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Mail-from: From reid@math.brown.edu Tue Aug 5 22:42:50 1997
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Date: Tue, 5 Aug 1997 22:45:08 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: more patterns at distance 20f
i can also give three more patterns that are exactly 20 face turns from
start. all three are very symmetric; they have 24 symmetries. the
symmetry group is "H" in dan hoey's taxonomy. in particular, all are
local maxima (in the quarter turn metric). see hoey and saxe's
"symmetry and local maxima" (december 14, 1980) for more info about this.
the first pattern is the composition of superfliptwist with pons asinorum.
you may recall that i suggested this pattern to dik winter when he was
looking for positions that couldn't be solved in 20 face turns or less.
he did succeed in solving it in 20f, using kociemba's algorithm, but it
took much longer than most other positions did.
the other two are inverses of one another. they can both be described
as the composition of superfliptwist with 6 H's. however, the patterns
"6 H's" and "superfliptwist" each come in two orientations. therefore,
fix your favorite orientation of 6 H's; now there are two different
orientations of superfliptwist which may be composed. this gives two
distinct patterns, and the positions are inverses.
by symmetry, we may assume that the first face turn of a maneuver for
any of these positions is either U or U2. to confirm that the pattern
superfliptwist . pons asinorum is not within 19f of start, we need to
search the positions
superfliptwist . pons asinorum . U
and superfliptwist . pons asinorum . U2
completely through depth 18f. similarly, for the second pattern,
two complete searches through depth 18f were required. the third
pattern is the same distance from start as is its inverse, so this one
doesn't require further testing.
my optimal solver did not find the minimal maneuvers for these, although
it certainly would have, if i'd let it continue searching long enough.
however, one can find 20f maneuvers using kociemba's algorithm:
superfliptwist . pons asinorum:
D' R' B' L2 U' L B' D' R' D' B2 D2 B' U D2 R2 F2 D' L' B' (20f)
superfliptwist . 6 H's:
B' L2 D B2 R' D2 F' L2 U' L' F' B U' R D' R2 F2 R2 U' D2 (20f)
it would be nice to find a position that was not within 20f of start.
of course, we don't know if any such positions exist. my guess is that
they do, but that's only a hunch. dik winter examined 9000 random
positions and found that they were all within 20f of start. therefore
the positions we're looking for are extremely scarce. i think that
looking at positions with a lot of symmetry seems to be the right way
to approach this. i've tested some of the most symmetric positions,
but each that i examined was solved in 20f or less.
mike