From BRYAN@wvnvm.wvnet.edu Sun Aug 21 09:57:44 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05642; Sun, 21 Aug 94 09:57:44 EDT Message-Id: <9408211357.AA05642@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 1055; Sun, 21 Aug 94 08:58:31 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 5634; Sun, 21 Aug 1994 08:58:31 -0400 X-Acknowledge-To: Date: Sun, 21 Aug 1994 08:58:30 EDT From: "Jerry Bryan" To: "Cube Lovers List" Subject: Analysis of Turns Towards Start for Whole Cube The following chart for level 0 through level 8 of the whole cube using Q-turns gives the number of Q-turns for each cube which will move towards Start. (I recently gave the same analysis for corners only.) Columns 9 through 12 are omitted from the chart since they contain all zeros. Any local maxima would appear in column 12. This adds one new level known not to contain any local maxima. It would be extremely interesting to be able to extend the chart at least to level 12 because level 12 is known to include a local maximum. As with the corners-only case, the chart contains almost all even numbers. However, unlike the corners-only case, the numbers do not cluster in the even columns. Rather, they cluster towards column 1. This means that (close to Start, at least) most cubes have only one "good" move that takes you closer to Start. It also serves to illustrate why "random" moves so quickly scramble the cube. Number of Q-turns which Move Closer to Start Level Total 0 1 2 3 4 5 6 7 8 Cubes 0 1 1 0 0 0 0 0 0 0 0 1 12 0 12 0 0 0 0 0 0 0 2 114 0 96 18 0 0 0 0 0 0 3 1068 0 912 144 12 0 0 0 0 0 4 10011 0 8544 1368 96 3 0 0 0 0 5 93840 0 80088 12816 912 24 0 0 0 0 6 878880 0 749376 120612 8640 252 0 0 0 0 7 8221632 0 7001712 1135104 82152 2664 0 0 0 0 8 76843595 0 65391504 10645824 777936 28200 48 56 0 27 Total 86049153 1 73232244 11915886 869748 31143 48 56 0 27 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow?