From alan@curry.epilogue.com Fri May 19 22:01:05 1995 Return-Path: Received: from curry.epilogue.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03671; Fri, 19 May 95 22:01:05 EDT Received: (from alan@localhost) by curry.epilogue.com (8.6.8/8.6.6) id WAA00372; Fri, 19 May 1995 22:01:05 -0400 Date: Fri, 19 May 1995 22:01:05 -0400 Message-Id: <19May1995.194423.Alan@LCS.MIT.EDU> From: Alan Bawden Sender: Cube-Lovers-Request@ai.mit.edu Reply-To: Cube-Lovers-Request@ai.mit.edu To: Cube-Lovers@ai.mit.edu Subject: The Cube-Lovers Archives expands Occasionally people have approached me and wondered if there was an FTP archive of Cube-related material (programs, documents, databases, pictures, whatever). I have always replied that the only archive I knew of was the archive of Cube-Lovers mail that I maintain. Since this question keeps coming up, there must be a need to be filled, so I propose to expand our archives to cover any additional Cube-related material that people might care to submit. If you would like to submit a contribution to this archive, please send mail to Cube-Lovers-Request@AI.MIT.EDU (please do -not- send mail to all of Cube-Lovers) and include: o The location where I can pick up the files you wish to contribute (preferably using anonymous FTP). o A brief description of your contribution, to be included in a master index file. I reserve the right to redescribe, repackage, rename, recompress or totally reject your contribution. Periodically I will announce new additions to the archive to all of Cube-Lovers. Currently the archives contain nothing other than the electronic mail archives. (Although I know of at least one potential contributor who's been waiting in the wings for a couple of months now...) Some of you will no doubt have forgotten where the archive is: Using FTP, connect to FTP.AI.MIT.EDU, login as "anonymous" (any password), and go to the directory "pub/cube-lovers". (From the World Wide Web, you can use the URL: "ftp://ftp.ai.mit.edu/pub/cube-lovers".) - Alan From BRYAN@wvnvm.wvnet.edu Sun May 21 15:24:47 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26938; Sun, 21 May 95 15:24:47 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0109; Sun, 21 May 95 07:18:50 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3804; Sun, 21 May 1995 07:18:50 -0400 Message-Id: Date: Sun, 21 May 1995 07:18:49 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: AntiSlice Under M-conjugacy (and a problem with slice) I have had some posts not get through. The following will serve to consolidate several of them. Some of this may be a repeat, but not all, I think. Mark Longridge's antislice results are as follows: > arrangements arrangements >Moves Deep (2q or anti-slice moves) (4q or double anti-slice moves) > > 0 1 1 > 1 6 9 > 2 27 51 > 3 120 265 > 4 423 864 > 5 1,098 1,785 > 6 1,770 2,017 > 7 1,650 1,008 > 8 851 144 > 9 198 > ----- ----- > 6,144 6,144 We have the following M-conjugacy results for 2q moves. Level Positions Local Maxima 0 1 0 1 1 0 2 3 0 3 10 0 4 37 0 5 93 1 6 166 2 7 147 7 8 89 12 9 21 21 ---- 568 The level 5 local maximum is (U'D')(FB)(FB)(UD)(L'R'). The position is not its own inverse, but we can use as an inverse (U'D')(FB)(FB)(UD)(LR). Hence, (U'D')(FB)(FB)(UD) forms a nice "middle" of the sequence. In fact, the (U'D')(FB)(FB)(UD) position in some ways seems more interesting than the local maximum itself. Does it already have a name? I have not verified if the length of the local maximum is 10q in G, nor if it is a local maximum in G. We have the following M-conjugacy results for 4q moves. Strong and weak local maxima are defined according to my preference. If you prefer Mike Reid's definition, ignore the "weak" column and read the "total" column as "weak". Level Positions Strong Weak Total Local Max Local Max Local Max 0 1 0 0 0 1 2 0 0 0 2 5 0 0 0 3 25 0 1 1 4 75 0 2 2 5 152 0 19 19 6 184 1 35 36 7 108 0 46 46 8 16 0 16 16 ---- 568 Back on the subject of the slice group, we have the following. Mark Longridge said: >By the way GAP gives NumberConjugacyClasses (slice) = 23 >In your calculations of M-conjugacy classes for the slice group the >total number of classes is 50, but I think GAP does not use >M-conjugates but C-conjugates instead. The NumberConjugacyClasses >function always thrashes with any larger groups unfortunately. >If you could easily tweak your program perhaps you could >verify my theory. Recall that in my work with , I had to use W3-conjugacy rather than M-conjugacy. The simplest explanation is that all M-conjugates of a position in are not in , and in particular the representative element might not be in . W3 is the largest subgroup of M such that all conjugates of are in . I flirted briefly with the notion that I might have the same problem with the slice group and the antislice group. But it seems immediate that M-conjugacy is appropriate for both slice and antislice. For example, think of applying M-conjugacy to all the individual 2q or 4q moves in a slice or antislice process. Clearly, the result is still in slice and antislice, respectively. I doubt that Mark's theory about GAP using C-conjugacy for slice instead of M-conjugacy is correct. I already have 50 positions to 23 for GAP, and using C-conjugacy would just make my results larger. For example, RL' and R'L are M-conjugate positions, but not C-conjugate positions. I don't have a clue why my results do not match GAP. I have double and triple checked my results, and they seem correct. For example, I can "expand" my conjugacy classes, and the results then match Mark's exactly. How does GAP's NumberConjugacyClasses function work? By that, I mean how does it know the subgroup with respect to which you are taking conjugacy classes (if my terminology is correct)? For example, how does it know to take C or M or whatever conjugates? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From mschoene@math.rwth-aachen.de Mon May 22 05:14:39 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25766; Mon, 22 May 95 05:14:39 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sDTYk-000MP6C; Mon, 22 May 95 11:13 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sDTYk-00025lC; Mon, 22 May 95 11:13 WET DST Message-Id: Date: Mon, 22 May 95 11:13 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Sun, 21 May 1995 07:18:49 -0400 (EDT) Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) Sorry, lately I didn't have any time to follow the discussions on Cube-Lovers (preparing the second upgrade of GAP 3.4 and working long hours for GAP 4.0). But Jerry Bryan's message talks about GAP's 'NumberConjugacyClasses' function. Jerry wrote I doubt that Mark's theory about GAP using C-conjugacy for slice instead of M-conjugacy is correct. I already have 50 positions to 23 for GAP, and using C-conjugacy would just make my results larger. For example, RL' and R'L are M-conjugate positions, but not C-conjugate positions. I don't have a clue why my results do not match GAP. I have double and triple checked my results, and they seem correct. For example, I can "expand" my conjugacy classes, and the results then match Mark's exactly. GAP's 'NumberConjugacyClasses' follows the general usage in group theory. The conjugacy class of an element of is the set of elements that are G-conjugated to (i.e., there exists an element in , such that ^-1 * * = ). Thus GAP is using -conjugacy classes. Since GAP is using a *larger* group, it is not surprising that GAP finds less conjugacy classes (if M were a subgroup of , then this had to be so, because in this case every M-conjugacy class would be a subset of a -conjugacy class). Jerry continued How does GAP's NumberConjugacyClasses function work? By that, I mean how does it know the subgroup with respect to which you are taking conjugacy classes (if my terminology is correct)? For example, how does it know to take C or M or whatever conjugates? It always takes the whole group itself as the acting group. With some related functions (e.g., 'ConjugacyClass' itself) you can specify that you want another group acting, but not with 'NumberConjugacyClasses'. Mark Longridge wrote (as cited by Jerry) In your calculations of M-conjugacy classes for the slice group the total number of classes is 50, but I think GAP does not use M-conjugates but C-conjugates instead. The NumberConjugacyClasses function always thrashes with any larger groups unfortunately. If you could easily tweak your program perhaps you could verify my theory. As I wrote above GAP does not use C-conjugates but -conjugates. Conjugacy classes in permutation groups are notoriously difficult. Computing the conjugacy classes of G (the full cube group) for example is absolutely impossible (without using some theory anyway). Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From mreid@ptc.com Mon May 22 17:31:26 1995 Return-Path: Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03820; Mon, 22 May 95 17:31:26 EDT Received: from ducie by ptc.com (5.0/SMI-SVR4-NN) id AA04356; Mon, 22 May 95 17:29:23 EDT Received: by ducie (1.38.193.4/sendmail.28-May-87) id AA21637; Mon, 22 May 1995 17:48:42 -0400 Date: Mon, 22 May 1995 17:48:42 -0400 From: mreid@ptc.com (michael reid) Message-Id: <9505222148.AA21637@ducie> To: cube-lovers@ai.mit.edu Subject: AntiSlice Under M-conjugacy (and a problem with slice) Content-Length: 1713 jerry writes (about the antislice group) > The level 5 local maximum is (U'D')(FB)(FB)(UD)(L'R'). [ ... ] > I have not verified if the length of the local maximum is 10q in G, > nor if it is a local maximum in G. this is exactly what i tried to explain in my recent posts: the latter statement follows from the former. and yes, its length is indeed 10q. it's pretty easy to find maneuvers which end in each quarter turn: (F'B') (UUDD) (FB) (RL) = (U'D') (FFBB) (UD) (R'L') = (R'L') (U'D') (FFBB) (UD) = (R'L') (UD) (FFBB) (U'D') = (RL) (F'B') (UUDD) (FB) = (RL) (FB) (UUDD) (F'B'). > We have the following M-conjugacy results for 4q moves. Strong > and weak local maxima are defined according to my preference. it seems like jerry's terminology is more reasonable, so i'll stop using mine. jerry's figures > Level Positions Strong Weak Total > Local Max Local Max Local Max > > 0 1 0 0 0 > 1 2 0 0 0 > 2 5 0 0 0 > 3 25 0 1 1 > 4 75 0 2 2 > 5 152 0 19 19 > 6 184 1 35 36 > 7 108 0 46 46 > 8 16 0 16 16 > ---- > 568 beg the obvious question: what is that strong local maximum, which is unique up to symmetry? mike From hoey@aic.nrl.navy.mil Tue May 23 13:11:33 1995 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17417; Tue, 23 May 95 13:11:33 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA02292; Tue, 23 May 95 13:11:28 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Tue, 23 May 95 13:11:27 EDT Date: Tue, 23 May 95 13:11:27 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9505231711.AA26574@sun13.aic.nrl.navy.mil> To: "Jerry Bryan" , cube-lovers@ai.mit.edu Subject: M-conjugacy vs. C-Conjugacy in the Slice group Jerry Bryan tried to communicate about this to cube-lovers, but there's apparently a technical difficulty. On 20 May, Jerry said: >I doubt that Mark's theory about GAP using C-conjugacy for slice >instead of M-conjugacy is correct. I already have 50 positions >to 23 for GAP, and using C-conjugacy would just make my results >larger. For example, RL' and R'L are M-conjugate positions, >but not C-conjugate positions. I emailed him to note to the contrary that RL' and R'L are indeed C-conjugates, for example under 180 degree rotation around the F-B axis. I did wonder, though, whether that meant that there could be C-conjugate slice positions that were not M-conjugate. He emailed me: > We can observe that R and R' are not C-conjugates, nor are L' and L, > which suckered me into stating that RL' and R'L are not. But > rewrite R'L as LR' since opposite face moves commute. Now, RL' > and LR' are clearly C-conjugate. > In fact, I have now verified with a quick search program that > all M-conjugates in the slice group are also C-conjugates. Hence, > there are 50 C-conjugate classes in slice, just as there are > 50 M-conjugate classes. > In retrospect, I don't think the search program was necessary.... and continues with an argument that did not convince me, but the following does: First, consider the central inversion v, which maps each point of the cube to its diametric opposite. Conjugation by v maps each face-turn (e.g. F) with its diametric opposite in opposite sense (B'). Since these are the pairs that constitute a slice move, and they commute, we have: v' FB' v = (v' F v) (v' B' v) = B' F = F B', and similarly for the other slice moves, showing that each slice move is its own v-conjugate. This extends to a proof that each position in the slice group is its own v-conjugate: v' s1 s2 ... sn v = (v' s1 v) (v' s2 v) ... (v' sn v) = s1 s2 ... sn. Suppose that we have two M-conjugate positions X, Y in the slice group. So X = m' Y m for some m in M. If m is in C, then X and Y are C-conjugate and we are done. Otherwise take the central inversion v; we know that mv is in