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Date: Mon, 21 Sep 1998 16:34:29 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Local Maxima which Fix the Corners, 12q from Start
To: Cube Lovers
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I am making a run to calculate God's Algorithm out to 12 moves
from Start in the quarter turn metric. It has been running
several weeks, and will probably run several more.
I have made some changes to my program to make it easier to
extract the positions for local maxima, and to checkpoint the
local maxima data. As a part of the checkpointing, I can
actually see the local maxima as they are generated without
having to wait for the program to end.
It is becoming apparent that there are a *lot* of local maxima
12q from Start. It is already known that there are only four
(unique to symmetry) which are 10q from Start (the shortest ones
in the quarter turn metric), and that there are none 11q from
Start. So I am a little surprised that I am seeing so many.
I have looked at quite a few of them, and most of them are not
all that interesting. But the ones which fix the corners are
all quite pretty. Because the positions are being produced in
lexicographic order, and because I am sorting by corners first,
edges second, the positions which fix the corners are the first
ones to appear. There are eight of them as follows.
1. F2 L2 F2 B2 R2 B2
2. F B' U2 D2 F' B R2 L2
3. F B R2 F' B' U D L2 U' D'
4. D' F B' R F R' F' B U F' U' D
5. F B R2 L2 F B U2 D2
6. R L' F2 B2 R L' F2 B2
7. F2 B2 U2 D2 R2 L2
8. R L' U D' F B' R2 L2 U D'
#1 is a 2-H pattern (only four edge cubies are moved).
#2 is a 4-H.
#3 moves four edge cubies, leaving eight of the nine
facelets the same color on four faces, and a solid color on the
other two faces.
#4 moves three edge cubies, leaving eight of the nine facelets
the same color on all six faces.
#5 has 2 H's, 2 checkerboards, and 2 solid faces -- with the
respective H's, checkerboards, and solid faces opposing each
other.
#6 has 4 H's and 2 checkerboards, with the 2 checkerboards
opposing each other.
#7 is the Pons Asinorum, and is included only for completeness
because we already knew that the Pons was a local maximum of
length 12q.
#8 has all six faces being sort of a "three colored
checkerboard".
Some of these positions may have appeared on Cube-Lovers in some
other context, but the only one I recognize for sure is the Pons.
In some ways, #4 is the most interesting to me, because it a
simple 3-cycle on the edges, and who would have thought that
such a position would turn out to be a local maximum? #1 and #3
both consist of two 2-cycles on the edges, and are about as
striking to me as is #4.
----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us