From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 20 14:36:11 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id OAA20857; Thu, 20 Aug 1998 14:36:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Aug 20 11:37:31 1998 Date: Thu, 20 Aug 1998 11:37:14 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: minimal maneuvers for X symmetric positions In-Reply-To: <199808182028.QAA24353@euclid.math.brown.edu> To: michael reid Cc: cube-lovers@ai.mit.edu Message-Id: On Tue, 18 Aug 1998 16:28:28 -0400 michael reid wrote: > local maxima in the quarter turn metric: > > #1, 4, 8, 9, 10, 15, 20, 21, 24, 25, 27, 29, 30, 31, 33, 34, 38, > 39, 40, 41, 42, 43, 46, 47, 48, 50, 51, 53, 55, 58, 60, 61, 62, > 64, 67, 68, 71, 72, 73, 75, 76, 77, 79, 82, 83, 86, 90, 91, 92, > 93, 94, 95, 96, 98, 102, 103, 105, 107, 108, 110, 111, 112, 113, > 114, 117, 118, 119, 121, 122, 123, 124. > > (strong) local maxima in the face turn metric: > > #8, 10, 11, 16, 29, 30, 31, 34, 38, 39, 43, 46, 51, 54, 58, 61, 64, > 71, 72, 75, 77, 79, 86, 91, 93, 94, 95, 98, 100, 103, 104, 107, 110, > 111, 114, 117, 118, 119, 121. I am curious how the local maxima were determined. 4-spot composed with superflip was based on sort of an "extended symmetry" argument, but what about all the others? If I had to guess, I would suspect that you found all minimal maneuvers for each position and observed that there was a maneuver terminating with each quarter (respectively, face) turn for each position. Or equivalently, perhaps you found all minimal maneuvers unique to symmetry for each position and observed that conjugation of the maneuvers would yield a maneuver terminating with each required kind of turn. Was it something like this? (All you would really need for the conjugation argument, since you already know that the maneuvers in question preserve the U-D axis, would be to find at least one minimal maneuver ending with any of {U, U', D, D'} and to find another minimal maneuver ending with any of {R, R', F, F', L, L', B, B'}.) It is interesting that you found strong local maxima in the face turn metric, rather than just "plain" local maxima. In my experience, finding strong local maxima with a computer search is easier than finding "plain" local maxima. Finding "plain" local maxima includes finding weak local maxima (where at least one face turn does not change the distance of the position from Start). If my guess about how you are identifying local maxima is correct, then your method would not identify weak local maxima. Finally, I have mused previously to Cube-Lovers that strong local maxima in the face turn metric may be extremely rare. I think I might be wrong. My God's algorithm searches in the face turn metric have already turned up more strong local maxima than I expected, and your search of the X-symmetric positions turned up more strong local maxima than I would have expected. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us