C. We also know that X = v' X v = v' m' Y m v = (mv)' Y (mv). So X and Y are C-conjugate in this case as well. QED. Note: "Being its own v-conjugate" might as well be called "being v-symmetric". Dan Hoey@AIC.NRL.Navy.Mil From BRYAN@wvnvm.wvnet.edu Tue May 23 14:39:03 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24587; Tue, 23 May 95 14:39:03 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa14844; 23 May 95 14:07 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3063; Tue, 23 May 95 14:04:04 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 6221; Tue, 23 May 1995 14:04:04 -0400 Message-Id: Date: Tue, 23 May 1995 14:04:03 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: M-conjugacy vs. C-Conjugacy in Slice and Antislice On 20 May, I said the following: >I doubt that Mark's theory about GAP using C-conjugacy for slice >instead of M-conjugacy is correct. I already have 50 positions >to 23 for GAP, and using C-conjugacy would just make my results >larger. For example, RL' and R'L are M-conjugate positions, >but not C-conjugate positions. We already know from Martin Schoenert that GAP is using neither M-conjugacy nor C-conjugacy, but -conjugacy. But my statement about C-conjugacy vs. M-conjugacy was completely incorrect in any case. Dan Hoey pointed out to me that RL' and R'L in fact *are* C-conjugates under 180 degree rotation around the U-D axis. We can observe that R and R' are not C-conjugates, nor are L' and L, which suckered me into stating that RL' and R'L are not. But rewrite R'L as LR' since opposite face moves commute. Now, RL' and LR' are clearly C-conjugate. In fact, I have now verified with a quick search program that all M-conjugates in the slice group are also C-conjugates. Hence, there are 50 C-conjugate classes in slice, just as there are 50 M-conjugate classes. In retrospect, I don't think the search program was necessary. Suppose X and Y are M-conjugates in the slice group. Then, they can be written as M-conjugate sequences. That is, they can be written so that the individual slice moves are respective M-conjugates for some fixed m in M. (The fact that it might be possible also to write them so that the individual slice moves are not respective M-conjugates for some fixed m in M is irrelevant.) Furthermore, write the sequence for X so that the clockwise half of each slice is written prior to the counter-clockwise half of the slice. The sequence for Y with individual slice moves being respective M conjugates of X may or not have this property. But if not, then simply reorder the halves of the slices of Y to put the clockwise half first, and Y will still be piecewise M-conjugate with X. Then, the piecewise M-conjugate slices are also C-conjugate, and therefore the X and Y positions are C-conjugate. Antislice is totally different. For example, RL is M-conjugate to R'L', but it not C-conjugate. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Tue May 23 14:43:48 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA24876; Tue, 23 May 95 14:43:48 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0566; Tue, 23 May 95 08:50:41 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 5304; Tue, 23 May 1995 08:50:41 -0400 Message-Id: Date: Tue, 23 May 1995 08:50:40 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: M-conjugacy vs. C-Conjugacy in Slice and Antislice On 20 May, I said the following: >I doubt that Mark's theory about GAP using C-conjugacy for slice >instead of M-conjugacy is correct. I already have 50 positions >to 23 for GAP, and using C-conjugacy would just make my results >larger. For example, RL' and R'L are M-conjugate positions, >but not C-conjugate positions. We already know from Martin Schoenert that GAP is using neither M-conjugacy nor C-conjugacy, but -conjugacy. But my statement about C-conjugacy vs. M-conjugacy was completely incorrect in any case. Dan Hoey pointed out to me that RL' and R'L in fact *are* C-conjugates under 180 degree rotation around the U-D axis. We can observe that R and R' are not C-conjugates, nor are L' and L, which suckered me into stating that RL' and R'L are not. But rewrite R'L as LR' since opposite face moves commute. Now, RL' and LR' are clearly C-conjugate. In fact, I have now verified with a quick search program that all M-conjugates in the slice group are also C-conjugates. Hence, there are 50 C-conjugate classes in slice, just as there are 50 M-conjugate classes. In retrospect, I don't think the search program was necessary. Suppose X and Y are M-conjugates in the slice group. Then, they can be written as M-conjugate sequences. That is, they can be written so that the individual slice moves are respective M-conjugates for some fixed m in M. (The fact that it might be possible also to write them so that the individual slice moves are not respective M-conjugates for some fixed m in M is irrelevant.) Furthermore, write the sequence for X so that the clockwise half of each slice is written prior to the counter-clockwise half of the slice. The sequence for Y with individual slice moves being respective M conjugates of X may or not have this property. But if not, then simply reorder the halves of the slices of Y to put the clockwise half first, and Y will still be piecewise M-conjugate with X. Then, the piecewise M-conjugate slices are also C-conjugate, and therefore the X and Y positions are C-conjugate. Antislice is totally different. For example, RL is M-conjugate to R'L', but it not C-conjugate. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Wed May 24 15:50:47 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA11496; Wed, 24 May 95 15:50:47 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa10690; 24 May 95 15:50 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2564; Wed, 24 May 95 15:46:46 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 6128; Wed, 24 May 1995 15:46:46 -0400 Message-Id: Date: Wed, 24 May 1995 15:46:45 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) In-Reply-To: Message of 05/22/95 at 17:48:42 from mreid@ptc.com On 05/22/95 at 17:48:42 mreid@ptc.com said: >> Level Positions Strong Weak Total >> Local Max Local Max Local Max >> >> 6 184 1 35 36 >beg the obvious question: what is that strong local maximum, >which is unique up to symmetry? I haven't verified in the antislice data base, but it *has* to be Pons Asinorum, accomplshed as six antislices. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Wed May 24 17:14:08 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14808; Wed, 24 May 95 17:14:08 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa11446; 24 May 95 16:40 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2982; Wed, 24 May 95 16:36:42 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7958; Wed, 24 May 1995 16:36:43 -0400 Message-Id: Date: Wed, 24 May 1995 16:36:30 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) In-Reply-To: Message of 05/22/95 at 11:13:00 from , Martin.Schoenert@math.rwth-aachen.de On 05/22/95 at 11:13:00 Martin Schoenert said: >GAP's 'NumberConjugacyClasses' follows the general usage in group theory. >The conjugacy class of an element of is the set of elements >that are G-conjugated to (i.e., there exists an element in , >such that ^-1 * * = ). Just to give an example that I am familiar with, suppose the group in question were M itself. Then, NumberConjugacyClasses should yield 10, because the 48 elements in M yield 10 conjugacy classes under M-conjugation. If anybody who has GAP also has defined M, you might give it a try. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From mschoene%math.rwth-aachen.de@samson.math.rwth-aachen.de Thu May 25 09:39:22 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10425; Thu, 25 May 95 09:39:22 EDT Received: from samson.math.rwth-aachen.de by MINTAKA.LCS.MIT.EDU id aa23524; 25 May 95 6:19 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sEa08-000MPEC; Thu, 25 May 95 12:18 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sEa07-00026WC; Thu, 25 May 95 12:18 WET DST Message-Id: Date: Thu, 25 May 95 12:18 WET DST From: "Martin Schoenert" To: Cube-Lovers@lcs.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Wed, 24 May 1995 16:36:30 -0400 (EDT) Subject: Re: Re: AntiSlice Under M-conjugacy (and a problem with slice) Jerry Bryan wrote Just to give an example that I am familiar with, suppose the group in question were M itself. Then, NumberConjugacyClasses should yield 10, because the 48 elements in M yield 10 conjugacy classes under M-conjugation. If anybody who has GAP also has defined M, you might give it a try. GAP also thinks that M has 10 conjugacy classes (under M-conjugation). Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From BRYAN@wvnvm.wvnet.edu Thu May 25 12:01:36 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19301; Thu, 25 May 95 12:01:36 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa25656; 25 May 95 11:55 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8346; Thu, 25 May 95 11:51:27 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7593; Thu, 25 May 1995 11:51:00 -0400 Message-Id: Date: Thu, 25 May 1995 11:50:55 -0400 (EDT) From: "Jerry Bryan" To: "Dan Hoey" , "Cube Lovers List" Subject: Re: M-conjugacy vs. C-Conjugacy in the Slice group In-Reply-To: Message of 05/23/95 at 13:11:27 from hoey@AIC.NRL.Navy.Mil I said: >> In fact, I have now verified with a quick search program that >> all M-conjugates in the slice group are also C-conjugates. Hence, >> there are 50 C-conjugate classes in slice, just as there are >> 50 M-conjugate classes. >> In retrospect, I don't think the search program was necessary.... On 05/23/95 at 13:11:27 hoey@AIC.NRL.Navy.Mil said: >and (Jerry) continues with an argument that did not convince me, but the >following does: I think I can both greatly simplify and greatly strengthen the argument that did not convince Dan. My argument is based on the idea (copied from _Symmetry and Local Maxima_) that M-conjugation can be viewed as a permutation on Q, the set of twelve quarter turns. Call the six clockwise quarter turns Q+ and the six counter-clockwise quarter turns Q-. We can observe that the 24 rotations in M all map Q+ to Q+ and map Q- to Q-, and that the 24 reflections in M all map Q+ to Q- and map Q- to Q+. We also note that in particular, the central inversion v is a reflection. Suppose X and Y are M-conjugates in with Y=m'Xm for some fixed m in M. Write X as pairs of quarter turns (each pair is a slice), and write Y as pairs of quarter turns which are respective M-conjugates (via the fixed permutation m) of the quarter turns in X. If the respective quarter turns have been mapped Q+ to Q+ and Q- to Q-, then m is a rotation and we are done. Otherwise, commute the halves of each slice in Y. We first note that so commuting is the identity on Y. We also note that so commuting is equivalent to performing the permutation operation v on Q, and is therefore equivalent to performing v-conjugation on Y. (In passing, we see that this effectively proves Dan's first point, namely that X=v'Xv for all X in . Given that, I would shorten the rest of Dan's argument by saying Y=m'Xm=v'('m'Xm)v=v'm'Xmv, and noting that either m or mv is a rotation). But having started with the "commuting the halves of slices" argument, I would continue as follows. Having commuted the halves of the slices, we still have an M-conjugate (and still the same M-conjugate) because commuting is equivalent to v-conjugation, v is in M, and v-conjugation is the identity in . Finally, having commuted the halves of the slices, we are now mapping Q+ to Q+ and Q- to Q-, so we have a rotation. