From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 1 23:44:12 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id XAA05471; Sat, 1 Aug 1998 23:44:12 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jul 31 22:28:10 1998 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