From @mitvma.mit.edu,@WVNVM.WVNET.EDU:BRYAN@WVNVM.WVNET.EDU Sun Dec 5 00:03:57 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 AA07956; Sun, 5 Dec 93 00:03:57 EST Message-Id: <9312050503.AA07956@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 3262; Sun, 05 Dec 93 00:04:01 EST Received: from WVNVM.WVNET.EDU (NJE origin MAILER@WVNVM) by MITVMA.MIT.EDU (LMail V1.1d/1.7f) with BSMTP id 3276; Sun, 5 Dec 1993 00:04:01 -0500 Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.1d/1.7f) with BSMTP id 5628; Sun, 5 Dec 1993 00:01:11 -0500 X-Acknowledge-To: Date: Sun, 5 Dec 1993 00:01:08 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: God's Algorithm for the Edges of the 3x3x3 Here are my God's Algorithm results for the edges of the 3x3x3 cube. I explained in the last post what I mean by 1152-fold symmetry. All results below are for 1152-fold symmetry. I am working on the 24-fold case, but I am not quite done. The 24-fold case is just a matter of determining the sizes of the equivalence classes in the 1152-fold case. One item of terminology needs to be explained. Several people, including myself, have posted results for the 2x2x2 cube and for the corners of the 3x3x3 cube. If you take the term "corners of the 3x3x3 cube" absolutely literally, it is completely isomorphic to the 2x2x2 cube. When people have posted results for the "corners of the 3x3x3 cube", they all (including myself) really mean "corners plus centers of the 3x3x3". See below: -------------- --------------------- ------------------- | x | x | | x | | x | | x | | x | | | | | | | | | | | | |-----|------- -------|------|------ ------|-----|------ | x | x | | | | | | | x | | | | | | | | | | | | | -------------- -------|------|------ ------|-----|------ 2x2x2 | x | | x | | x | | x | | | | | | | | | --------------------- ------------------- Corners of 3x3x3 Corners + Centers Thus, when I say I have solved the "edges of the 3x3x3", I need to clarify what I mean. I have solved the "edges without the centers". I suppose my next project will be "edges with the centers". Unfortunately, "edges with the centers" is a twenty-four times larger problem than is "edges without the centers". "Edges without the centers" took about a year running 24 hours a day, 7 days a week, on two machines. I am going to have to rethink "edges with the centers" before I start. I don't want it to take 24 years. --------------------- ------------------- | | x | | | | x | | | | | | | | | | -------|------|------ ------|-----|------ | x | | x | | x | x | x | | | | | | | | | -------|------|------ ------|-----|------ | | x | | | | x | | | | | | | | | | --------------------- ------------------- Edges without Centers Edges with Centers Results using q-turns only Distance Number of from Start Nodes using 1152-fold Symmetry 0 1 1 1 2 5 3 25 4 215 5 1,860 6 16,481 7 144,334 8 1,242,992 9 10,324,847 10 76,993,295 11 371,975,385 12 382,690,120 13 8,235,392 14 54 15 1 Results using q-turns and h-turns Distance Number of from Start Nodes using 1152-fold Symmetry 0 1 1 2 2 9 3 75 4 919 5 11,344 6 139,325 7 1,664,347 8 18,524,022 9 167,864,679 10 582,489,607 11 80,930,364 12 314 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?