From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Fri Dec 17 14:25:29 1993 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 AA08679; Fri, 17 Dec 93 14:25:29 EST Message-Id: <9312171925.AA08679@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 7700; Fri, 17 Dec 93 14:25:17 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 8890; Fri, 17 Dec 1993 14:25:14 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 2064; Fri, 17 Dec 1993 14:22:36 -0500 X-Acknowledge-To: Date: Fri, 17 Dec 1993 14:22:34 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Size of the Cube Group In-Reply-To: Message of 12/17/93 at 11:21:51 from , BRYAN%WVNVM.BITNET@mitvma.mit.edu On 12/17/93 at 11:21:51 Jerry Bryan said: >However, the alert reader should have noticed a problem. Why did I >not divide by 2 to take into account the fact that odd edge >permutations can only occur with odd corner permutations and vice >versa? Actually, I did, but the division by 2 cancelled. The reason >it canceled is slightly tricky. Also, remember that we are talking >about equivalence classes, not specific cube configurations. Any >equivalence class has both even and odd members, depending on how ^^^^^^^^^^^^^^^^^^^^^^^^^ >the members are rotated. Hence, any corner equivalence class can be >matched up with any edge equivalence class, assuming the rotations >are compatible. But you still have to worry about "dividing by 2", >as follows. It is pretty bad when you have to followup with corrections to your own posts. I hurried to complete the previous post before lunch, and just didn't think clearly enough -- till I had time to think *during* lunch. Let's try this again. A qturn of the whole cube (a 90 degree rotation of the whole cube) is odd. However, if you think of a qturn rotation of the whole cube as disjoint between edges and corners, a qturn rotation of the corners is even, and a qturn rotation of the edges is odd. Hence, for any equivalence class of the corners under M, either the whole equivalence class is even, or the whole equivalence class is odd. For any equivalence class of the edges under M, half of the equivalence class is even and half is odd. Thus, any equivalence class of the corners can occur with any equivalence class of the edges, but with only half the members of the edge equivalence class -- namely those with the same parity. I believe my calculations were correct, but a piece of the justification was not. I hope I am not still missing something. You do have to "divide by 2", and my calculations do indeed "divide by 2" as previously described, but the parity of edges vs. the parity of corners was incorrect in the previous post. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?