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From @mail.uunet.ca:mark.longridge@canrem.com Thu May 25 19:56:07 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16697; Thu, 25 May 95 19:56:07 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <173224-6>; Thu, 25 May 1995 19:55:56 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA17756; Thu, 25 May 95 19:50:36 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E382E; Thu, 25 May 95 18:46:58 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: GAP notes From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1142.5834.0C1E382E@canrem.com> Date: Thu, 25 May 1995 18:41:00 -0400 Organization: CRS Online (Toronto, Ontario) On 05/22/95 at 11:13:00 Martin Schoenert said: >GAP's 'NumberConjugacyClasses' follows the general usage in > group theory. >The conjugacy class of an element of is the set of elements >that are G-conjugated to (i.e., there exists an element in , >such that ^-1 * * = ). On 05-24-95 (18:16) Jerry Bryan said: >Just to give an example that I am familiar with, suppose the group >in question were M itself. Then, NumberConjugacyClasses should yield >10, because the 48 elements in M yield 10 conjugacy classes under >M-conjugation. If anybody who has GAP also has defined M, you >might give it a try. Ok... let's define C in the context of GAP: c := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19) (20,12,36,28)(21,13,37,29) (46,48,43,41)(44,47,45,42)(38,30,22,14)(39,31,23,15)(40,32,24,16), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35) (2,18,42,39)(7,23,47,34) (30,32,27,25)(28,31,29,26)(19,43,38,3) (21,45,36,5) (24,48,33, 8) );; M is the same as C but with the central reflection: m := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19) (20,12,36,28)(21,13,37,29) (46,48,43,41)(44,47,45,42)(38,30,22,14)(39,31,23,15)(40,32,24,16), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35) (2,18,42,39)(7,23,47,34) (30,32,27,25)(28,31,29,26)(19,43,38,3) (21,45,36,5) (24,48,33, 8), (1,8)(3,6)(2,7)(4,5) (17,24)(19,22)(18,23)(20,21) (9,16)(11,14)(10,15)(12,13) (25,32)(27,30)(26,31)(28,29) (33,40)(35,38)(34,39)(36,37) (41,48)(43,46)(42,47)(44,45) );; Then we have Size (c) = 24 NumberConjugacyClasses (c) = 5 Size (m) = 48 NumberConjugacyClasses (m) = 10 These results concur with Dan's message from Tue, 28 Dec 93 18:40:52 EST from the archives. We can also use GAP to calculate the size of the M-conjugacy class of a given element: Size (ConjugacyClass (m, cross4)) = 3 Here we see there are three possible 4 Cross order 2 patterns. I've tried dabbling in some GAP programming. Say we are looking for an element in the slice group with 4 variants under M-conjugacy.... a := 0; x := 0; z := Elements (slice); repeat a := a+1 x := Size (ConjugacyClass (m, Random (slice))); until a = 768 or x = 4 This short program found no elements of size 4 in the slice group. -> Mark <- From BRYAN@wvnvm.wvnet.edu Fri May 26 19:37:17 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16994; Fri, 26 May 95 19:37:17 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9666; Fri, 26 May 95 19:33:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1582; Fri, 26 May 1995 19:33:51 -0400 Message-Id: Date: Fri, 26 May 1995 19:33:50 -0400 (EDT) From: "Jerry Bryan" To: , "Cube Lovers List" Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) In-Reply-To: Message of 05/24/95 at 17:14:09 from mreid@ptc.com On 05/24/95 at 17:14:09 mreid@ptc.com said: >hi jerry, >you said >> On 05/22/95 at 17:48:42 mreid@ptc.com said: >> >> >> Level Positions Strong Weak Total >> >> Local Max Local Max Local Max >> >> >> >> 6 184 1 35 36 >> >> >beg the obvious question: what is that strong local maximum, >> >which is unique up to symmetry? >> >> I haven't verified in the antislice data base, but it *has* to be >> Pons Asinorum, accomplshed as six antislices. >no it hasn't. this is the equivalent of the face turn metric, >i.e. your "level" is half the face turn distance. (RL and RRLL >each count as 1, which is half their face turn count.) >then pons asinorum is at level 3, and gives the weak local maximum >at that level. Mike is correct. Try instead, (L2R2)(U2D2)(F'B')(U2D2)(L2R2)(F'B'). This is a very pretty pattern which may well have a name, but I don't know what the name is. Also, it is its own inverse. Is the length 12h in ? Is it a local maximum (strong or otherwise) in ? Is the length 20q in ? Is it a local maximum in ? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From mreid@ptc.com Tue May 30 15:39:31 1995 Return-Path: Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25072; Tue, 30 May 95 15:39:31 EDT Received: from ducie by ptc.com (5.0/SMI-SVR4-NN) id AA11138; Tue, 30 May 95 15:37:29 EDT Received: by ducie (1.38.193.4/sendmail.28-May-87) id AA15624; Tue, 30 May 1995 12:29:29 -0400 Date: Tue, 30 May 1995 12:29:29 -0400 From: mreid@ptc.com (michael reid) Message-Id: <9505301629.AA15624@ducie> To: cube-lovers@ai.mit.edu Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice) Content-Length: 1649 jerry writes [ ... ] > >> >> Level Positions Strong Weak Total > >> >> Local Max Local Max Local Max > >> >> > >> >> 6 184 1 35 36 > >> > >> >beg the obvious question: what is that strong local maximum, > >> >which is unique up to symmetry? [ ... ] > Try instead, (L2R2)(U2D2)(F'B')(U2D2)(L2R2)(F'B'). > > This is a very pretty pattern which may well have a name, but I > don't know what the name is. Also, it is its own inverse. > > Is the length 12h in ? Is it a local maximum (strong or otherwise) > in ? Is the length 20q in ? Is it a local maximum in ? no, yes (otherwise), no, and yes, respectively. we have seen this pattern several times recently. this is one of those positions with 16 symmetries. i called it "four pluses" in my message of may 11 (although i gave it in a different orientation) ) four pluses ( R2 F2 R2 U'D F2 R2 F2 UD' ) in fact, this maneuver is minimal in both the quarter turn and the face turn metrics, so its length is 16q, 10f. it is a weak local maximum in the face turn metric; one can check that no minimal maneuver ends with the face turn R. however, using the 16 symmetries which preserve the U-D axis, and inversion, we can give minimal maneuvers which end with turns of any of the six faces. this shows that it's a weak local maximum in the face turn metric. local maximality in the quarter turn metric follows in a similar manner. also, mark pointed out on april 16 that this position lies in the center of the antislice group. mike From @mail.uunet.ca:mark.longridge@canrem.com Sat Jun 3 03:39:09 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03320; Sat, 3 Jun 95 03:39:09 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <174124-7>; Sat, 3 Jun 1995 03:40:48 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA02923; Sat, 3 Jun 95 03:35:21 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E504D; Sat, 3 Jun 95 03:29:35 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Super Groups From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1148.5834.0C1E504D@canrem.com> Date: Sat, 3 Jun 1995 04:14:00 -0400 Organization: CRS Online (Toronto, Ontario) Notes on the various Super-Groups --------------------------------- I have calculated the size of the super-groups for various subgroups of the cube. I have suffixed the standard group names with the letter c to show that the centre orientations are significant. The groups are (ranked smallest to largest): Size (slice) = 768 Size (slicec) = 24,576 Size (slicec) / Size (slice) = 32 The following reference confirms this calculation and expounds further on the nature of the slice group... The Slice Group in Rubik's Cube, by David Hecker, Ranan Banerji Mathematics Magazine, Vol. 58 No. 4 Sept 1985 Size (antisl) = 6,144 Size (antislc) = 49,152 Size (antislc) / Size (antisl) = 8 Size (sq) = 663,552 Size (sqc) = 5,308,416 Size (sqc) / Size (sq) = 8 Size (ur) = 73,483,200 Size (urc) = 587,865,600 Size (urc) / Size (ur) = 8 Size (cube) = 43,252,003,274,489,856,000 Size (cubec) = 88,580,102,706,155,225,088,000 Size (cubec) / Size (cube) = 2,048 The case of the super squares group (sqc) is interesting. It is only possible to rotate opposite centres 180 degrees. There are actually 8 centres in the super square's group: (1 way) Identity (1 way) All 6 centres rotated 180 degrees (3 ways) 2 opposite centres rotated 180 degrees (3 ways) 2 pairs of opposite centres rotated 180 degrees -> Mark <- From @mail.uunet.ca:mark.longridge@canrem.com Sun Jun 4 03:28:58 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18076; Sun, 4 Jun 95 03:28:58 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <173779-8>; Sun, 4 Jun 1995 03:30:39 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA25003; Sun, 4 Jun 95 03:25:11 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E5238; Sun, 4 Jun 95 03:22:02 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Super Squares Group From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1150.5834.0C1E5238@canrem.com> Date: Sun, 4 Jun 1995 04:21:00 -0400 Organization: CRS Online (Toronto, Ontario) Way back on Thu Aug 20 20:10:13 1992 Mike wrote: > R2 F2 B2 L2 U2 L2 F2 B2 R2 ~ D2 , > so that = . I bet he didn't realize at the time he was finding a minimal sequence to rotate 2 centres in the super square's group! If we tack D2 on the end we get the sequence: R2 F2 B2 L2 U2 L2 F2 B2 R2 D2 = turn U & D centres 180 degrees in 10 face turns. -> Mark <- From bagleyd@source.asset.com Wed Jun 7 17:31:08 1995 Return-Path: Received: from source.asset.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA12024; Wed, 7 Jun 95 17:31:08 EDT Received: by source.asset.com (AIX 3.2/UCB 5.64/4.03) id AA36858; Wed, 7 Jun 1995 17:07:22 -0400 Date: Wed, 7 Jun 1995 17:07:22 -0400 From: bagleyd@source.asset.com (David A. Bagley) Message-Id: <9506072107.AA36858@source.asset.com> To: Cube-Lovers@ai.mit.edu Subject: Dinosaur Rubik's Cube Hi I just just added new modes to my xdino puzzle. It is the Rubik's Dinosaur Puzzle recently mentioned. (A cube with diagonal X cuts.) In addition to the Period 3 movement it now has a Period 2 movement with the faces cut up like: ___ |\ /| | X | |/ \| --- as opposed to just \ / X / \ In the Period 3 movement the cube turns around a corner while in Period 2 movement it turns around the center of an edge. Of course if you want to make it harder there is a "Both" mode where you can have both turning modes at once. I would like to thank Derek Bosch for suggesting the Period 2 movement -> Bosch's Cube :) I spent a good 10 minutes trying to solve Bosch's Cube and did not get anywhere. The Period 3 seems easier. Be the first in the Universe to solve it. :) All those who have X should be able to run it. There are many other puzzles in the collection as well. Any problems with the compilation or bugs ... let me know. Cheers, --__--------------------------------------------------------------- / \ \ / David A. Bagley \ | \ \ / bagleyd@source.asset.com | | \//\ Some days are better than other days. | | / \ \ -- A short lived character of Blake's 7 | \ / \_\puzzles Available at: ftp.x.org/contrib/games/puzzles / ------------------------------------------------------------------- From norgomez@itecs5.telecom-co.net Mon Jun 12 22:31:21 1995 Return-Path: Received: from ITECS5.TELECOM-CO.NET by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA11033; Mon, 12 Jun 95 22:31:21 EDT Date: Mon, 12 Jun 1995 21:28:16 -0400 Message-Id: <95061221281669@itecs5.telecom-co.net> From: norgomez@itecs5.telecom-co.net (ORLANDO GOMEZ CAMACHO, BOGOTA-COLOMBIA.) To: JCCANNEK@ukcc.uky.edu, MTY.ITESM.MX@ukcc.uky.edu, commune-list@stealth.acf.nyu.edu, COMP-CEN%UCCVMA.BITNET@vm1.nodak.edu, CONFOCAL%UBVM.BITNET@vm1.nodak.edu, CW-MAIL%TECMTYVM.BITNET@vm1.nodak.edu, CWIS-L%WUVMD.BITNET@vm1.nodak.edu, DB2-L%AUVM.BITNET@vm1.nodak.edu, DBASE-L%TECMTYVM.BITNET@vm1.nodak.edu, JO%ILNCRD.BITNET@cunyvm.cuny.edu, COMPOS01%ULKYVX.BITNET@cunyvm.cuny.edu, CRTNET%PSUVM.BITNET@cunyvm.cuny.edu, CUMREC-L%NDSUVM1.BITNET@cunyvm.cuny.edu, CVNET%YORKVM1.BITNET@cunyvm.cuny.edu, Cyber-L%Bitnic.BITNET@cunyvm.cuny.edu, DANCE-L%HEARN.BITNET@cunyvm.cuny.edu, Comp-Soc@limbo.intuitive.com, concrete-blonde@ferkel.ucsb.edu, CONS-L%MCGILL1.BITNET@cornellc.cit.cornell.edu, 441495@acadvm1.uottawa.ca, com-priv@psi.com, cjr2@cornell.edu, CORPORA-REQUEST@x400.hd.uib.no, CORRYFEE@hasara11.telecom-co.net, CPAE@catfish.valdosta.peachnet.edu, CPE-LIST@uncvm1.oit.unc.edu, CREA-CPS@nic.surfnet.nl, CREWRT-L@mizzou1.missouri.edu, CROMED-L@aearn.bitnet, CSNET-FORUM@sh.cs.net Subject: Oportunidades de Negocios X-Vms-To: @LISTA1.DIS ====================================================================== OPORTUNIDADES DE NEGOCIOS SOLO PARA PERSONAS RESIDENTES EN LOS ESTADOS UNIDOS, CON BUENOS CONTACTOS EN EMPRESAS DE ALTA TECNOLOGIA EN INFORMATICA, OFIMATICA Y TELECOMUNICACIONES ----------------------------------------------------------------------- Apreciados amigos de la lista: Nuestra empresa ha sido contratada por una firma que edita una publicaciOn especializada en las Areas de Computadoras, Telecomunicaciones, OfimAtica y Servicios relacionados que circula en Venezuela, Ecuador, Peru, Brazil y Colombia, constituyEndose en un medio Unico de informaciOn sobre estos tOpicos en las naciones descritas. Para comercializar ESPACIOS PUBLICITARIOS de esta publicaciOn, se requiere de una compan~Ia o persona que maneje presupuesto de negocios, preferible- mente con capacidad econOmica propia e idoneidad para contratar. Es nece- sario que la persona o empresa resida en los Estados Unidos de AmErica. A continuaciOn se presentan las caracterIsticas principales de la publica- ciOn: * CirculaciOn efectiva en Brazil, Colombia, Ecuador, Peru y Venezuela. * A los anunciadores se les proveen en medio magnEtico, las bases de datos con informaciOn importante de estos paises. A las Empresas y Usuarios de cada naciOn tambiEn se les harA entrega de este material que incluirA informaciOn sobre proveedores de bienes y servicios en los sectores de Telecomunicaciones, Software, Equipos de Oficina, Computadores, y Suministros. * En la publicaciOn figuran empresas Norteamericanas que fabrican, representan y proveen equipos, programas y servicios en las areas de telecomunicaciones, sector electronico, de comunicacion, computacion perifericos, componentesm accesorios, suministros, servicios y areas afines * El tiraje es de Cuarenta Mil (40,000) ejemplares para ser distribui- dos en forma gratuita. * El perfil de los anunciadores corresponde a compan~Ias Estadouniden- ses de estos sectores que se encuentren en ampliar y consolidar su operacionalidad en estos paises. La Empresa OFRECE: * Exclusividad * Participacion atractiva sobre los negocios * Excelente calidad editorial y material informativo y util para generar nuevas y mejores oportunidades de negocios en forma proactiva. ============================================================================== A las personas interesadas, se les ruega contactarnos por esta misma via, indicando los siguientes datos: Nombre : e-mail : Direccion: Ciudad : Telefono : Fax : ============================================================================== Mayores informes: COMUNICACIONES INTERACTIVAS Contacto : Orlando Gomez Camacho e-mail : norgomez@itecs5.telecom-co.net Voice Mail: (+571) 5002072 ------------------------------------------------------------------------------ From MULL4195@splava.cc.plattsburgh.edu Wed Jun 14 16:17:44 1995 Return-Path: Received: from splava.cc.plattsburgh.