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Date: Mon, 30 Jun 1997 21:37:39 -0400
From: michael reid
To: cube-lovers@ai.mit.edu, jbryan@pstcc.cc.tn.us
Subject: example of a local maximum whose inverse is not a local maximum
jerry bryan asks if the inverse of a local maximum is necessarily a local
maximum. the following example shows that this need not be the case.
the interesting "six-two-one" pattern is produced by the sequence
B U2 F2 R U' R' B' R' U F2 U2 (15q)
this position has six symmetries, generated by the cube rotation C_UFR
and central reflection. therefore we also have the maneuvers
L F2 R2 U F' U' L' U' F R2 F2
D R2 U2 F R' F' D' F' R U2 R2
F' D2 B2 L' D L F L D' B2 D2
R' B2 L2 D' B D R D B' L2 B2
U' L2 D2 B' L B U B L' D2 L2
for the same position. it is not hard to check (by computer) that
these are minimal maneuvers. note that for each quarter turn, we
have a maneuver that ends with that quarter turn. thus, from this
position, any quarter turn brings us closer to start, so our position
is a local maximum.
consider now the inverse position; it is produced by
U2 F2 U' R B R U R' F2 U2 B' (15q)
it is not hard to check (by computer) that applying the quarter turn B'
to this moves us further from start (16q), so this position is not locally
maximal.
note that this is already in the archives; i first reported it on april 20,
1995 in my message "correction and an interesting example"
mike