From mark.longridge@canrem.com Wed Feb 7 03:00:03 1996 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA13842; Wed, 7 Feb 96 03:00:03 EST Received: by canrem.com (PCB-UUCP 1.1f) id 20A152; Wed, 7 Feb 96 02:54:24 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Cube Musings From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1289.5834.0C20A152@canrem.com> Date: Wed, 7 Feb 96 02:48:00 -0500 Organization: CRS Online (Toronto, Ontario) Further Cube Musings ==================== We usually think of positions antipodal to start only, but there are positions antipodal to any given position. Given a small enough subgroup of the cube, i.e. one which we can exhaustively study, it is not hard to determine some examples. Let's use the square's group and the good ol' pons asinorum. Pons is antipodal to position X. Pons + X = Antipode (let's use position p135) p135 2 X, 4 T L2 B2 D2 F2 T2 F2 T2 L2 T2 D2 F2 T2 L2 D2 F2 Solving for position X is easy enough.... X = Antipode - Pons Position X = F2 D2 L2 D2 F2 L2 T2 F2 T2 F2 T2 F2 L2 The idea of (-1) * pons or (-pons) is equivalent to the inverse of pons, since (+pons) + (-pons) = identity. So Pons and Position X are antipodes of each other. Using this straightforward method we can find an antipode to any position in the square's group, or for any other positions in another small subgroup. This brings up the idea of a "Rubik's Tour". Such a tour would touch on a set of interesting patterns within a given subgroup, or potentially the entire cube group. Of course, "God's Tour" would not only touch on all the interesting patterns, it would also sequence all the patterns AND orient them in space such that the number of q turns would be minimal for the tour! I am currently working on "God's Tour" for some of the lesser subgroups, touching on say a dozen patterns for the square's group. If humans and computers ever resolve "God's Algorithm" there is some solace that there are problems even more intractible. Hmmmm, I just had a thought. It would probably be best to group all the patterns closer to start and work outwards towards the more antipodal ones. With the smaller groups a "Total Tour" would be possible! Visit all elements! -> Mark <-