From @mail.uunet.ca:mark.longridge@canrem.com Fri Sep 2 00:08:42 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 AA03291; Fri, 2 Sep 94 00:08:42 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <95284-1>; Fri, 2 Sep 1994 00:08:46 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA06043; Fri, 2 Sep 94 00:05:44 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1AD333; Thu, 1 Sep 94 23:45:13 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: < U, R > revisited From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.796.5834.0C1AD333@canrem.com> Date: Thu, 1 Sep 1994 22:56:00 -0400 Organization: CRS Online (Toronto, Ontario) Analysis of the 3x3x3 group (continued) ---------------------------------- branching Moves Deep arrangements (q only) factor 0 1 1 -- 1 4 5 4 2 10 15 2.5 3 24 39 2.4 4 58 97 2.416 5 140 237 2.413 6 338 575 2.414 7 816 1,391 2.414 8 1,970 3,361 2.414 9 4,756 8,117 2.414 10 11,448 19,565 2.407 11 27,448 47,013 2.401 ML's Conjecture: The < U, R > group is >=20 turns deep in qt metric UR Reflective processes: (in the q metric) A different sort of symmetry which I started to investigate, having been inspired by my friend who solves his cube 2 adjacent faces last! These are the only UR reflective processes at 10 q turns: U3 R1 U1 R1 (U2) R3 U3 R3 U1 = R3 U1 R1 U1 (R2) U3 R3 U3 R1 (10) U1 R3 U3 R3 (U2) R1 U1 R1 U3 = R1 U3 R3 U3 (R2) U1 R1 U1 R3 (10) Here is the obvious one we all know: ( U2 R2 ) ^ 3 = ( R2 U2 ) ^ 3 (12) I liked this pattern in particular... U1 R1 U2 R3 U2 R3 U2 R1 U1 = R1 U1 R2 U3 R2 U3 R2 U1 R1 (12) I hope to have an algorithm to plumb the depths of the < U, R > group soon. Amusingly my friend complained about not been able solve the cube completely as he was stuck in a position with 2 flipped edges. After watching him squirm for a few weeks I did tell him you can't flip edges in the < U, R > group! ;-> Congrats to Dan Hoey, Dik Winter, Jerry Bryan and Ludwig Plutonium for making it into the 1994 Internet White Pages! I'm in good company. -> Mark <- Email: mark.longridge@canrem.com P.S. I just read the last J.B. post and see I've been somewhat overshadowed. Ok let's see some antipodes! At least our results are the same though. So, ummmm I guess ML's conjecture is correct!