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