From cube-lovers-errors@mc.lcs.mit.edu Fri Apr 30 20:25:46 1999 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 UAA07983 for ; Fri, 30 Apr 1999 20:25:46 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Apr 30 07:35:29 1999 Date: Fri, 30 Apr 1999 12:33:43 +0100 From: David Singmaster To: whuang@ugcs.caltech.edu Cc: cube-lovers@ai.mit.edu Message-Id: <009D768E.2615E470.8@ice.sbu.ac.uk> Subject: RE: Reinventing (and some edge-flipping techniques) I've just spent three days being a Rubik's Cube demonstrator at a trade fair, using the method given the my Step by Step solution in the middle of the fifth edition of my Notes. The problems that Wei-Hwa Huang has described are ones that I found in trying to develop an easy method. Basically there is no 'simple' way to flip edges, where 'simple' means easily understood and remembered. There are simple ways to move edges, move corners and twist corners. Consequently I decided to get the edge orientations correct at the beginning of work on the last face, so that I wouldn't have to worry about what happened to the rest of the face. In case you haven't got my notes at hand, I used BLUL'U'B' which is a simple conjugate of the commutator [L,U]. This exchanges the four U corners as two pairs of exchanges and cycles three U edge, but effectively flips two of them on the way. Using the inverse process BULU'L'B' does the same thing, but one of them effectively flips two adjacent edges and the other flips two opposite edges. Two applications will flip all four edges. Then we know simple processes which do 3-cycles or pairs of 2-cycles of edges, preserving orientation, or of corners and we have simple processes for twisting corners. So we could carry out these three steps in any order giving six possible algorithms and I'm pretty sure I received examples of all of these. Indeed, of one considers doing the last face as having a fourth stage of orienting edges, there are 24 possible algorithms and at one point I was classifying algorithms into these 24 cases. I don't think I kept up with this long enough to have all 24 cases, but I expect all of them exist! Some personal comments and recollections, much of which is in my Notes. The ideas of monotwist and monoflip, though blindingly obvious, took well over a year to emerge! Despite the fact that lots of quite bright mathematicians were working on the cube (e.g. Conway, Penrose, Rubik), I remember first hearing about the idea in Jan 1980 (?) from Peter McMullen who said they were using the idea at Cambridge. At first it seemed unreasonable as we generally were looking for moves that only affected the U face and the mono-moves are almost all elsewhere. However once I realised that the idea gives a way of building simple moves, I realised that the commutator [F,R] was a mono-move in the L face and its square was a mono-twist in the L face. The Cambridge group had been using shorter, but less simple, moves. With these mono-moves, it was now pretty easy to build the algorithm that is my Step by Step solution and I think I did it within a few weeks as I recall the 5th ed. of my Notes was produced by March. (Remebering dates from 20 years ago is always a bit dodgy - check what's in the Notes, which I don't have a copy of here.) DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk