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Date: Thu, 14 Aug 1997 18:21:24 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: patterns with 24-fold symmetry
i've finished computing minimal maneuvers for those positions with
24-fold symmetry. these positions were classified by dan hoey and
jim saxe in their note "symmetry and local maxima." there are 24
such positions; they form an abelian subgroup of type 6, 2, 2.
we may take as generators superfliptwist, pons asinorum, and 6 H's.
of these 24 positions, 4 have 48-fold symmetry; i'll include these
here as well. the other 20 positions occur in 10 pairs which differ
only in orientation; i.e. there are 10 "patterns".
some of these maneuvers were found earlier by others; i'll acknowledge
this to the extent that i'm aware of it. in addition, most maneuvers
were found by kociemba's algorithm, and a few by my optimal solver,
which is based on the same ideas.
positions with 48-fold symmetry
start (0q, 0f) no turns needed
superflip
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)
U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f)
U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f)
dik winter was the first to find a 20f maneuver.
pons asinorum
U2 D2 F2 B2 R2 L2 (12q, 6f)
F B' U2 D2 R2 L2 F B' (12q)
F B' U' D F B' U' D F B' U' D (12q)
this last maneuver is due to dan hoey.
pons asinorum composed with superflip
B' D' L' F' D' F' B U F' B R2 L U D' F L U R D (20q, 19f)
F' U' B' R' F R L' D' R L' U D' L' U D' F R B U F (20q)
B' R' F' U' F R L' D' R L' U D' L' U D' F U F R B (20q)
B' R' B' D' F U' D L' U' D R L' U' R L' F D B R B (20q)
R U R B R' U' D F U' D F B' D F B' R' B' R' U' R' (20q)
U D F R L' F B' L D2 R L F' B' U' L2 F B' U2 L' (19f)
U D F' B' L' U2 F' B L2 U' R' L' F' U' D F' B D' L2 (19f)
U2 R F U F B' L' D' F B' L B R L' U D2 B' R' U2 (19f)
U2 R F U2 D' R' L F' L' F B' U L F B' D' B' R' U2 (19f)
U2 R U2 D2 R U' L' U B R F2 U' D B' R' F' D B' L2 (19f)
jerry bryan found the 20q maneuvers.
positions with 24-fold symmetry
superfliptwist
U R F' B U' D' F U' D F L F' L' U R D F U R L (20q, 20f)
herbert kociemba was the first to find a 20f maneuver.
supertwist
U R' B D B U L D B' D2 R U' F L F R D L F' L2 (22q)
B' L2 U D R2 B' D2 F2 D' R2 F B L2 D' B2 U2 (16f)
dik winter first found the 16f maneuver.
6 H's
D2 L2 B2 U2 D2 B2 R2 D2 (16q, 8f)
jim saxe found this maneuver.
superflip composed with 6 H's
U F' L' F' B U R F' B U' B' U D' R2 L' B U' (18q, 17f)
superfliptwist composed with pons asinorum
U F B D R L U' F2 B2 R L D' F B D R L D F' B' (22q, 20f)
dik winter was the first to find a 20f maneuver.
supertwist composed with pons asinorum
F L D F U' B2 R F R' F' R F L2 U' R D B R (20q)
B2 L U2 F' B' U2 R' F2 L2 F' U2 R' L' U2 B R2 (16f)
superfliptwist composed with 6 H's (type 1)
U F B U' R L U F B R2 L2 D' F B U' R L D' R' L' (22q, 20f)
superfliptwist composed with 6 H's (type 2)
inverse of type 1
supertwist composed with 6 H's (type 1)
U2 L U B D L U B' R' L' F' D R U F D L' U2 (20q)
L' B2 U' D' B2 R' U2 L2 U B2 R L F2 U R2 U2 (16f)
supertwist composed with 6 H's (type 2)
inverse of type 1
some of these maneuvers have some symmetry. i find the maneuver for
superfliptwist composed with pons asinorum especially interesting.
it is composed of: twists of the U or D face, and antislices
along the R-L and F-B axes:
U (FB) D (RL) U' (F2B2) (RL) D' (FB) D (RL) D (F'B')
therefore, when we conjugate this maneuver by the cube rotation C_U2,
we get the same maneuver! the maneuver for superfliptwist composed with
6 H's has the same type of symmetry.
mike