From cube-lovers-errors@oolong.camellia.org Mon Jun 30 21:57:52 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id VAA19066; Mon, 30 Jun 1997 21:57:52 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org Message-Id: <199707010132.VAA26778@life.ai.mit.edu> Date: Mon, 30 Jun 1997 21:37:39 -0400 From: michael reid To: cube-lovers@ai.mit.edu, jbryan@pstcc.cc.tn.us Subject: example of a local maximum whose inverse is not a local maximum jerry bryan asks if the inverse of a local maximum is necessarily a local maximum. the following example shows that this need not be the case. the interesting "six-two-one" pattern is produced by the sequence B U2 F2 R U' R' B' R' U F2 U2 (15q) this position has six symmetries, generated by the cube rotation C_UFR and central reflection. therefore we also have the maneuvers L F2 R2 U F' U' L' U' F R2 F2 D R2 U2 F R' F' D' F' R U2 R2 F' D2 B2 L' D L F L D' B2 D2 R' B2 L2 D' B D R D B' L2 B2 U' L2 D2 B' L B U B L' D2 L2 for the same position. it is not hard to check (by computer) that these are minimal maneuvers. note that for each quarter turn, we have a maneuver that ends with that quarter turn. thus, from this position, any quarter turn brings us closer to start, so our position is a local maximum. consider now the inverse position; it is produced by U2 F2 U' R B R U R' F2 U2 B' (15q) it is not hard to check (by computer) that applying the quarter turn B' to this moves us further from start (16q), so this position is not locally maximal. note that this is already in the archives; i first reported it on april 20, 1995 in my message "correction and an interesting example" mike