From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 22 19:50:07 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id TAA19137; Tue, 22 Sep 1998 19:50:02 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Mon, 21 Sep 1998 16:34:29 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Local Maxima which Fix the Corners, 12q from Start To: Cube Lovers Message-Id: I am making a run to calculate God's Algorithm out to 12 moves from Start in the quarter turn metric. It has been running several weeks, and will probably run several more. I have made some changes to my program to make it easier to extract the positions for local maxima, and to checkpoint the local maxima data. As a part of the checkpointing, I can actually see the local maxima as they are generated without having to wait for the program to end. It is becoming apparent that there are a *lot* of local maxima 12q from Start. It is already known that there are only four (unique to symmetry) which are 10q from Start (the shortest ones in the quarter turn metric), and that there are none 11q from Start. So I am a little surprised that I am seeing so many. I have looked at quite a few of them, and most of them are not all that interesting. But the ones which fix the corners are all quite pretty. Because the positions are being produced in lexicographic order, and because I am sorting by corners first, edges second, the positions which fix the corners are the first ones to appear. There are eight of them as follows. 1. F2 L2 F2 B2 R2 B2 2. F B' U2 D2 F' B R2 L2 3. F B R2 F' B' U D L2 U' D' 4. D' F B' R F R' F' B U F' U' D 5. F B R2 L2 F B U2 D2 6. R L' F2 B2 R L' F2 B2 7. F2 B2 U2 D2 R2 L2 8. R L' U D' F B' R2 L2 U D' #1 is a 2-H pattern (only four edge cubies are moved). #2 is a 4-H. #3 moves four edge cubies, leaving eight of the nine facelets the same color on four faces, and a solid color on the other two faces. #4 moves three edge cubies, leaving eight of the nine facelets the same color on all six faces. #5 has 2 H's, 2 checkerboards, and 2 solid faces -- with the respective H's, checkerboards, and solid faces opposing each other. #6 has 4 H's and 2 checkerboards, with the 2 checkerboards opposing each other. #7 is the Pons Asinorum, and is included only for completeness because we already knew that the Pons was a local maximum of length 12q. #8 has all six faces being sort of a "three colored checkerboard". Some of these positions may have appeared on Cube-Lovers in some other context, but the only one I recognize for sure is the Pons. In some ways, #4 is the most interesting to me, because it a simple 3-cycle on the edges, and who would have thought that such a position would turn out to be a local maximum? #1 and #3 both consist of two 2-cycles on the edges, and are about as striking to me as is #4. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us