From cube-lovers-errors@mc.lcs.mit.edu Thu Aug 14 18:29:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA12833; Thu, 14 Aug 1997 18:29:05 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Thu Aug 14 18:19:32 1997 Message-Id: <199708142216.SAA16395@life.ai.mit.edu> 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