From cube-lovers-errors@mc.lcs.mit.edu Tue Dec 1 19:18:47 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil (sun28.aic.nrl.navy.mil [132.250.84.38]) by mc.lcs.mit.edu (8.9.1a/8.9.1-mod) with SMTP id TAA20686 for ; Tue, 1 Dec 1998 19:18:47 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Mon, 30 Nov 1998 23:13:51 -0500 From: michael reid Message-Id: <199812010413.XAA11878@euclid.math.brown.edu> To: cube-lovers@ai.mit.edu Subject: new types of cyclic shifters a few months ago, i introduced the position superflip composed with four spot and showed that it had a new type of cyclic shifting property. i've now found some new ways to generalize cyclic shifting, and this in turn suggests some new positions to consider. first, some brief review. the position superflip is central, so it commutes with all turns. therefore, if x y produces superflip, so does y x . we can shift one turn at a time, removing the first turn of the sequence and shifting it to the end. in other words, if m produces superflip, then x m = m x for all turns x . clearly, any position m with the property that x m = m x for all turns x , is central, so the only such positions are superflip and start. i showed in an earlier message, that if x is any turn and m is superflip composed with four spot, then x m = m y , where y is x conjugated by the cube rotation C_U2 . more generally, we can ask for positions m such that for any turn x , there is another turn y satisfying x m = m y . for such a position, we can cyclically shift any maneuver, one turn at a time, by replacing the turn x^(-1) at the beginning with the corresponding y^(-1) at the end. some other positions with this property are: four spot, six spot, six spot composed with superflip. a new way to generalize this is to consider positions m such that for any turn x , we have x m = n x , where n is the same pattern as m , but perhaps in a different orientation. for such a position, we can cyclically shift any maneuver, by shifting the first turn to the end, and then conjugating by the appropriate cube symmetry. for example, consider the position in which the UFR corner is twisted clockwise, and the other seven corners are twisted counterclockwise. (i'll call this "1-7-twist" for now, but this pattern needs a better name.) this position is created by U F2 B' U B U D2 R2 U2 B U' D2 B U F2 B L2 B2 now, cyclically shift the U at the beginning to the end to get F2 B' U B U D2 R2 U2 B U' D2 B U F2 B L2 B2 U which produces a different orientation of the same position; this time, the ULF corner is twisted clockwise. now conjugate this maneuver by C_U to get R2 L' U L U D2 B2 U2 L U' D2 L U R2 L F2 L2 U which produces the original position, in its original orientation. actually, there are 3 cube symmetries by which one could conjugate, since the position has 3-fold symmetry. another position with this type of cyclic shifting property is 1-7-twist composed with superflip. we can combine both types of generalizations, and ask for positions m that have the property that for any turn x , we have x m = n y , where y is another turn, and n is the same pattern as m , but perhaps in a different orientation. for such positions, we can cyclically shift any maneuver by replacing x^(-1) at the beginning of the maneuver by the corresponding y^(-1) at the end, and then conjugating the whole maneuver by the appropriate cube symmetry. two such positions are: 1-7-twist composed with four spot, and 1-7-twist composed with four spot composed with superflip. here's all the examples of cyclic shifters that i know, along with minimal maneuvers: 1. central positions start (0q*, 0f*) superflip R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q*, 22f) F' B' D2 L' B2 L2 F2 U' D B' D2 R L D' F2 U' L2 D' F2 D' (20f*, 28q) 2. four spot F2 B2 U D' R2 L2 U D' (12q*, 8f*) six spot F B' U D' R L' F B' (8q*, 8f*) four spot composed with superflip U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f) F U2 R L D F2 U R2 D F2 D F' B' U2 L F2 R2 B2 U' D (20f*, 28q) six spot composed with superflip R' U D R' U F' D R' B U' L' U' F' D F' B' D' R' F D F D' R2 (24q*, 23f) U2 F B' R F L2 F2 D B2 D2 R2 B' L2 F' D2 R2 D' B R B2 (20f*, 30q) 3. 1-7-twist F R' U' L' F' U' B' L' U' R2 F L' D' R' F' D' B' R' D' L2 (22q*, 20f) F R2 L' F L F B2 U2 F2 L F' B2 L F R2 L D2 L2 (18f*, 26q) 1-7-twist composed with superflip F R F' R U B L D B D' L' D L F B' R B D F R' (20q*, 20f) F R L' B L D F' L2 B D2 B' L' F2 B' D B U L B (19f*, 22q) 4. 1-7-twist composed with four spot F B2 L' U' B' L U' D L2 U R' U' R F' U' L' F D B' U' (22q*, 20f*) 1-7-twist composed with four spot composed with superflip F U' R' L F' U R L' U2 D' B R L F' R D' R' F2 L U' (22q*, 20f*) as usual, i give a maneuver which is minimal in both metrics whenever this is possible. i don't claim that i've found all positions in these categories, but these are all that i know. if you find any others, they'd be good candidates for positions far from start. mike