From cube-lovers-errors@mc.lcs.mit.edu Thu Jan 22 12:46:32 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA17608; Thu, 22 Jan 1998 12:46:31 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Jan 22 00:44:32 1998 Date: Thu, 22 Jan 1998 00:43:14 -0400 (EDT) From: Jerry Bryan Subject: Re: Face Turns Nine Moves from Start In-Reply-To: To: Cube-Lovers Message-Id: On Mon, 12 Jan 1998, Jerry Bryan wrote: > I have not yet installed the logic to detect weak local maxima. The logic > to detect strong local maxima is installed with an interesting result. Two > patterns were detected at 9f from Start which are strong local maxima. > Regrettably, I have no idea what they are. I will have to add something > to the program to print out strong local maxima when they are detected. > All I know is that the patterns are at least "somewhat symmetric" in that > they collectively represent only 32 positions. I have not yet added the logic for weak local maxima, but further perusing of my printout from the run which has already been made does yield a bit of confirmation to some previous results reported by others. At each distance from Start, my program summarizes the number of patterns and positions by the symmetry class (one of the 33 symmetry classes of M, the group of 48 symmetries of the cube). Hence, I can easily look for "highly symmetric" positions based on the symmetry class. Of the 72 positions defined as q-transitive by Jim Saxe and Dan Hoey in Symmetry and Local Maxima, only 4 of them show up in the search through 9f. One of them is at 0f (Start), one of them is at 6f (Pons Asinorum, a weak local maximum, only the six half turns move closer to Start), and two of them are at 8f (the two conjugate 6-H positions, weak local maxima with only the six half turns moving closer to Start). We therefore know that the two patterns at 9f which are strong local maxima are not q-transitive, and cannot be shown to be local maxima by symmetry considerations alone. Strictly speaking, we already knew all this based on Dan's study of Pons Asinorum from many years ago, and based on Mike Reid's recent studies of highly symmetric positions with his optimal cube solver. Jim Saxe found the 8f processes for the 6-H positions many years ago, but I do not believe that they were shown to be minimal until Mike's recent work. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990