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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