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29070; Wed, 14 Jun 95 16:17:44 EDT Received: from splava.cc.plattsburgh.edu by splava.cc.plattsburgh.edu (PMDF V4.2-11 #3312) id <01HRPAOJ7EWG8WYOF1@splava.cc.plattsburgh.edu>; Wed, 14 Jun 1995 16:17:58 EST Date: Wed, 14 Jun 1995 16:17:58 -0500 (EST) From: John Mullen Subject: To: Cube-Lovers@ai.mit.edu Message-Id: <01HRPAOJ87UA8WYOF1@splava.cc.plattsburgh.edu> Organization: SUNY at Plattsburgh, New York, USA X-Envelope-To: Cube-Lovers@ai.mit.edu X-Vms-To: IN%"Cube-Lovers@ai.mit.edu" X-Vms-Cc: MULL4195 Mime-Version: 1.0 Content-Transfer-Encoding: 7BIT signoff cube-lovers From comnetlu!georges.helm@eo.net Thu Jun 15 15:38:16 1995 Return-Path: Received: from eol1 (eol1.eo.lu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10184; Thu, 15 Jun 95 15:38:16 EDT Received: from comnetlu.UUCP by eol1 with UUCP (Smail3.1.28.1 #4) id m0sMKk2-000bKNC; Thu, 15 Jun 95 21:38 MET DST To: cube-lovers@ai.mit.edu Subject: Andras Mezei's book From: georges.helm@comnet.eo.lu (GEORGES HELM) Message-Id: <8AB54D5.0063000243.uuout@comnet.eo.lu> Date: Thu, 15 Jun 95 20:37:00 +1 Organization: ComNet Luxembourg BBS Reply-To: georges.helm@comnet.eo.lu (GEORGES HELM) X-Mailreader: PCBoard Version 15.21 X-Mailer: PCBoard/UUOUT Version 1.10 Mark Longridge wrote > Does anyone on Cube-Lovers have that book? Yes, I do. The illustrations cover: Rubik himself, Polytoys, Konsumex, cubes, books, cube-related items (stickers, pencils, lp's, earrings, t-shirts...) There is an article on a competition in 1982 in Budapest. And many more articles I don't understand a word of. Georges From comnetlu!georges.helm@eo.net Fri Jun 16 03:20:00 1995 Return-Path: Received: from eol1 (eol1.eo.lu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19818; Fri, 16 Jun 95 03:20:00 EDT Received: from comnetlu.UUCP by eol1 with UUCP (Smail3.1.28.1 #4) id m0sMVh1-000bKLC; Fri, 16 Jun 95 09:19 MET DST To: Cube-Lovers@ai.mit.edu Subject: Andras Mezei's book From: georges.helm@comnet.eo.lu (GEORGES HELM) Message-Id: <8AB622C.006300026F.uuout@comnet.eo.lu> Date: Fri, 16 Jun 95 09:16:00 +1 Organization: ComNet Luxembourg BBS Reply-To: georges.helm@comnet.eo.lu (GEORGES HELM) X-Mailreader: PCBoard Version 15.21 X-Mailer: PCBoard/UUOUT Version 1.10 I have the book. - Georges From BRYAN@wvnvm.wvnet.edu Fri Jun 16 14:10:52 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17367; Fri, 16 Jun 95 14:10:52 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 6542; Fri, 16 Jun 95 14:11:03 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2667; Fri, 16 Jun 1995 14:11:03 -0400 Message-Id: Date: Fri, 16 Jun 1995 14:11:02 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima There are four of them unique up to M-conjugacy (more information to come). Distance Cubes Branch Lcl {m'Xm} Branch Ratio Local from Factor Max (M-Conj. Factor of Max Start Classes) Cubes to Classes 0 1 1 0 1 12 12.000 0 1 1.000 12.000 0 2 114 9.500 0 5 5.000 22.800 0 3 1,068 9.368 0 25 5.000 42.720 0 4 10,011 9.374 0 219 8.760 45.712 0 5 93,840 9.374 0 1,978 9.032 47.442 0 6 878,880 9.366 0 18,395 9.300 47.778 0 7 8,221,632 9.355 0 171,529 9.325 47.931 0 8 76,843,595 9.347 0 1,601,725 9.338 47.976 0 9 717,789,576 9.341 0 14,956,266 9.338 47.993 0 10 6,701,836,858 9.337 42 139,629,194 9.336 47.997 4 11 62,549,615,248 9.333 1,303,138,445 9.333 47.9992 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Fri Jun 16 14:17:43 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17624; Fri, 16 Jun 95 14:17:43 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 6602; Fri, 16 Jun 95 14:17:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 2819; Fri, 16 Jun 1995 14:17:52 -0400 Message-Id: Date: Fri, 16 Jun 1995 14:17:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima Search Matrix The individual cells in this chart give numbers of M-conjugacy classes. The local maxima are in column 12. Still more information to come. Number of Moves Which Go Closer to Start Level Total 0 1 2 3 4 5 6 7 8 9 1 1 1 Classes 0 1 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 5 0 2 3 0 0 0 0 0 0 0 0 0 0 3 25 0 20 4 1 0 0 0 0 0 0 0 0 0 4 219 0 182 34 2 1 0 0 0 0 0 0 0 0 5 1978 0 1677 280 20 1 0 0 0 0 0 0 0 0 6 18395 0 15642 2561 184 8 0 0 0 0 0 0 0 0 7 171529 0 145974 23773 1721 61 0 0 0 0 0 0 0 0 8 1601725 0 1362579 222235 16241 663 1 3 0 3 0 0 0 0 9 14956266 0 12719643 2077549 153026 5954 74 15 2 3 0 0 0 0 10 139629194 0 118711701 19418503 1438825 58862 925 318 11 37 0 8 0 4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Sat Jun 17 00:35:23 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23622; Sat, 17 Jun 95 00:35:23 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 9876; Sat, 17 Jun 95 00:35:32 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 4531; Sat, 17 Jun 1995 00:35:32 -0400 Message-Id: Date: Sat, 17 Jun 1995 00:35:31 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 10q Local Maxima Positions 1. (UU) (F'B')(RL)(RL)(FB) 2. (UD') (F'B')(RL)(RL)(FB) 3. (UD) (F'B')(RL)(RL)(FB) 4. (UD')(FB')(LR')(FB')(FB') #4 is the one Mike Reid already found in the slice group. #3 is the one I already found in the antislice group. #1, #2, and #3 are obviously closely related. #1 and #2 appear not to be in either slice or antislice, but I have been fooled before by alternative sequences which yield the same position. #1, #2, and #3 all have the property that |{m'Xm}|=6 and |Symm(X)|=8. As has already been discussed, #4 has the property that |{m'Xm}|=24 and |Symm(X)|=2. The symmetry groups for #1, #2, and #3 are of a type Dan Hoey's taxonomy calls class P, class S, and class AX, respectively. These particular classes are hard to describe succinctly without introducing a lot of notation. But in all three cases, the symmetry groups (subgroups of M such that X=m'Xm} consist of four rotations and four reflections, and have as an axis of symmetry one of the three major axes of the cube (U-D, F-B, or R-L). There are three groups P1, P2, P3 with axis of symmetry U-D, F-B, and R-L, respectively, and similarly for S1, S2, and S3, and for AX1, AX2, and AX3. For #4, we have Symm(X)=HV in Dan's taxonomy, where HV={i,v}, and where i is the identity in M and v is the central inversion in M. If proper typography were available, the i and the v would be upper case script letters to follow Frey and Singmaster. There are relatively few positions in all of cube space for which Symm(X)=Pi or Symm(X)=Si or Symm(X)=AXi (i in 1..3). There are only 10 P positions through level 10 in the search tree (of which just one is a local maximum). There is only one S position through level 10, and only one AX position through level 10, both of which are of course local maxima. The positions are not Q-transitive, but the positions look "symmetric", and they fulfill the (incorrect) intuition that "symmetric" positions must be local maxima. We have no reason to say that other P or S or AX positions further down the search will be local maxima. I find position #4 extremely intriguing. In general, HV is not very strong symmetry, and there are relatively speaking, quite a few HV positions in cube space. We could create an HV position as follows. Put any edge cubie anywhere (say UF in RD). Put the "opposite" cubie in the "opposite" cubicle (DB in LU in this case). Continue for the remaining edge cubies, and then do the same thing for the corners, remembering only to make sure the edges and corners have the same parity. You can easily make an HV position that looks quite "random" to the casual glance, and in fact most HV positions don't look very "symmetric". But Mike's position looks very "symmetric" at a casual glance, as if its symmetry must be much stronger than HV. I certainly would not have expected to find an HV position as a local maximum close to Start. I think the "look" of Mike's position as "symmetric", and the fact that it is a local maximum close to Start are related. Without getting too long winded, I think the reasons are two-fold. First, the corners and edges have much stronger symmetry separately than they do collectively. Second, the symmetry looks much stronger if you ignore the centers (i.e., if you ignore the rotational positioning of the cubies), perhaps in the sense of Dan's CSymm function. For example, the corners are properly positioned with respect to each other, even though they are not properly positioned with respect to the fixed face centers. In the next few days, I intend to calculate Symm(X) for the corners and edges separately for Mike's position, as well as calculating CSymm(X) for the corners and edges separately and combined. I think the results will be enlightening. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From @mail.uunet.ca:mark.longridge@canrem.com Sat Jun 17 22:00:06 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03704; Sat, 17 Jun 95 22:00:06 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <177474-1>; Sat, 17 Jun 1995 21:03:06 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA05634; Sat, 17 Jun 95 20:57:30 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E7553; Sat, 17 Jun 95 19:59:47 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Crazy Corner Pattern From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1161.5834.0C1E7553@canrem.com> Date: Sat, 17 Jun 1995 05:38:00 -0400 Organization: CRS Online (Toronto, Ontario) On Mon, 17 Jan 1994 09:06:59 EST, Jerry wrote: > > Counting M-conjugacy classes of the corners of Rubik's cube > ----------------------------------------------------------- > > M-Class Number Number > Size of of > Classes Elements > > 1 1 = 1 Start > 2 1 = 2 +4 -4 Twist > 3 3 = 9 > 4 1 = 4 > 6 34 = 204 > 8 33 = 264 > 12 301 = 3612 > 16 104 = 1664 > 24 9064 = 217536 > 48 1832428 = 87956544 > > Total 1841970 88179840 I'm trying to find the 1 pattern with M-class size of 4 of the corners group. The only pattern that I can find is 4 alternate corners twisted clockwise which is in the twist orbit. It does not seem to be any pattern with just corners twisted in place. Jerry, if you could *please* identify this pattern before I go nuts...... ! -> Mark <- From BRYAN@wvnvm.wvnet.edu Sun Jun 18 00:35:58 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09680; Sun, 18 Jun 95 00:35:58 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 1914; Sat, 17 Jun 95 17:18:01 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1612; Sat, 17 Jun 1995 17:18:01 -0400 Message-Id: Date: Sat, 17 Jun 1995 17:18:00 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: A Third Way to Calculate the Real Size of Cube Space? We define the real size of cube space to be the number of M-conjugate classes {m'Ym} for m in M, set of 48 rotations and reflections of the cube, and for Y in G. Dan Hoey has calculated the real size of cube space using the Polya-Burnside theorem. Dan and I (mostly Dan) have also calculated the same result using exhaustive computer search. The computer search is much less elegant than the Polya-Burnside results, but the search does provide additional information, such as the number of positions associated with each symmetry group. The results from the computer search have not yet been posted to the list, but a draft paper is in progress. In the meantime, it occurs to me that perhaps -- but only perhaps -- there is a third way to calculate the real size of cube space. The third way would not require (much) computer searching, but would provide the same level of detail about number of positions per symmetry group as does the full blown search. The idea is based on a posting from Mike Reid. Mike calculated the number of positions in G whose symmetry preserves the U-D axis. Such positions have a symmetry group which is called X1 in Dan's taxonomy. For these positions, we say Symm(Y)=X1, where in general for Y in G we have Symm(Y) is the set (and group) of all m in M such that Y=m'Ym. X1 contains sixteen elements (eight rotations and eight reflections), and preserves the U-D axis. X2 and X3 are conjugate subgroups of X1 and similarly preserve the F-B and the R-L axes, respectively. If Y is