From @mail.uunet.ca:mark.longridge@canrem.com Tue Apr 25 23:05:02 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 AA29970; Tue, 25 Apr 95 23:05:02 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <194293-8>; Tue, 25 Apr 1995 23:06:32 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA03179; Tue, 25 Apr 95 23:01:31 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1DE11D; Tue, 25 Apr 95 22:50:25 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: More Cube Orders From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1107.5834.0C1DE11D@canrem.com> Date: Tue, 25 Apr 1995 23:27:00 -0400 Organization: CRS Online (Toronto, Ontario) I said: > Still trying to find a pattern which will > result in 4 distinct ways, but I am not optimistic. Jerry adds: > As one more followup, for each symmetry group order in the above list, > there exists at least one cube. > That is, 96 of the 98 subgroups are symmetry groups for at > least one cube. The two "missing" subgroups -- A and C -- are of > order 24. But there is a third subgroup -- H -- of order 24 > (H is the set of 12 even rotations and 12 odd reflections), and there > are cube positions whose symmetry subgroup is H. Hence, there are > cube positions for every symmetry subgroup order. Well, I figure Jerry is correct and so I kept looking for the magic pattern which transforms 4 ways... Number of different Pattern patterns ------- --------- ... 4 6 flip (UF, UR, FR, DB, DL, BL) ... So there are 4 types of this 6 flip. Jerry has said before: > I believe that Dan and I have solved (sort of independently, and sort > of working together) the problem you pose (and I give Dan the bulk > of the credit). That is, how many cubs are there in each symmetry > group and each symmetry class? That sounds harder. Looks like I am specifying only the index of the symmetry subgroup... perhaps it makes sense to find out exactly which subgroup of M is the symmetry group of my positions. It all sounds vaguely familar.... but it will try again tomorrow. -> Mark <-