From BRYAN@wvnvm.wvnet.edu Tue Dec 27 22:46:45 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA13741; Tue, 27 Dec 94 22:46:45 EST Message-Id: <9412280346.AA13741@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2172; Tue, 27 Dec 94 14:26:46 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9642; Tue, 27 Dec 1994 14:26:46 -0500 X-Acknowledge-To: Date: Tue, 27 Dec 1994 14:26:45 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Squares Group On 9 Aug 1994, Mark Longridge posted God's Algorithm results for the squares group consisting entirely of 180 degree turns. Mark invited corroboration. The following will serve to verify Mark's results, and will also provide the same information for the M-conjugacy classes of . Notice that the ratio of cube positions to M-conjugacy classes never gets very close to 48. Hence, there are a significant number of positions in at each level of the search tree that are at least somewhat "symmetrical". M Branching Cube Branching Ratio of Level Conjugate Factor Positions Factor Cubes to Classes M Classes 0 1 1 1 1 1 1 6 6 6 2 2 2 27 4.5 13.5 3 5 2.5 120 4.4444 24 4 18 3.6 519 4.3250 28.8333 5 56 3.1111 1932 3.7225 34.5000 6 162 2.8929 6484 3.3561 40.0247 7 482 2.9753 20310 3.1323 42.1369 8 1258 2.6100 55034 2.7097 43.7472 9 2627 2.0882 113892 2.0695 43.3544 10 4094 1.5584 178495 1.5672 43.5992 11 4137 1.0105 179196 1.0039 43.3154 12 2231 0.5393 89728 0.5007 40.2187 13 548 0.2456 16176 0.1803 29.5182 14 114 0.2080 1488 0.0920 13.0526 15 16 0.1404 144 0.0968 9 Total 15752 663552 42.1249 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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