From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 9 18:35:15 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id SAA01664; Mon, 9 Nov 1998 18:35:13 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Wed, 04 Nov 1998 16:41:26 -0500 (Eastern Standard Time) From: Jerry Bryan Subject: Local Maxima Whose Inverses are not Local Maxima To: Cube Lovers Message-Id: On 30 June 1997 I reported that if you could find a local maximum whose inverse was not a local maximum, then you could also find a longer local maximum. For example, suppose x is a local maximum in the quarter turn metric and x' is not. Then, there exists q in Q such that |x'q| = |x'| + 1 = |x| + 1. But we know that q'x is a local maximum and we also know that |q'x| = |x| + 1 because |q'x| is the same as |x'q|. Because we now have at 12q a good number of local maxima whose inverses are not local maxima as specimens, I have begun to wonder if the same process might be able to be repeated several times to yield progressively longer local maxima. For example, if x is a local maximum and (q1)x is a local maximum, might also (q2)(q1)x be a local maximum and also (q3)(q2)(q1)x etc. It seems to me that good candidates to investigate in this regard might be those local maxima at 12q whose inverses have a very small maximality. For example, if x is a local maximum where the maximality of x' is 2 (and there are several such cases), then we know that there are 10 local maxima of the form qx. I am not sure if I have time to investigate this question further, but I certainly would love to hear from anyone who has the time and the computing resources to do so. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us