense that a cube with the bottom layer solved is "somewhat solved", but a position in still looks pretty scrambled. There is some disagreement among Cube-Lovers as to whether you can look at a scrambled cube and determine easily whether it is in or not. I will leave that question unaddressed for the purposes of this note. The real point is that suppose that you were in and by some clever strategy or other managed to get your cube into . You would have made some hard won progress. Furthermore, you could solve the cube without giving up any of your hard won progress because you could solve the cube without making any more 90 degree F and B moves. This is very much unlike the situation of solving by layer, where inevitably you must give up some hard won progress. It was thinking along these lines that led me to think in terms of closed and open groups, namely those where you can or cannot proceed without giving up any of your hard won progress. The Thistlethwaite algorithm tells you how to get from to . It continues by a progression of subgroups that goes something like and then on its way to Start. So Thistelthwaite is trying to get into a position where 90 degree turns are no longer necessary and the solution can be completed using only 180 degree moves. Let's go back to the subgroup just for a minute, where ={I,U,U2,U'}. is a subgroup of where ={I,U2}. As a silly example of the Thistlethwaite technique, we could go from to and then on to I. For example, suppose we were at U, which is in . Since we are bound and determined to get into , we could make the move U which takes us into and we could complete the solution by making the move U2. Hence, we have solved the position U with two moves (namely U U2) when one would have sufficed (namely U'). As silly as this example is, it is illustrative of the way in which Thistlethwaite's method is suboptimal, and how Thistlethwaite's method can be improved. Finally, I think the most elegant sequence of closed subgroups is more in the vein of starting with a corner and working your way out from the corner by layer. For example, you might first solve a 2x2x2 subcorner of a 3x3x3, then solve the other three faces. This approach does not inherently involve any preference for 180 degree turns. I like it because I do not like counting 180 degree turns. The trouble with this approach is that, for example, is an awfully big group to solve all at one go. Mike Reid has suggested breaking down into etc. to make the problem manageable, but then we are back into using 180 degree turns. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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