X1-symmetric, then we have {m'Ym}={Y1,Y2,Y3}. One of the Yi is Y and is X1-symmetric, one of the Yi is X2-symmetric, and the other Yi is X3-symmetric. Mike determined (without computer search) that there are 128 X1-symmetric positions. Since four of the positions are also M-symmetric, we have 124 positions Y for which Symm(Y)=X1. Similar results hold for X2 and X3. Hence, there are 124 M-conjugacy classes containing cubes for which Symm(Y)=Xi, or perhaps we might say for which SymmClass(Y)=X. The important fact here is that we have determined that there are 124 M-conjugacy classes for symmetry class X without having to do a computer search. If we could similarly determine the number of K-symmetric positions for each of the 98 subgroups K of M without computer search, then we could calculate the real size of cube space. You really only have to determine the size of 33 subgroups. Just as the solution for X1 also gave us the solution for X2 and X3, similarly the solution for any subgroup provides the solution for all conjugate subgroups, and there are 33 classes of conjugate subgroups. I usually get myself in trouble when I delve too much into things I don't understand, but let's try a few examples. The subgroup HV={i,v} is easy to understand, where v is the central inversion. For the edges, the number of HV-symmetric positions should be 24*20*16*12*8*4. That is, put the first cubie anywhere (24 possibilities) which dictates the location of the respective "opposite" cubie. There are then 20 possibilities for the location of the third cubie which again dictates the position of the respective "opposite" cubie, and so forth. In the same manner, the number of HV-symmetric corner positions is 24*18*12*6. The number of HV-symmetric positions is then (24*20*16*12*8*4)*(24*18*12*6)/2 to take parity into account. Now we have the rub. In order to calculate the positions for which Symm(Y)=HV, we must subtract out the HV-symmetric positions which have stronger symmetry than HV, just as we subtracted out the M-symmetric positions in Mike's X1 case. But to do so, we cannot take the subgroups of M in isolation. We have to do them all, starting with M and working our way down. (And HV is pretty far down the food chain.) Some of the subgroups I can do pretty easily, and for others I have not a clue. Recall that A is the subgroup of M consisting of the 24 even rotations and reflections and that C is the subgroup of M consisting of the 24 rotations (12 even and 12 odd). As long ago as _Symmetry and Local Maxima_, Dan Hoey and Jim Saxe determined that there are only four A-symmetric positions and only four C-symmetric positions, namely the four that are also M-symmetric. Hence, there are no positions for which Symm(Y)=A nor for which Symm(Y)=C. But I haven't a clue how they knew, nor how to go about constructing an A-symmetric or a C-symmetric position from scratch. You can't get very far with my proposal unless you can figure out how to construct K-symmetric positions for any K. For one more example, consider H, the set of 12 even rotations and 12 odd reflections. I know from computer search and also from _Symmetry and Local Maxima_ that there are 24 H-symmetric positions, of which 4 are M-symmetric and 20 are H-symmetric without also being M-symmetric. The 20 H-symmetric but not M-symmetric positions form 10 M-conjugacy classes for which we would say SymmClass(Y)=H. It ought to be easy to derive this result without a computer search, but again I confess I haven't a clue as to go about constructing the 24 H-symmetric positions from scratch. Well, I could cheat and look up the Class H positions in _Symmetry and Local Maxima_, but what about the classes that haven't been figured out yet? Also, I could cheat and use the results from computer search, but that's hardly the point. One final point: just as Mike's 128 X1-symmetric positions formed a group, similarly the set of K-symmetric positions form a group for all 98 possible values of K. We have to be a little careful with our terminology. The X1-symmetric positions form a group, as do the X2-symmetric and the X3-symmetric positions. But if we want to talk about the X-symmetric positions, we no longer have a group. For example, we do not in general have closure when forming the composition of X1-symmetric positions with X2-symmetric positions. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Sun Jun 18 08:48:57 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA17825; Sun, 18 Jun 95 08:48:57 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 3647; Sun, 18 Jun 95 08:25:50 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 8731; Sun, 18 Jun 1995 08:25:50 -0400 Message-Id: Date: Sun, 18 Jun 1995 08:25:49 -0400 (EDT) From: "Jerry Bryan" To: Subject: Re: Crazy Corner Pattern In-Reply-To: Message of 06/17/95 at 05:38:00 from mark.longridge@canrem.com On 06/17/95 at 05:38:00 mark.longridge@canrem.com said: >On Mon, 17 Jan 1994 09:06:59 EST, Jerry wrote: >> >> Counting M-conjugacy classes of the corners of Rubik's cube >> ----------------------------------------------------------- >> >> M-Class Number Number >> Size of of >> Classes Elements >> >> 1 1 = 1 Start >> 2 1 = 2 +4 -4 Twist >> 3 3 = 9 >> 4 1 = 4 >> 6 34 = 204 >> 8 33 = 264 >> 12 301 = 3612 >> 16 104 = 1664 >> 24 9064 = 217536 >> 48 1832428 = 87956544 >> >> Total 1841970 88179840 >I'm trying to find the 1 pattern with M-class size of 4 of the >corners group. >The only pattern that I can find is 4 alternate corners twisted >clockwise which is in the twist orbit. >It does not seem to be any pattern with just corners twisted in >place. >Jerry, if you could *please* identify this pattern before I go >nuts...... ! The position is called T-symmetric in Dan's taxonomy (actually, there are four T subgroups of M, T1 through T4). The symmetry is related to opposite corners, e.g., the UFL-DRB axis, which is why there are four T subgroups. Also, the position is Q-transitive, so you can check it out in _Symmetry and Local Maxima_. I quote ".... each edge on the girdle may be swapped with the diametrically opposite edge, provided that the corners on the girdle are swapped with their opposites as well." Here, you would fix the edges and pay attention only to the swapping of the corners. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Sun Jun 18 15:55:14 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA03583; Sun, 18 Jun 95 15:55:14 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4771; Sun, 18 Jun 95 15:55:23 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3480; Sun, 18 Jun 1995 15:55:23 -0400 Message-Id: Date: Sun, 18 Jun 1995 15:55:22 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/17/95 at 17:18:00 from BRYAN@wvnvm.wvnet.edu I should have said "a fourth way", I think. Martin Schoernert performed the same calculation with GAP. Hence, we have three ways in hand: 1) Dan's Polya-Burnside method, 2) Martin's GAP calculations, and 3) brute force computer search. My new proposal would then be a fourth way. Here is a question for Martin: is there any way with GAP to calculate the number of M-conjugacy classes associated with each symmetry class? It is this additional information about the "real cube space" which *is* available via computer search, and for which I am proposing an alternative which does not involve computer search. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From @mail.uunet.ca:mark.longridge@canrem.com Sun Jun 18 23:54:16 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23716; Sun, 18 Jun 95 23:54:16 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <177456-5>; Sun, 18 Jun 1995 23:55:59 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA07947; Sun, 18 Jun 95 23:50:23 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E77E1; Sun, 18 Jun 95 23:47:31 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Re: Crazy Corner Pattern From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1167.5834.0C1E77E1@canrem.com> Date: Mon, 19 Jun 1995 00:36:00 -0400 Organization: CRS Online (Toronto, Ontario) >>I'm trying to find the 1 pattern with M-class size of 4 of the >>corners group. >>The only pattern that I can find is 4 alternate corners twisted >>clockwise which is in the twist orbit. >>It does not seem to be any pattern with just corners twisted in >>place. >>Jerry, if you could *please* identify this pattern before I go >>nuts...... ! >The position is called T-symmetric in Dan's taxonomy (actually, >there are four T subgroups of M, T1 through T4). The symmetry >is related to opposite corners, e.g., the UFL-DRB axis, >which is why there are four T subgroups. >Also, the position is Q-transitive, so you can check it out >in _Symmetry and Local Maxima_. I quote ".... each edge on the >girdle may be swapped with the diametrically opposite edge, >provided that the corners on the girdle are swapped with >their opposites as well." Here, you would fix the edges and >pay attention only to the swapping of the corners. Hmmmmm, that's very interesting. Below is my interpretation of "...corners on the girdle are swapped with their opposites as well." Let's call it pattern X. D U D U U U D U D F L B R F L B R F L B R L L L F F F R R R B B B F L B R F L B R F L B R U D U D D D U D U If I understand the terminology correctly, then for this pattern X = |{m'Xm}|=3 and |Symm(X)|=16, same as the 4 spot. Also X = |{c'Xc}|=3. But perhaps this is not the same pattern.... Let's call this next cube arrangement pattern Y. F U U U U U D U L R L B R F B D R R B B D L L L F F F R R R B B B U L L F F U L R F L B F D D B D D D R D U Y = |{m'Ym}|=4 and |Symm(Y)|=12, and Y=|{c'Yc}| = 4 This pattern was created by swapping 3 pairs of diametrically opposite corners, which is in the *swap* orbit on a normal cube, but since we are dealing with corners of Rubik's cube and ignoring edges we can realize permutations with an odd number of pairs of corners swapped. -> Mark <- From BRYAN@wvnvm.wvnet.edu Mon Jun 19 09:57:45 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14163; Mon, 19 Jun 95 09:57:45 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8039; Mon, 19 Jun 95 09:57:53 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 6101; Mon, 19 Jun 1995 09:57:54 -0400 Message-Id: Date: Mon, 19 Jun 1995 09:57:53 -0400 (EDT) From: "Jerry Bryan" To: Subject: Re: Crazy Corner Pattern In-Reply-To: Message of 06/19/95 at 00:36:00 from mark.longridge@canrem.com Perhaps I should have quoted a little more from _Symmetry and Local Maxima_. Here is the T-symmetric position given by Saxe and Hoey. In their position, the UFL and RBD corners are in place, and the other three pairs are swapped. The "girdle" includes the three pairs that are swapped. Hence, there is an axis of symmetry along the UFL-RBD axis. The odd number of swaps is compensated by an odd number in the edges. The compensation is not required for corners only. R D D U U D U U B D L L F F D L L F L F F R L L F F B L R R B B F R R B R B B U R R B B U U U L U D D F D D = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Mon Jun 19 14:04:49 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29637; Mon, 19 Jun 95 14:04:49 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0056; Mon, 19 Jun 95 13:41:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 3719; Mon, 19 Jun 1995 13:41:52 -0400 Message-Id: Date: Mon, 19 Jun 1995 13:41:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: 3x3x3 Cubes for Sale For the first time in years, I have seen 3x3x3 cubes for sale in a store. (For American readers, the store is Hills, which is a regional chain which competes against WalMart and Kmart. The price is $8.97 -- U.S. dollars.) The box has a note signed by Erno Rubik, and gives the proper size of the cube group. Also, and I couldn't see inside the box to verify this, the Face centers seemed to be marked in such a way as to support the Supergroup. Rubik's note about the size of the problem says it is 4^4 times bigger than the regular problem. The manufacturer (or maybe distributor?) is Matchbox Toys (or is it Matchbook Toys?). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From news@nntp-server.caltech.edu Mon Jun 19 18:36:29 1995 Return-Path: Received: from piccolo.cco.caltech.