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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
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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