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Date: Fri, 31 Jul 1998 22:28:02 -0400
From: michael reid
Message-Id: <199808010228.WAA11081@euclid.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: all 24q maneuvers for superflip
with my new optimal solver, i've calculated all 24q maneuvers for
superflip. there are three transformations we can apply to a
maneuver for superflip, none of which change its length.
we may conjugate by any cube symmetry.
we may cyclically permute the maneuver, i.e. replace
sequence_1 sequence_2 by sequence_2 sequence_1
we may invert the maneuver.
in a previous message (august 7, 1997), i showed that, using these
three transformations, any maneuver for superflip can be transformed
into one that begins with one of the ten sequences
U R2 U D' R U D R U D R' U R F
U R F' U R' F U R' F' U' R F' U' R' F'
my program took 101 hours to exhaustively search these ten cases.
there are four inequivalent maneuvers; two were previously known:
R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q*)
U R2 F' R D' L B' R U' R U' D F' U F' U' D' B L' F' B' D' L' (24q*)
the two new ones are:
U D' R F U' D' L D' F R U' R U' D' F U' F L B' U F' B' L B' (24q*)
U D' R F' D L' B L' U' R' D' B' U' D L' F D' R B' R U L D B (24q*)
this last one can be written as
(U D' R F' D L' B L' U' R' D' B' R_rl)^2
where R_rl denotes reflection through the R-L plane.
we can also count the total number of 24q maneuvers for superflip.
note that U2 = U U also is U' U' , so can be cyclically shifted
in an extra way. similarly, U D' = D' U , so this also accounts
for an extra cyclic shift. and the same is true for U' D'.
the total number of maneuvers therefore is
28 * 24 * 2 + 28 * 48 * 2 + 28 * 48 * 2 + 26 * 24 * 2 = 7968
where the first factor is the number of cyclic shifts, the second
factor is the number of cube symmetries we can apply, and the third
factor is 2, for inversion. the first and last maneuvers only get
a factor of 24 for the number of cube symmetries, because a cyclic
shift by 12q gives the same maneuver in a different orientation.
the total number of 24q sequences is 274575811926317204506464.
the total number of even positions is 21626001637244928000.
so even positions have an average of 12696.56 different 24q maneuvers.
mike