From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 6 11:16:57 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA01366; Wed, 6 Aug 1997 11:16:57 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From reid@math.brown.edu Tue Aug 5 22:42:50 1997 Message-Id: <199708060239.WAA23888@life.ai.mit.edu> Date: Tue, 5 Aug 1997 22:45:08 -0400 From: michael reid To: cube-lovers@ai.mit.edu Subject: more patterns at distance 20f i can also give three more patterns that are exactly 20 face turns from start. all three are very symmetric; they have 24 symmetries. the symmetry group is "H" in dan hoey's taxonomy. in particular, all are local maxima (in the quarter turn metric). see hoey and saxe's "symmetry and local maxima" (december 14, 1980) for more info about this. the first pattern is the composition of superfliptwist with pons asinorum. you may recall that i suggested this pattern to dik winter when he was looking for positions that couldn't be solved in 20 face turns or less. he did succeed in solving it in 20f, using kociemba's algorithm, but it took much longer than most other positions did. the other two are inverses of one another. they can both be described as the composition of superfliptwist with 6 H's. however, the patterns "6 H's" and "superfliptwist" each come in two orientations. therefore, fix your favorite orientation of 6 H's; now there are two different orientations of superfliptwist which may be composed. this gives two distinct patterns, and the positions are inverses. by symmetry, we may assume that the first face turn of a maneuver for any of these positions is either U or U2. to confirm that the pattern superfliptwist . pons asinorum is not within 19f of start, we need to search the positions superfliptwist . pons asinorum . U and superfliptwist . pons asinorum . U2 completely through depth 18f. similarly, for the second pattern, two complete searches through depth 18f were required. the third pattern is the same distance from start as is its inverse, so this one doesn't require further testing. my optimal solver did not find the minimal maneuvers for these, although it certainly would have, if i'd let it continue searching long enough. however, one can find 20f maneuvers using kociemba's algorithm: superfliptwist . pons asinorum: D' R' B' L2 U' L B' D' R' D' B2 D2 B' U D2 R2 F2 D' L' B' (20f) superfliptwist . 6 H's: B' L2 D B2 R' D2 F' L2 U' L' F' B U' R D' R2 F2 R2 U' D2 (20f) it would be nice to find a position that was not within 20f of start. of course, we don't know if any such positions exist. my guess is that they do, but that's only a hunch. dik winter examined 9000 random positions and found that they were all within 20f of start. therefore the positions we're looking for are extremely scarce. i think that looking at positions with a lot of symmetry seems to be the right way to approach this. i've tested some of the most symmetric positions, but each that i examined was solved in 20f or less. mike