From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Mon Jan 10 23:08:35 1994 Return-Path: <@mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26716; Mon, 10 Jan 94 23:08:35 EST Message-Id: <9401110408.AA26716@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 5041; Mon, 10 Jan 94 23:08:38 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 4686; Mon, 10 Jan 1994 23:08:38 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 9272; Mon, 10 Jan 1994 23:06:01 -0500 X-Acknowledge-To: Date: Mon, 10 Jan 1994 23:06:00 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: |G\M| - Some Trivial Partial Results It occurs to me that a small part of my incorrect attempt during December to calculate |G\M| can be salvaged. In particular, for those cases where B-conjugate classes are of order 1152, the calculations are trivial. About 99.9923% of the edge conjugate classes and about 96.924% of the corner conjugate classes are of order 1152, so we can calculate the correct number of M-conjugates of G for a very large percentage of the cases. Consider some fixed X in GC\B and some fixed Y in GE\B where |BClass(X)|=1152 and |BClass(Y)|=1152. Form BClass(X) * BClass(Y). Now, |BClass(X) * BClass(Y)| = |BClass(X)| * |BClass(Y)| / 2 = 1152 * 1152 / 2. (The division by 2 takes care of parity). Finally, form (BClass(X) * BClass(Y))\M, and we have |(BClass(X) * BClass(Y))\M| = |BClass(X) * BClass(Y)| / 48 = (1152 * 1152) / 2 / 48 = 13,824. We know the number of BClasses of GC of order 1152 from computer search (namely 75,392), and we know the number of BCLasses of GE of order 1152 from computer search (namely 851,493,140). Hence, for the special case of both BClasses being of order 1152, we have the total number of elements of G\M being 851,493,140 * 75,392 * 13,824 = 887,442,335,689,605,120. We can derive similar results if only one of BCLass(X) and BClass(Y) are of order 1152. For example, there are 4 elements of GE\B for which |BClass(Y)|=24. Choose such a Y, and choose X in GC\B such that |BClass(X)|=1152. Form BClass(X) * BClass(Y). It will be the case that |BClass(X) * BClass(Y)| = 1152 * 24 / 2 = 13,824. Hence, |(BClass(X) * BClass(Y))\M| = 13,824/48 = 288. There are 75,392 values of X for which |BClass(X)|=1152, 4 values of Y for which |BCLass(Y)|=24, and hence there are 75,392 * 4 * 288 = 86,851,584 elements of G\M of the form BClass(X) * BClass(Y) for which |BCLass(X)| = 1152 and |BClass(Y)| = 24. There are nineteen cases in all for which at least one of BClass(X) and BCLass(Y) are of order 1152, and this note calculates only two of the nineteen. Completing the other seventeen would be trivial but tedious. However, a total solution to the problem will require coming up with some way to deal with the cases where neither |BClass(X)|=1152 nor |BClass(Y)|=1152. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?