From cube-lovers-errors@mc.lcs.mit.edu Sun Aug 23 01:09:13 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id BAA00571; Sun, 23 Aug 1998 01:09:03 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Aug 22 20:16:23 1998 Date: Sat, 22 Aug 1998 19:27:50 -0400 From: michael reid Message-Id: <199808222327.TAA13228@cauchy.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: Re: minimal maneuvers for X symmetric positions jerry asks > 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? yes, this is essentially what i did. i added automatic symmetry reduction to my program (this was a challenge to program, but it makes things so much more convenient). so now the program finds all minimal maneuvers up to symmetry, from which local maxima can be spotted easily. i did not find all minimal maneuvers for #91 (superflip composed with four spot) nor for #117 in the quarter turn metric, because these are too far from start (26q, 24q respectively). so for these positions, which are locally maximal, it suffices to find minimal maneuvers ending with each quarter turn. as you see, symmetry is helpful here. also, all the X symmetric positions have order 2, so any maneuver can be inverted. this is also helpful. > 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. yes, this is exactly correct. i will leave it to someone who's more interested in "weak" local maxima to determine those. mike