From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 1 10:26:49 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id KAA13649; Tue, 1 Sep 1998 10:26:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Tue, 01 Sep 1998 09:59:50 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Strong Local Maxima 9f and 10f from Start To: Cube Lovers Message-Id: #1. D2 L2 B2 F2 U2 B2 F2 U2 R2 U2 (10f*) #2. D2 F2 L2 D' U L2 F2 D' U' (9f*) #3. U2 B2 L2 D U' R2 B2 D' U' (9f*) #4. B2 D' U' L2 B2 L2 D' U' B' F' (10f*) #5. L2 U2 F2 L2 D2 F2 U2 R2 B' F' (10f*) #6. D' U' B2 R2 D2 L2 D U B' F' (10f*) #7. D U L2 D2 R2 F2 D' U' B' F' (10f*) #8. B2 F2 D U' B' F L R' D U' (10f*) This completes the list of strong local maxima 9f and 10f from Start in the face turn metric. I posted #1, #2, and #3 previously, but the rest are new. 9f is the shortest strong local maximum. I continue to think that all eight of these positions share a special kind of symmetry that is related to the fact that they are strong local maxima, but I can't quite get my arms around a good description for this symmetry. Generally speaking, they look more symmetric if you look at corner cubies and edge cubies separately than if you look at them in combination. Also, they look more symmetric if you look only at the colors of the facelets (looking at two dimensional 3x3 faces) rather than if you look at the location of entire cubies. They do all share the following in common. Looking just at the colors of the facelets, all pairs of opposed 3x3 faces have the same pattern for all eight positions. Hence, there are (up to) three different face patterns for each position. Also, if the cube is colored according the "opposite faces differ by yellow" convention, then the pairs of opposed face patterns for all eight positions are the "yellow complements" of each other. Finally, all the face patterns (and some of them are fairly complicated, having as many as four colors) are symmetric with respect a reflection across either a vertical or horizontal axis of the 3x3 square making up the face. Even though none of these strong local maxima are q-transitive in the classic Saxe-Hoey sense, the "face symmetry" they all share seems too unusual to me to be just a coincidence. I think #8 is an especially interesting position. All six faces have the same face pattern, sort of a three colored checkerboard (if that is not a contradiction in terms). The position is basically Pons Asinorum with the edge and corner cubies rotated as a unit along a major diagonal axis relative to the fixed face centers. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us