From cube-lovers-errors@mc.lcs.mit.edu Fri Aug 22 21:50:40 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA04121; Fri, 22 Aug 1997 21:50:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Fri Aug 22 21:15:50 1997 Message-Id: <199708222345.TAA12112@life.ai.mit.edu> Date: Fri, 22 Aug 1997 19:51:01 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: minimal maneuvers for T-symmetric positions dan hoey denotes by T a subgroup of size 12 of the symmetries of the cube that preserves one of the long diagonals. there are 4 conjugates of T , each preserves a different long diagonal. i'll choose the UFR - DLB diagonal. (note that "preserving" allows the diagonal to be reversed in direction.) in "symmetry and local maxima," hoey and saxe classify all cube positions with T-symmetry. for each choice of a T subgroup, there are 16 such positions. they form a commutative subgroup of the cube group of type 2, 2, 2, 2. (this means that it is isomorphic to a product C_2 x C_2 x C_2 x C_2 of cyclic groups of order 2.) we may take generators of these cyclic factors to be superflip, pons asinorum, and the two positions (UB+) (UL+) (FL+) (FD+) (RD+) (RB+) (number 1 below) and (DRF, UBL) (FLD, BUR) (LUF, RBD) (UB, DF) (UL, DR) (FL, BR) (number 5 below). 4 of these positions have more symmetry, namely the subgroup generated by superflip and pons asinorum. for the other 12 positions, minimal maneuvers are given below. i've also given a maneuver that is minimal in both metrics, whenever such a maneuver exists. 1. B U L' F' U R U2 D2 F' L U' B' L D R2 L2 B2 (22q, 17f) 2. F U D' R2 U2 R' B' U' F R' D R L' F U' F U' R' (20q, 18f) 3. U R U' F D R L' B' L' F R F B' U' L' D B' D' (18q, 18f) 4. D' R' U B' D' R' L F L B' R' F B' U L D' F U' D2 (20q, 19f) 5. D' B' D' R' B L B U' B U R D R L D R' L2 D (19q) B2 L U' L D R' L' D2 R U L' B2 U R2 U2 F2 U (17f) 6. U L U D F B' U' D L2 F U D B' R L B' U' D' F U (21q) D' L F' B' L F2 B2 U R L' U D' L F' R2 L2 F2 U' D2 (19f) 7. F U D L2 F2 L2 U' F B D F' B' D' F' (17q, 14f) 8. U F B' L U F B' L D F D' R2 L F U' B' L F2 R (21q) U B R' F2 U' D' L2 D2 R2 B2 L' F' B R2 F2 R' D F (18f) 9. U B U2 L F' B2 U' F' B L U' B' D' F R U B R L' (21q) U F2 D B' U' B2 R B2 D' F2 U' D2 B2 L' U2 B D2 (17f) 10. U F B D' L' U' B' L' F R' L' D L U F U B D B R' D' (21q) U F U D' B' D2 R U D R2 D2 B R L2 F2 B' D R2 L' (19f) 11. U R F' B D B' U' F B' D' F R L' D2 R D' B' U' D F' (21q) D' F2 U2 B2 R F' L U' F2 B R' F' D L2 D R2 F2 U' F2 (19f) 12. U B' D' R F' R' F' R2 B' D F' B2 D B' L' U F B R' (21q) U F2 B R2 F2 B' U L U B' U2 R2 L' U B U' L' U' (18f) mike