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA15966; Mon, 19 Jun 95 18:36:29 EDT Received: from gap.cco.caltech.edu by piccolo.cco.caltech.edu with ESMTP (8.6.7/DEI:4.41) id PAA27121; Mon, 19 Jun 1995 15:36:21 -0700 Received: by gap.cco.caltech.edu (8.6.7/DEI:4.41) id PAA16086; Mon, 19 Jun 1995 15:36:18 -0700 To: mlist-cube-lovers@nntp-server.caltech.edu Path: whuang From: whuang@cco.caltech.edu (Wei-Hwa Huang) Newsgroups: mlist.cube-lovers Subject: Re: 3x3x3 Cubes for Sale Date: 19 Jun 1995 22:36:17 GMT Organization: California Institute of Technology, Pasadena Lines: 25 Message-Id: <3s4u51$fmk@gap.cco.caltech.edu> References: Nntp-Posting-Host: accord.cco.caltech.edu X-Newsreader: NN version 6.5.0 #12 (NOV) "Jerry Bryan" writes: >For the first time in years, I have seen 3x3x3 cubes for sale in a >store. (For American readers, the store is Hills, which is a regional >chain which competes against WalMart and Kmart. The price is $8.97 -- >U.S. dollars.) The box has a note signed by Erno Rubik, and gives >the proper size of the cube group. Also, and I couldn't see inside >the box to verify this, the Face centers seemed to be marked in such >a way as to support the Supergroup. Rubik's note about the size of >the problem says it is 4^4 times bigger than the regular problem. >The manufacturer (or maybe distributor?) is Matchbox Toys (or is it >Matchbook Toys?). What you saw was "Rubik's Cube 4th Dimension." This was released some time in the early 90s. The only difference between this and the typical 3x3x3 is that four faces have pictures on their center squares, requiring the solver to orient them correctly. I belive last year there was a European Rubik Puzzle series that rereleased the cube, along with other rubik puzzles. -- -- Wei-Hwa Huang (whuang@cco.caltech.edu) Homepage (under construction): http://www.ugcs.caltech.edu/~whuang/ I have a proof that NP = P; however, it requires exponential time to write down. From pbeck@pica.army.mil Tue Jun 20 08:00:48 1995 Return-Path: Received: from COR6.PICA.ARMY.MIL by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19315; Tue, 20 Jun 95 08:00:48 EDT Date: Tue, 20 Jun 95 8:00:16 EDT From: Peter Beck (FSAC) To: cube-lovers@ai.mit.edu Subject: cube availability Message-Id: <9506200800.aa03200@COR6.PICA.ARMY.MIL> In Feb at the NY international toy show the KOOSH people told me that they had bought the marketing rights for RUBIK items from Western Publishing (the Matchbox LABEL). They had new packaging and were changing the items to be sold (eg, they said they were going to sell Rubuik's amgic). So the items seen in Hills with Matchbox packaging is old stock that will soon no longer be around. THE FUTURE IS PUZZLING, BUT CUBING IS FOREVER !!! pete beck From mschoene@math.rwth-aachen.de Tue Jun 20 08:40:27 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21537; Tue, 20 Jun 95 08:40:27 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sO2aR-000MP0C; Tue, 20 Jun 95 14:39 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sO2aQ-00026zC; Tue, 20 Jun 95 14:39 WET DST Message-Id: Date: Tue, 20 Jun 95 14:39 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Sun, 18 Jun 1995 15:55:22 -0400 (EDT) Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? Jerry wrote in his message of 1995/06/17 We define the real size of cube space to be the number of M-conjugate classes {m'Ym} for m in M, set of 48 rotations and reflections of the cube, and for Y in G. Dan Hoey has calculated the real size of cube space using the Polya-Burnside theorem. Dan and I (mostly Dan) have also calculated the same result using exhaustive computer search. The computer search is much less elegant than the Polya-Burnside results, but the search does provide additional information, such as the number of positions associated with each symmetry group. The results from the computer search have not yet been posted to the list, but a draft paper is in progress. In the meantime, it occurs to me that perhaps -- but only perhaps -- there is a third way to calculate the real size of cube space. The third way would not require (much) computer searching, but would provide the same level of detail about number of positions per symmetry group as does the full blown search. ... detailed description of the method deleted ... And he continued in his message of 1995/06/18 I should have said "a fourth way", I think. Martin Schoernert performed the same calculation with GAP. Hence, we have three ways in hand: 1) Dan's Polya-Burnside method, 2) Martin's GAP calculations, and 3) brute force computer search. My new proposal would then be a fourth way. Well, I don't know about *four* ways. Dan used the Polya-Burnside theorem. That is, he computed the number of M-conjugacy classes as the average number of fixed points of the elements of M w.r.t. to their action on G. He computed the number of fixed points of an element m using clever arguments about the cycle structure of elements of G that m would fix. I simply observed that the number of fixed points of an element m is the size of the centralizer of in G, and then used GAP to compute those. So I don't think it is correct to call this a way of its own. The method you propose is indeed different from Dan's method that uses Polya-Burnside. I can't figure out how the brute force computer search works. So I can't tell whether it is really different from the other methods (and if indeed it is a method to compute the real size of the cube ;-). Jerry, could you say a little bit more about this computation? It appears to me that Dan and Jim Saxe must have realized all the important pieces for your new method when they wrote their seminal ``Symmetry and Local Maxima (long message)'' message of 1980/12/14. As Jerry points out, they did calculate the important values for 9 of the 33 conjugacy classes of subgroups of M (those whose sizes are a multiple of 12). It is neither clear from their message how they found those 9 classes (in fact they apparently found all 98 subgroups of M), nor how they computed the numbers of elements of G that have a specific subgroup of M as symmetry group. Perhaps Dan can say a little bit more about this? Jerry continued in his message of 1995/06/18 Here is a question for Martin: is there any way with GAP to calculate the number of M-conjugacy classes associated with each symmetry class? It is this additional information about the "real cube space" which *is* available via computer search, and for which I am proposing an alternative which does not involve computer search. Of course there is ;-). Given a subgroup H of M, the centralizer of H in G is the set of all those elements of G that commute with all elements of H. But this is of course simply the set of those elements of G that have either H or a larger group as their symmetry group. So we can compute the numbers of elements of G with symmetry group H by computing the size of the centralizer of H in G, and then subtracting the numbers of those elements that have a symmetry group that is a proper supergroup of H. This is easy if we compute those numbers for all subgroups of M, from the larger subgroups down to the smaller. Of course it is not neccessarry to do this for all 98 subgroups of M, but only for one subgroup for each of the 33 conjugacy classes. Then if we simply divide the number of elements with symmetry group H by the index of H in M, we obtain the number of M-conjugacy classes into which those elements fall. As a GAP program this looks as follows # compute the conjugacy classes of subgroups of M classes := ConjugacyClassesSubgroups( M ); numbers := []; # for all conjugacy classes of subgroups of M for i in [Length(classes),Length(classes)-1..1] do # select a representative for this conjugacy class rep := Representative( classes[i] ); # compute how many elements have at least this symmetry group number := Size( Centralizer( G, rep ) ); # subtract the number of elements that have a larger symmetry group for k in [Length(classes),Length(classes)-1..i+1] do for sub in Elements( classes[k] ) do if IsSubgroup( sub, rep ) then number := number - numbers[k]; fi; od; od; # store the number numbers[i] := number; # print the number of the class Print( i, ":\t" ); # the size of the subgroups in the class Print( Size(rep), "\t" ); # the number of subgroups in the class Print( Size(classes[i]), "\t" ); # the number of elements whose symmetry group lies in the class Print( Size(classes[i]), " * ", number, "\t" ); # and the number of M-conjugacy classes of those elements Print( Size(classes[i]), " * ", number, " / ", Index(M,rep), "\n" ); od; *Do not try this with GAP 3.4.2 (our latest release).* GAP 3.4.2 contains several naive functions for permutation groups, that cause this computation to take a very long time. But with GAP 3.5 (our current development version), this produces in about a minute the following table. CLASS SIZE LENGHT NUMBER REAL NAME 33: 48 1 1 * 4 1 * 4 / 1 (M) 32: 24 1 1 * 0 1 * 0 / 2 (C) 31: 24 1 1 * 0 1 * 0 / 2 (AM) 30: 24 1 1 * 20 1 * 20 / 2 (H) 29: 16 3 3 * 124 3 * 124 / 3 (X1,X2,X3) 28: 12 4 4 * 12 4 * 12 / 4 (T1,T2,T3) 27: 12 1 1 * 48 1 * 48 / 4 26: 8 3 3 * 384 3 * 384 / 6 25: 8 3 3 * 1408 3 * 1408 / 6 24: 8 3 3 * 2944 3 * 2944 / 6 23: 8 3 3 * 1920 3 * 1920 / 6 22: 8 3 3 * 384 3 * 384 / 6 21: 8 3 3 * 896 3 * 896 / 6 20: 8 1 1 * 11892 1 * 11892 / 6 19: 6 4 4 * 416 4 * 416 / 8 18: 6 4 4 * 32 4 * 32 / 8 17: 6 4 4 * 7740 4 * 7740 / 8 16: 4 6 6 * 96232 6 * 96232 / 12 15: 4 6 6 * 96256 6 * 96256 / 12 14: 4 3 3 * 92928 3 * 92928 / 12 13: 4 3 3 * 437504 3 * 437504 / 12 12: 4 3 3 * 574208 3 * 574208 / 12 11: 4 3 3 * 1163520 3 * 1163520 / 12 10: 4 3 3 * 144640 3 * 144640 / 12 9: 4 3 3 * 62208 3 * 62208 / 12 8: 4 1 1 * 280272 1 * 280272 / 12 7: 3 4 4 * 3770864 4 * 3770864 / 16 6: 2 6 6 * 424415168 6 * 424415168 / 24 5: 2 6 6 * 2547748032 6 * 2547748032 / 24 4: 2 3 3 * 15285460992 3 * 15285460992 / 24 3: 2 3 3 * 18342768640 3 * 18342768640 / 24 2: 2 1 1 * 45862360944 1 * 45862360944 / 24 1: 1 1 1 * 43252003109885814336 1 * 43252003109885814336 / 48 As expected the numbers in the fourth column add to the size of G. And the numbers in the fifth column add to 901083404981813616, the real size of the cube (|M\MG/M|). For those classes that I could identify I have added their names. If somebody could describe Dan's taxonomy, I will name the other classes as well. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From mschoene@math.rwth-aachen.de Tue Jun 20 09:29:59 1995 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23574; Tue, 20 Jun 95 09:29:59 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0sO2bG-000MPBC; Tue, 20 Jun 95 14:40 MET DST Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0sO2bF-00026zC; Tue, 20 Jun 95 14:40 WET DST Message-Id: Date: Tue, 20 Jun 95 14:40 WET DST From: "Martin Schoenert" To: Cube-Lovers@ai.mit.edu Cc: BRYAN@wvnvm.wvnet.edu In-Reply-To: "Jerry Bryan"'s message of Sun, 18 Jun 1995 15:55:22 -0400 (EDT) Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? Looking through the old messages about the real size of the cube group, it appeared to me that no one has shown a proof for the Polya-Burnside theorem. Since it is not difficult to prove, I decided to write one up. In the following I will use TeX notation for formulae, i.e., formulae are included in '$' signs, '{}' are used to group terms, '^' is used for superscripts, and '_' for subscripts. If $g \in G$, then I denote the set of elements that are really equivalent to $g$ by $g^M$. Jerry denotes this set by {m'gm}, but $g^M$ is the more common notation in group theory. The sum $\sum_{g \in h^M}{1/|g^M|}$ is simply 1, since it is the sum over all elements in one M-conjugacy class (h^M) of 1 over the length of that M-conjugacy class. Thus the sum $\sum_{g \in G}{1/|g^M|}$ is the number of M-conjugacy classes. Now we need a standard lemma from group theory, which tells us that the length of a class $g^M$ of an element $g$ under the action of a group $M$ is equal to the size of the group $M$ divided by the size of the subgroup of those elements of $M$ that fix $g$ (more precisely the index of that subgroup in $M$, since the lemma is true, even if $M$ is infinite). So using Jerry's notation this lemma gives $1/|g^M| = |Symm(g)|/|M|$. Applying that to the above formula we see that the number of M-conjugacy classes in $G$ is $\sum_{g \in G} {|Symm(g)|/|M|}$ Or, after a trivial change, $1/|M| \sum_{g \in G} {|Symm(g)|}$. Assume that $(g^m == g)$ is 1 if $g^m$ is equal to $g$ and 0 otherwise. Then we have $|Symm(g)| = \sum_{m \in M}{(g^m == g)}$. Thus the number of M-conjugacy classes is $1/|M| \sum_{g \in G} \sum_{m \in M} {(g^m == g)}$. Now we can simply change the order of the two summations, so we get $1/|M| \sum_{m \in M} \sum_{g \in G} {(g^m == g)}$. But of course $\sum_{g \in G} {(g^m == g)}$ is obviously the number of fixpoints of $m$. So we obtain the Polya-Burnside lemma: ``The number of M-conjugacy classes is the average number of fixpoints of the elements of $M$ w.r.t. their operation on $G$''. However, here the operation is special, so we can simplify even further. $g^m$ here means $m^{-1} g m$, so $(g^m == g)$ means $(m^{-1} g m == g)$, which is equivalent to $(m == g^{-1} m g)$ (multiply the equation first by $m$ and then by $g^{-1}$ from the left), which is $(m == m^g)$. So the number of M-conjugacy classes is $1/|M| \sum_{m \in M} \sum_{g \in G} {(m == m^g)}$. But $\sum_{g \in G} {(m == m^g)}$ is simply the size of the subgroup of those elements in $G$ that fix $m$. This is the centralizer of $m$ in $G$. So the number of M-conjugacy classes is finally $1/|M| \sum_{m \in M} |Centralizer(G,m)|$. This is the formulation that I used to compute the real size of the cube group with GAP. