From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sat Dec 4 23:18:20 1993 Return-Path: <@mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05338; Sat, 4 Dec 93 23:18:20 EST Message-Id: <9312050418.AA05338@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 3107; Sat, 04 Dec 93 23:18:20 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 2822; Sat, 4 Dec 1993 23:18:20 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 5413; Sat, 4 Dec 1993 23:15:32 -0500 X-Acknowledge-To: Date: Sat, 4 Dec 1993 23:15:30 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: God's Algorithm for the Corners of the 3x3x3 Here are my God's Algorithm results for the corners of the 3x3x3 cube. I explained in the last post what I mean by 1152-fold symmetry and 24-fold symmetry. The 1152-fold symmetry is what I actually calculated. In this particular case, I did not do the 24-fold symmetry calculations myself based on the size of the equivalence classes as I did with the 2x2x2 cube. Rather, I went back and found the figures in the Cube-Lover archives (Dik Winter's post). Results Using Both q-turns and h-turns Distance Number of Number of from Nodes using Nodes using Start 1152-fold 24-fold symmetry symmetry 0 1 1 1 2 18 2 9 243 3 71 2,874 4 637 28,000 5 4,449 205,416 6 24,629 1,168,516 7 113,049 5,402,628 8 433,611 20,776,176 9 947,208 45,391,616 10 316,823 15,139,616 11 1,481 64,736 Results Using Only q-turns Distance Number of Number of from Nodes using Nodes using Start 1152-fold 24-fold symmetry symmetry 0 1 1 1 1 12 2 5 114 3 24 924 4 149 6,539 5 850 39,528 6 4,257 199,926 7 16,937 806,136 8 57,800 2,761,740 9 180,639 8,656,152 10 466,052 22,334,112 11 676,790 32,420,448 12 392,558 18,780,864 13 45,744 2,166,720 14 163 6,624 Results Using Only h-turns Distance Number of from Nodes using Start 1152-fold symmetry 0 1 1 1 2 2 3 4 4 3 It turns out that the maximum distance from Start is the same for the corners of the 3x3x3 cube as it is for the 2x2x2 cube. I found this rather surprising, although the archives of Cube-Lovers do provide a reasonable explanation. I am just going to have to go back and read it five or ten times until I fully understand it. In any case, I was curious about the following question. Suppose you are N moves from Start on the corners of the 3x3x3. How many moves from Start would you be on the 2x2x2 if the 2x2x2 was in the same configuration as the corners of the 3x3x3 where you currently were. As it turns out, I stored the results for the 2x2x2 in the same data base as I stored the results for the corners of the 3x3x3, so the question was easy to answer. Here are the results. Corresponding Distances from Start Using Both q-turns and h-turns 2x2x2 Corner of 3x3x3 Number Distance from Distance from of Nodes Start Start 0 0 1 2 2 4 2 1 1 2 2 2 3 4 4 3 2 2 5 3 12 4 18 5 3 3 3 55 4 106 5 41 4 4 508 5 457 6 38 5 5 3,948 6 1,237 7 2 6 6 23,354 7 1,992 8 20 7 7 111,055 8 3,242 9 20 8 8 430,349 9 5,460 10 62 9 9 941,728 10 3,770 11 20 10 10 312,991 11 45 11 11 1,416 Corresponding Distances from Start Using Only q-turns 2x2x2 Corner 3x3x3 Number Distance from Distance from of Nodes Start Start 0 0 1 2 1 4 2 6 1 1 1 1 3 2 5 2 2 2 4 4 10 6 6 3 3 22 5 46 7 4 4 4 137 6 145 5 5 802 7 356 6 6 4,105 8 474 7 7 16,577 9 83 8 8 57,326 10 24 12 24 9 9 180,556 11 148 10 10 466,028 12 192 11 11 676,642 13 144 12 12 392,342 13 13 45,600 14 14 163 Corresponding Distances from Start Using Only h-turns 2x2x2 Corner of 3x3x3 Number Distance from Distance from of Nodes Start Start 0 0 1 2 1 1 1 1 3 1 2 2 1 3 3 3 4 4 3 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?