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Date: Mon, 12 Jan 1998 23:25:18 -0400 (EDT)
From: Jerry Bryan
Subject: Face Turns Nine Moves from Start
To: Cube-Lovers
Message-Id:
I have some new search results for the face turn metric. Here is a
summary of the new search.
Face Turns Patterns Positions Branching Positions/
from Start Factor Patterns
0 1 1 1.000
1 2 18 18.000 9.000
2 9 243 13.500 27.000
3 75 3240 13.333 43.200
4 934 43239 13.345 46.294
5 12077 574908 13.296 47.604
6 159131 7618438 13.252 47.875
7 2101575 100803036 13.231 47.965
8 27762103 1332343288 13.217 47.991
9 366611212 17596479795 13.207 47.998
The results at 8f and 9f from Start are new. Previously, the face turn
metric had only been searched through 7f from Start. All the results in
terms of patterns (M-conjugacy classes) are new. Previously, the face
turn metric had been searched only in terms of positions.
Note that the branching factor does not change very much. We already know
(or strongly suspect by statistical arguments based on the results of
Kociemba, Winter, Reid, and Korf) that it cannot change much this close to
Start. Otherwise, the mode of the distribution would be greater than the
18f which is strongly suspected to be the case.
I have not yet installed the logic to detect weak local maxima. The logic
to detect strong local maxima is installed with an interesting result. Two
patterns were detected at 9f from Start which are strong local maxima.
Regrettably, I have no idea what they are. I will have to add something
to the program to print out strong local maxima when they are detected.
All I know is that the patterns are at least "somewhat symmetric" in that
they collectively represent only 32 positions.
I have begun to suspect that strong local maxima are fairly rare in the
face turn metric. Recall that a strong local maximum is one where all 18
face turns carry the cube closer to Start. A weak local maximum, by
contrast, is a local maximum where at least one face turn leaves the cube
the same distance from Start. If I have not made a mistake in analyzing
them (which is entirely possible), the only one of Mike Reid's "highly
symmetric" positions which is a strong local maximum is superflip. Even
Pons Asinorum is not a strong local maximum.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990