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany From BRYAN@wvnvm.wvnet.edu Tue Jun 20 18:04:35 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA25438; Tue, 20 Jun 95 18:04:35 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8764; Tue, 20 Jun 95 13:48:52 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 4745; Tue, 20 Jun 1995 13:48:53 -0400 Message-Id: Date: Tue, 20 Jun 1995 13:48:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/20/95 at 14:39:00 from , Martin.Schoenert@Math.RWTH-Aachen.DE On 06/20/95 at 14:39:00 Martin Schoenert said: >I can't figure out how the brute force computer search works. >So I can't tell whether it is really different from the other methods >(and if indeed it is a method to compute the real size of the cube ;-). >Jerry, could you say a little bit more about this computation? I look at corners and edges separately, and then combine the results. I can't speak for how Dan did his corner search and his edge search, but I can describe mine. Our results do match, which is always nice. Dan did most of the figuring out of "combining the results" for corners and edges. Conceptually, I look at every single corner position X and calculate Symm(X), and I look at every single edge position Y and calculate Symm(Y). In practice, there are some important shortcuts. I have a data base containing "every" corner position and a second data base containing "every" edge position with "every" defined in the following sense. The set of all positions is partitioned into equivalence classes of the form {m'Xmc} for the corners and {m'Ymc} for the edges, for m in M (48 rotations and reflections) and c in C (24 rotations). The data bases contain a representative element from each equivalence class. For all the cases where |{m'Xmc}|=1152 and |{m'Ymc}|=1152, no searching is required. For these cases, we know *a priori* that there are 24 M-conjugacy classes containing 48 elements each, and that for each of the 1152 positions we have Symm(X)=I and Symm(Y)=I. Fortunately, for the vast majority of the cases (over 97% for corners and well over 99% for edges), the so-called B-class size is 1152. Suppose the representative for {m'Xmc} is V and for {m'Ymc} is W. For the remaining cases, the idea is to use Vc and Wc for each fixed c in C as a base or representative element for an M-conjugacy class of the form {m'(Vc)m} and {m'(Wc)m}. The tricky part here is that while Vc and Vd are distinct when c and d in C are not equal, nonetheless Vc and Vd may be M-conjugate. Hence, we use each Vc and Wc which are distinct up to M-conjugacy as a representative element for an M-conjugacy class. Conceptually, we calculate Symm(T) for each T in {m'(Vc)m} for each fixed c (and for Vc distinct up to M-conjugacy) and summarize the results. In practice, it is sufficient to calculate Symm(Vc). Here comes another tricky part (at least it was until I figured it out). Initially, I assumed that Symm(T) was the same for all T in {m'(Vc)m}. However, such is not the case. Rather, if you calculate each Symm(T), you will discover that each one will be a group from a class of conjugate groups, and that each group in the class of conjugate groups will appear an equal number of times. Given that you have picked an arbitrary representative from the M-conjugacy class, you don't know which conjugate group you are going to get when you calculate Symm(m'(Vc)m). I got around this issue originally by calculating a representative group (the 98 subgroups of M then have 33 representative groups, one for each of the 33 symmetry classes). In the work I am doing now, I map directly from each of the 98 subgroups to their respective symmetry class. The exhaustive search then consists of performing the above calculations for each B-class of the form {m'Xmc} and {m'Ymc} and summarizing the results (that is, counting all the M-conjugacy classes). In addition to an overall total, you summarize by symmetry class, which gives you the additional information I have talked about, over and above Dan's Polya-Burnside results. At this point, the summary results give you the real size of the corners only problem and the real size of the edges only problem. While you are at it, you have to count the M-conjugacy classes for even positions and odd positions separately in order to put corners and edges together properly. Finally, you put the corners and edges together in all possible ways. The putting together of the corners and edges is described in the draft I have mentioned, so I will just wait until the draft is ready before posting the rest. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From hoey@aic.nrl.navy.mil Thu Jun 22 03:44:23 1995 Return-Path: Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA02954; Thu, 22 Jun 95 03:44:23 EDT Received: by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA25013; Thu, 22 Jun 95 03:43:59 EDT Date: Thu, 22 Jun 95 03:43:59 EDT From: hoey@aic.nrl.navy.mil (Dan Hoey) Message-Id: <9506220743.AA25013@Sun0.AIC.NRL.Navy.Mil> To: Cube-Lovers@ai.mit.edu, "Martin Schoenert" Subject: Ways to Calculate the Real Size of Cube Space? In-Reply-To: Martin Schoenert says: > I can't figure out how the brute force computer search works. and while Jerry Bryan gives one answer, I have another. For you see, I also ran a brute force computer search for the symmetry classes, too. And while it agrees with Jerry's answers, mine was a significantly different algorithm, which I discuss a little a the end of this message. > It appears to me that Dan and Jim Saxe must have realized all the > important pieces for your new method when they wrote their seminal > ``Symmetry and Local Maxima (long message)'' message of 1980/12/14. > As Jerry points out, they did calculate the important values for > 9 of the 33 conjugacy classes of subgroups of M (those whose sizes > are a multiple of 12). It is neither clear from their message how > they found those 9 classes (in fact they apparently found all 98 > subgroups of M), nor how they computed the numbers of elements of > G that have a specific subgroup of M as symmetry group. > Perhaps Dan can say a little bit more about this? First, to find all the subgoups of M. I represented the elements of M as a list of permutations on faces, found easily enough by finding the closure of the generators. Then I did a depth-first search for subgroups, branching by computing the closure of the current subgroup with each possible element not in that subgroup, and cutting off the search on previously-seen subgroups. It's quick, it's dirty, it works. I found the nine subgroups of order a multiple of 12, as shown in the Hasse subgroup diagram: order 48 . . . M_ /|\\_ / | \ \_ / | \ \_ order 24 . A . H . C \_ \ | / \_ \ | / \_ \|/ \ order 12 . . . E . . . . .T[1..4] (except we called E "AC" back then). The trick then was to find all the E-symmetric positions and all the T1-symmetric positions; the tasks of finding the full symmetry group of such positions and counting the positions for T2, T3, and T4 were straightforward. The best tool we had for figuring out symmetric positions was essentially the one I wrote about in ``The real size of cube space'' on 4 Nov 94. For a subgroup J of M, if a position g is J-symmetric, then g must commute with each operation m in J. Recall: The fundamental principle we use in finding whether g commutes with m can be found by examining the cycles of m. Suppose m permutes a cycle (c1,c2,...,c[k-1],ck), so that c2=m(c1), c3=m(c2), ..., ck=m(c[k-1]), and c1=m(ck). For g to commute with m, we have g(c2)=m(g(c1)), g(c3)=m(g(c2)), ..., g(ck)=m(g(c[k-1])), and g(c1)=m(g(ck)). So (g(c1),g(c2),...,g(ck)) is also a cycle of m. Thus g must map each k-cycle of m to another k-cycle of m, and in the same order. The orientation question was a lot more difficult, so we ran through a bunch of little results. The following is a cleaned-up sample of the sort of arguments, as I remember them. In it FRD is an unoriented corner cubie/cubicle and FRD.D is its down-facing color-tab/facicle. ================================================================ Lemma 1: Suppose X and Y are corners, and m is in C, m(X)=Y. Suppose g(X)=X and g(Y)=Y, and g commutes with m, Then g applies the same twist to corners X and Y. Proof: Let TX,TY be the clockwise 120-degree rotation of corners cubies X and Y, respectively. Then m(TX(.))=TY(m(.)), as can be seen by the fact that we could apply a twist to X in place (TX) before moving it to Y with m, or we can perform the same twist on Y (TY) after moving it. So if g(X)=TX^k(X), then g(Y)=g(m(X))=m(g(X))=m(TX^k(X))=TY^k(m(X))=TY^k(Y). performing the same twist on Y as on X, QED. Lemma 2: If g is E-symmetric, then each corner cubie remains in its home cubicle (not considering orientation). Proof: Supposing otherwise, take a moved cubie (without loss of generality) to be FRD, and suppose (w.l.o.g.) it moves to one of locations FDL, FLT, or BTL. Case 1. If g(FRD)=FDL, consider operation m to be the 120-degree rotation about FRD. m(FRD)=FRD, m(FDL)=FTR. So g(FRD)=g(m(FRD))=m(g(FRD))=m(FDL)=FTR, contradicting g(FRD)=FDL. Case 2. If g(FRD)=FLT, then take m as in case 1; m(FLT)=BRT, so g(FRD)=g(m(FRD))=m(g(FRD))=m(FLT)=BRT, contradicting g(FRD)=FDL. Case 3. If g(FRD)=BTL, then g(FRD.F) is BTL.B, BTL.T, or BTL.L. Case 3a. If g(FRD.F)=BTL.B, then g(FRD.R)=BTL.T, by clockwise adjacency. But m(FRD.F)=FRD.R and m(BTL.B)=BTL.L, and g(FRD.R)=g(m(FRD.F))=m(g(FRD.F))=m(BTL.B)=BTL.L contradicts G(FRD.F)=BTL.B. Cases 3b and 3c work the same way. The contradictions establish that FRD does not move, QED. Lemma 3: If g is E-symmetric, then the twists of the four corner cubies FRD, FLT, BLD, and BRT agree with each other, and and the other four also agree with each other. Proof: For any two of the four corners (e.g. FRD, FLT), there is a 120-degree rotation in E taking one to the other (e.g. the rotation about FTR). Lemma 1 applies immediately to show the twists agree, QED. Lemma 4. If g is E-symmetric, then the corner cubies are all solved, or are rotated alternately in opposite directions. Proof: From Lemma 2, all the cubies are in their home cubicles. If one of the sets from Lemma 3 is twisted, their total twist is the twist of a single cubie (since there are four of them) and so must be counteracted by having the other set twisted in the opposite direction, which is alternate corners twisted oppositely; otherwise all corner are solved, QED. Lemma 5. Let T1 refer to the group that fixes the FRD-BTL axis. Then any T1-symmetric position g must keep the FRD and BTL cubies in their solved position, and rotated by the same amount. Proof: Let m be the 120-degree rotation about FRD, which is in T1. Since FRD and BTL are the only 1-orbits of m, they are kept in place or swapped. From the proof of Lemma 2 (which uses the same m) they cannot be swapped. Otherwise, the two cubies are kept in place, and 180 degree rotation about the FL-BR axis, also in T1 fulfills the requirements of Lemma 1 to show they will both be rotated the same amount, QED. ================================================================ That's about all I feel like remembering and formalizing right now. As you can see, it's long, mechanical, and boring. That's why we never got around to writing it all down. Early last year I wrote a computer program to find *all* the symmetry groups of *all* the positions. The first part did the corners and edges separately, counting the number of positions for each symmetry group and permutation parity. For each of the 8! or 12! permutations, I checked to see if the permutation commuted with some nontrivial operation of M; if not, I just counted the appropriate number of I-symmetric positions. Otherwise I applied orientations and counted up the symmetry groups of each possible orientation. (It could probably have been made to go faster, by cutting off partial permutation or orientation generation as soon as all the non-trivial operations were ruled out.) In the above counts, I also kept track each time of whether the permutation was even or odd. Then after I had the count of even and odd permutations for corners and edges in each symmetry group, I had a program that intersected the symmetry groups, and for each pair of subgroups J,K, and each parity P, I added Corners[J,P] * Edges[K,P] to Whole[J intersect K, P], with some fancy footwork so I only needed to deal with conjugacy classes for J and K. Then for each K, the number of whole K-symmetric positions was Whole[K,Even]+Whole[K,Odd]. The program finished about the time I realized the application of the Polya-Burnside theorem. Then for most of the year, I put off writing it all up. I have Jerry to thank for reminding me to get with it, and for useful comments and discussions on the early drafts. Dan From @secyt.gov.ar,@recom.edu.ar:administ@marben.recom.edu.ar Mon Jun 26 11:47:47 1995 Return-Path: <@secyt.gov.ar,@recom.edu.ar:administ@marben.recom.edu.ar> Received: from secyt.secyt.gov.ar ([200.9.244.2]) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05732; Mon, 26 Jun 95 11:47:47 EDT Received: by secyt.gov.ar with UUCP id <12188>; Mon, 26 Jun 1995 12:43:07 -0300 Received: from marben.recom.edu.ar by recom.recom.edu.ar with bsmtp (Smail3.1.28.1 #2) id m0sQFwa-0003qDC; Mon, 