From BRYAN@wvnvm.wvnet.edu Wed Aug 31 17:17:08 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA18279; Wed, 31 Aug 94 17:17:08 EDT Message-Id: <9408312117.AA18279@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 5436; Wed, 31 Aug 94 15:23:48 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7373; Wed, 31 Aug 1994 15:23:48 -0400 X-Acknowledge-To: Date: Wed, 31 Aug 1994 15:23:47 EDT From: "Jerry Bryan" To: Subject: Re: < U, R> Group In-Reply-To: Message of 08/09/94 at 01:48:00 from mark.longridge@canrem.com On 08/09/94 at 01:48:00 mark.longridge@canrem.com said: > Analysis of the 3x3x3 group > ---------------------------------- > branching >Moves Deep arrangements (q only) factor > 0 1 -- > 1 4 4 > 2 10 2.5 > 3 24 2.4 > 4 58 2.416 > 5 140 2.413 > 6 338 2.414 > 7 816 2.414 > 8 1,970 2.414 > program starts to really bog down after this... > I leave it to Jerry or Dan to check my results. I checked up to 2 >moves deep by hand and verified 10 different positions. Ok, here it is. This search is narrower and deeper than any I have ever done before. Frey and Singmaster give a good bit of attention in their book, pointing out that it is trickier than it might first appear. It is called the Two-Generator Group. The size of the group can be calculated as (7!5!/2)(3^6/3) = 73,483,200. The 3^6 factor accounts for twisting the corners, but there is no 2^n factor as the edges cannot be flipped. These results are in terms of Q turns without any conjugate class checking. I would regard the following as open problems: local maxima, results with Q+H turns, and results in terms of conjugate classes. In this particular case, it would not be M-conjugates. I would have to look at Dan Hoey's 98 subgroups of M to see which subgroup applies to . 0 1 1 4 4 2 10 2.5 3 24 2.4 4 58 2.416 5 140 2.413 6 338 2.414 7 816 2.414 8 1,970 2.414 9 4,756 2.414 10 11,448 2.407 11 27,448 2.398 12 65,260 2.378 13 154,192 2.363 14 360,692 2.339 15 827,540 2.294 16 1,851,345 2.237 17 3,968,840 2.144 18 7,891,990 1.988 19 13,659,821 1.755 20 18,471,682 1.352 21 16,586,822 0.898 22 8,039,455 0.485 23 1,511,110 0.188 24 47,351 0.031 25 87 0.002 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow?