From @mail.uunet.ca:mark.longridge@canrem.com Sun Oct 23 03:41:06 1994 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 AA09529; Sun, 23 Oct 94 03:41:06 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <86581-4>; Sun, 23 Oct 1994 03:41:30 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA05154; Sun, 23 Oct 94 03:38:20 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1B6751; Sun, 23 Oct 94 03:23:51 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Cross and X's From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.814.5834.0C1B6751@canrem.com> Date: Sun, 23 Oct 1994 02:36:00 -0400 Organization: CRS Online (Toronto, Ontario) ----------------------------- Possible Legal Cross Patterns ----------------------------- Plummer Cross (6 Cross order 3) = 8 patterns Christman Cross (6 Cross order 2) = 6 patterns 4 Cross order 2 (sq group) = 3 patterns 4 Cross order 4 = 6 patterns ----------- 14 Six Cross + 9 Four Cross = 23 total legal Cross patterns There are 0 total cross patterns in the swap orbit ------------------------- Possible Legal X Patterns ------------------------- 6 X order 3 = 8 patterns 6 X order 6 = 8 patterns 6 X order 2 (sq group) = 1 pattern 4 X order 2 (sq group) = 3 patterns 2 X order 2 (sq group) = 3 patterns ----------- 17 Six X + 3 Four X + 3 Two X = 23 total legal X patterns For a while I thought that [6 x order 3] combined with the [2 x pattern] would make a new sort of [6 x order 6], but combining [6 x order 3] with the [2 x pattern] is essentially the same as combining 6 x order 3 with the pons asinorum or 6 x order 2. ------------------------------ Possible Swap-Orbit X Patterns ------------------------------ 6 X order 2 = 6 patterns 6 X order 4 = 6 patterns 4 X order 2 = 6 patterns 4 X order 4 = 6 patterns ----------- 12 Six X + 12 Four X = 24 total swap-orbit X patterns Some description of the swap-orbit patterns is in order. The 6 X order 2 pattern has a 2-cycle of opposite edges and 2 sets of 2-cycles of adjacent edges. The 6 X order 4 pattern has a 2-cycle of opposite edges and a 4-cycle of edges of adjacent faces. The 4 X order 2 has 2 sets of 2-cycles of adjacent edges. The 4 X order 4 has a 4-cycle of edges of adjacent faces. To make any of these swap-orbit patterns one would have to first exchange any 2 edge cubies. Interestingly, a thin line 6 X order 3 is possible on the 5x5x5 cube. No process as yet.... -> Mark <- Email: mark.longridge@canrem.com