26 Jun 95 12:19 ARG Received: by marben.recom.edu.ar (UUPC/extended 1.11q(RAN-0.95)); Mon, 26 Jun 1995 11:57:35 ARG Received: by marben.recom.edu.ar (UUPC/extended 1.11q(REC-1.30)); Mon, 26 Jun 1995 11:57:35 ARG Date: Mon, 26 Jun 1995 08:57:35 -0300 From: "Marcelo Daniel Benveniste " Message-Id: <2feecadf.marben@marben.recom.edu.ar> To: cube-lovers@life.ai.mit.edu X-Mailer: RECMAIL [UUPC/extended 1.11q(REC-1.30)] Subject: Subscription request Please, subscribe-me to this list --- Marcelo Daniel Benveniste administ@marben.recom.edu.ar From BRYAN@wvnvm.wvnet.edu Tue Jun 27 23:22:41 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23088; Tue, 27 Jun 95 23:22:41 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 1105; Tue, 27 Jun 95 23:22:51 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9315; Tue, 27 Jun 1995 23:22:51 -0400 Message-Id: Date: Tue, 27 Jun 1995 23:22:50 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Constructing K-symmetric Cubes This is a followup on several recent messages concerning the question of K-symmetric cubes, where K is one of the 98 subgroups of M. We recall that a permutation is a special kind of function, namely a one-to-one and onto function on a set. A very common technique used with functions is to restrict the domain to a (usually proper) subset of the original domain. In the paradigm of a function as a general rule, the general rule is applied to the subset of the domain to obtain the restriction of the function. In the paradigm of a function as a set of ordered pairs, the restriction is simply a (usually proper) subset of the set of ordered pairs. A restriction of a permutation is usually not a permutation (certainly not a permutation on the original domain), but it is still a function. We will treat a permutation on the cube as a set of restrictions (functions) of the several cubicles. We take as our first example the function UFL->UFL. Unlike cycle notation, it is not assumed that the other cubicles are fixed; rather, the other function values are undefined (e.g., URF->?). Let X be any function (not necessarily a permutation) whose domain is some subset of the cubicles. We define Symm(X) in the standard fashion -- Symm(X) is the set of all m in M such that m'Xm=X. If X is the function UFL->UFL, we have Symm(X)=AT4, not Symm(X)=M as you might expect. AT4 is a subgroup in Dan's taxonomy containing six elements, and which has an axis of symmetry along the UFL-DBR axis. This definition of Symm(X) perhaps requires a minor bit of justification. In a function composition such as FG (left-to- right notation) or G(F(x)) (right-to-left "calculus" notation), it is sometimes taken as a convention that the range of F must match the domain of G. But we can also take the restriction of G to the intersection of the range of F with the domain of G, and we do so. Having done so, Symm(X) is well defined. We wish to build an M-symmetric permutation containing the function UFL->UFL, but Symm(UFL->UFL)=AT4 is not a very auspicious start. Rather, we define the conditions under which a function is K-symmetric is follows. A function X is K-symmetric if the union of the K-conjugates k'Xk is a function. Given this definition, the only M-symmetric function on UFL is in fact UFL->UFL, so we really have made a good start. Furthermore, any M-symmetric permutation that contains UFL->UFL must also contain all the M-conjugates of UFL->UFL, and the union of the M-conjugates is simply the identity permutation on the corners. The fact that Symm(UFL->UFL)=AT4 can be of some benefit in our investigations. In particular, the fact that |AT4|=6 means that there are 8 M-conjugates, so taking all the M-conjugates of UFL->UFL means that all 8 corners are specified. Our next example will be UFL->LUF (a twist of the corner). In this case, we have Symm(X)=ET4. ET4 is the subgroup of M in Dan's taxonomy which contains 3 elements including the identity plus the 1/3 and 2/3 rotations around the UFL-DBR axis. Of more import, UFL->LUF is not M-symmetric. However, it is both C-symmetric (C is the set of 24 rotations) and H-symmetric (H is the set of 12 even rotations and 12 odd reflections). As an aside, we note that it is not A-symmetric, where A is the set of 24 even rotations and reflections. But since it is both C-symmetric and H-symmetric, there is not a unique largest subgroup K for which we can say it is K-symmetric. It is easy to see that UFL->LUF is not M-symmetric. The set of M-conjugates contains both UFL->LUF and UFL->FLU, so the union of the M-conjugates is not a function. We can see the same thing from the fact that |ET4|=3. Since |ET4|=3, there are 16 M-conjugates, but there are only 8 corners to represent the 16 M-conjugates. Since UFL->LUF is C-symmetric, let's see if we can build a C-symmetric permutation. There are 8 C-conjugates (a good start!), and the 8 C-conjugates twist each of the 8 corners by 1/3 in the same direction. Hence, this is a C-symmetric but not M-symmetric permutation. Of course, it is an "illegal" position in the sense that it is not in the same orbit as Start. In the same manner, we can build a K-symmetric permutation for any K. We start with a K-symmetric function on a single cubicle. (A function which is K-symmetric is L-symmetric for any L which is a subgroup of K). We include all K-conjugates. If the cube is completely specified, we stop. Otherwise, we choose another K-symmetric function for any previously unspecified cubicle, add in the new K-conjugates, and so forth, repeating until the entire permutation is specified. Needless to say, this construction process suffers from not preserving orbit. Additional steps must be taken to assure that the constructed position is in the desired orbit (usually, the Start orbit). And some orbits do not have representatives from some subgroups, for example it is well known that there are no C-symmetric but not M-symmetric permutations in the Start orbit. To use Dan's adjectives, this process very quickly can become long, mechanical, and boring. But I now see how to build a K-symmetric permutation for any K. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From BRYAN@wvnvm.wvnet.edu Wed Jun 28 12:52:18 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA20782; Wed, 28 Jun 95 12:52:18 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4454; Wed, 28 Jun 95 10:09:51 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 0089; Wed, 28 Jun 1995 10:09:52 -0400 Message-Id: Date: Wed, 28 Jun 1995 10:09:51 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Re: A Third Way to Calculate the Real Size of Cube Space? In-Reply-To: Message of 06/20/95 at 14:39:00 from , Martin.Schoenert@Math.RWTH-Aachen.DE I guess I am going to have to break down and get a copy of GAP. It is truly impressive how much GAP can do so easily. My interpretation of Martin's GAP program is that it implements the general outline of the algorithm I described, except that GAP was able to calculate the number of K-symmetric permutations in a very simple and direct way, whereas I was going to have to puzzle each one out by hand. The heart of Martin's program appears to be the following, and I have a couple of questions. > # compute how many elements have at least this symmetry group > number := Size( Centralizer( G, rep ) ); The first question is: how does the Size function work? As a simpler example than the one above, what if you simply say Size(G)? I am naively assuming that G is specified to GAP in terms of generators only, and that it makes no attempt to actually represent each element of G (too big!). And I have seen snippets of GAP libraries for G posted by Mark Longridge, and they look like generators. I have been in several group theory books lately, and I don't recall seeing a general algorithm presented for determining the size of a finite group based on its generators. The second question is like unto the first: how does the Centralizer function work? In this particular case, we don't really need the Centralizer, we only need the Size of the Centralizer, but the question remains in either case. Surely, GAP does not literally try each element of G and each element of rep to see which elements commute (too big again). So what is the general algorithm? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU From rodrigo@lsi.usp.br Wed Jun 28 17:15:45 1995 Return-Path: Received: from ofelia.lsi.usp.br (lsi.poli.usp.br) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06724; Wed, 28 Jun 95 17:15:45 EDT Received: from mozart.lsi.usp.br (mozart.lsi.usp.br [143.107.3.237]) by ofelia.lsi.usp.br (8.6.12/8.6.9) with ESMTP id SAA21488; Wed, 28 Jun 1995 18:13:40 -0300 Received: (from rodrigo@localhost) by mozart.lsi.usp.br (8.6.12/8.6.9) id VAA12159; Wed, 28 Jun 1995 21:12:54 GMT Date: Wed, 28 Jun 1995 14:12:53 +48000 From: Rodrigo de Almeida Siqueira To: steinark@ifi.uio.no, bagleyd@source.asset.com, Reinaldo Augusto da Costa Bianchi Cc: cube-lovers@life.ai.mit.edu Subject: Rubik Cube... for Windows ? In-Reply-To: <199506281753.OAA19377@jaguar.lsi.usp.br> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII On Wed, 28 Jun 1995, steinark@ifi.uio.no wrote: > id: Email: steinark@ifi.uio.no > comments: > Just thought I would mention a Windows version Rubic Cube > program. It's located at several ftp sites and the file > to look for is called 'cubic.zip'. Has a far better > look-and-feel than XRubic I think. It has its own solving > algorithm, but this can be rather slow... > Check it out. If you don't find it I can probably send it > to you some how. > > Steinar Hello Steinar, Would you tell me where (ftp site) can I find cubic.zip ? I would like to try it and put a link in the "Robot can play with the Cube Web homepage" (http://www.lsi.usp.br/~daia/celula/cubo/) Thank you, Rodrigo Siqueira rodrigo@lsi.usp.br (http://www.lsi.usp.br/usp/rod/rod.html) From @mail.uunet.ca:mark.longridge@canrem.com Mon Jul 3 14:15:37 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06664; Mon, 3 Jul 95 14:15:37 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <181844-6>; Mon, 3 Jul 1995 14:17:04 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA20578; Mon, 3 Jul 95 14:11:10 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E9C05; Mon, 3 Jul 95 14:04:58 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Antislice Patterns From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1181.5834.0C1E9C05@canrem.com> Date: Mon, 3 Jul 1995 14:53:00 -0400 Organization: CRS Online (Toronto, Ontario) Patterns in the Anti-Slice Group -------------------------------- p4 8 flip (Op sides) (R1 L1 U1 D1 F1 B1) ^2 (12) p10a pons asinorum (L3 R1 U3 D1)^3 (12) p16a 4 cross order 2 F1 B1 U1 D1 L2 R2 U1 D1 F1 B1 U2 D2 (12) p17 4 diagonal (F1 B1 R1 L1) ^3 (12) p18a 4 diagonal,2 cross (F1 B1 R3 L3) ^3 (12) p22 2 DOT, 2 Stripe R1 L1 U2 D2 R3 L3 (6) p64a 4 Z F1 B1 L3 R3 F1 B1 L1 R1 F3 B3 L1 R1 (12) p143 Pinwheels F1 B1 L1 R1 F3 B3 U3 D3 L1 R1 U1 D1 (12) p175a 6 H order 2 U3 D3 L3 R3 F2 B2 U2 D2 L3 R3 U1 D1 (12) p198a 2 X, 4 Diag no C L1 R1 F1 B1 L3 R3 F3 B3 L1 R1 F1 B1 (12) p201 Pinwheels + Pons L1 R1 F3 B3 L1 R1 U3 D3 F1 B1 U3 D3 (12) p201 is a quite interesting position. The square's group equivalent is no shorter in q turns: p175 6 H order 2 type 2 U2 B2 L2 U2 D2 L2 F2 U2 (8) Note that p201 = |{m'Xm}|=2 and |Symm(X)|=24. From @mail.uunet.ca:mark.longridge@canrem.com Mon Jul 3 14:15:29 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA06658; Mon, 3 Jul 95 14:15:29 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <181839-4>; Mon, 3 Jul 1995 14:16:59 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA20583; Mon, 3 Jul 95 14:11:12 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1E9C06; Mon, 3 Jul 95 14:04:58 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Crazy Corner Pattern Revisited From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1182.5834.0C1E9C06@canrem.com> Date: Mon, 3 Jul 1995 14:59:00 -0400 Organization: CRS Online (Toronto, Ontario) Jerry writes: > In their position, the UFL and RBD corners are >in place, and the other three pairs are swapped. The "girdle" >includes the three pairs that are swapped. Hence, there >is an axis of symmetry along the UFL-RBD axis. The odd number of >swaps is compensated by an odd number in the edges. The compensation >is not required for corners only. > > R D D > U U D > U U B > > D L L F F D L L F L F F > R L L F F B L R R B B F > R R B R B B U R R B B U > > U U L > U D D > F D D Much is explained! (And this is all the way back in file "cube01" in the archives, Date: 14 December 1980 1916-EST). What does the 1916- mean in front of EST??? If we put the edges in place in Jerry's (and Dan's) position then we have a position M-conjugate to the one I posted. (re-posted below). F U U U U U D U L R L B R F B D R R B B D L L L F F F R R R B B B U L L F F U L R F L B F D D B D D D R D U Jerry Continues: >...there is an axis of symmetry along the UFL-RBD axis. The odd >number of swaps is compensated by an odd number in the edges. >The compensation is not required for corners only. Great!