From hoey@aic.nrl.navy.mil Tue Jan 14 12:49:25 1992 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA13585; Tue, 14 Jan 92 12:49:25 EST Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA27636; Tue, 14 Jan 92 12:49:17 EST Return-Path: Received: by sun13.aic.nrl.navy.mil; Tue, 14 Jan 92 12:49:16 EST Date: Tue, 14 Jan 92 12:49:16 EST From: hoey@aic.nrl.navy.mil Message-Id: <9201141749.AA27091@sun13.aic.nrl.navy.mil> To: Cube-Lovers@life.ai.mit.edu Cc: "Allan C. Wechsler" Subject: A new upper bound: 91 QT Keywords: Upper-Bounds, Thistlethwaite I just wrote a quick program to count the number of QT to move from the full cube group to the subgroup generated by . Thistlethwaite computed that this takes at least 7 HT in the worst case. The surprisingly good result is that it also takes only 7 *QT* in the worst case. This reduces the upper bound I posted Friday to 91 QT. I had wondered if the worst cases could be reduced by choosing a different pair of faces to restrict to half-twists. Unfortunately, the all-edges-flipped position is one of those that requires at least 7 HT (and so 7 QT), and by symmetry it cannot be improved. Allan C. Wechsler analyzed his own cube-solving method, finding that: > For example, flipping two edges in place takes 22 qtw. This can be done in 16 QT, though I don't know if that is the best known. Any pair can be flipped with a conjugate of the 14 QT slice mono-op FOTAROFATO-RAM TAFORATOFA-ROM (FT'RF'T L'R B'TR'BT' LR'). Adjacent and antipodal pairs require the introduction of a non-cancelling QT in the conjugator. > Obviously a lot could be gained from tweaking the earlier part of > the algorithm to guarantee that I don't need to do this at the end. Probably, but it's hard to make that guarantee. The problem is that unless you flip edges in place with no other action (the very problem you're trying to avoid) you may affect the later choices in the algorithm, making the earlier tweaks wrong for that branch of the algorithm. For instance, the 7-QT method my program found solves the orientation of all the edges (using a particular non-standard labeling of the orientation that, when solved, is invariant under F^2, B^2, L, R, T, and D). But it may permute edges, and permute and twist corners, so it may not form a useful part of an arbitrary cube-solving algorithm. It works in Thistlethwaite's only because he is careful in all branches of the rest of the algorithm never to mix up the orientation of those edges. Dan Hoey Hoey@AIC.NRL.Navy.Mil