From BRYAN@wvnvm.wvnet.edu Sun Feb 19 23:21:13 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA27346; Sun, 19 Feb 95 23:21:13 EST Message-Id: <9502200421.AA27346@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2606; Sun, 19 Feb 95 23:20:23 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 5418; Sun, 19 Feb 1995 23:20:23 -0500 X-Acknowledge-To: Date: Sun, 19 Feb 1995 23:20:22 EST From: "Jerry Bryan" To: "Cube Lovers List" Subject: Qturn Lengths of M-Symmetric Positions 1. The length of Start is of course 0 qturns. 2. The length of Pons Asinorum is of course 12 qturns. This result has been known since 1980. However, in the process of testing out half-depth searches, I tested Pons Asinorum and encountered a minor surprise. There are five "halfway" positions which are unique up to M-conjugacy. I only expected three. They are: a. (RRLL)(FF) expected -- continue BB etc. b. (RRLL)(FB) expected -- continue FB etc. c. (RRLL)(FB') expected -- continue FB' etc. d. (FB')(RRLL) a surprise to me e. (RL')(FB')(RL') a surprise to me 3. The length of Pons Asinorum composed with Superflip is 20 qturns. Half-depth searches through level 9 found nothing. A half-depth search at level 10 found ten "halfway" positions which are unique up to M-conjugacy. I have my usual trouble of spinning tapes containing representative elements in order to find the processes, but I should have them in a couple of days or so. I expect we will find that many (or all) of them are really closely related, differing only by commuting in fairly trivial ways, just as do the five half-way positions for Pons Asinorum. 4. The length of Superflip is 24 qturns. Half-depth searches through level 11 found nothing, so the length is greater than 22. Mike Reid has found a Superflip process of length 24. Hence, the length is 24. It would be more satisfying if I could perform a half-depth search to level 12, but the problem is just too big. Hence, I have no idea how many "halfway" positions there are which are unique up to M-conjugacy. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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