From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 20:49:17 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA28649; Wed, 1 Oct 1997 20:49:16 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From roger.broadie@iclweb.com Wed Oct 1 19:12:26 1997 From: roger.broadie@iclweb.com (Roger Broadie) To: Subject: Re: 4x4x4 solution Date: Thu, 2 Oct 1997 00:09:47 +0100 Message-Id: <19971002000735.AAA23683@home> I'm tempted to try a little more analysis of the parity constraints on the 4x4x4 cube, though no doubt it's all been done before. As der Mouse said, A slice turn produces a 4-cycle on the edges and two 4-cycles on the face centres; a face turn produces a 4-cycle on the face centres and two 4-cycles on the edges (and a 4-cycle on the corners, which may or may not be relevant). I think it is very relevant. We can set the effects out as follows: Turn Piece Cycle(s) Parity ------ ------- -------- ------- Slice edge 1x4 odd centre 2x4 even Face edge 2x4 even centre 1x4 odd corner 1x4 odd The consequence is that the parity of the centre pieces depends entirely on the number of face turns - any slice turns do not affect the parity of these pieces since the changes they introduce will be of even parity. For face turns, the changes to the parity of the corner pieces and the centre pieces are the same. Hence if the corner pieces are in place, the centres will be in an even permutation, and that will not be changed even if the edge pieces are in an odd permutation, which was the essence of Clive McCaig's original question. Nor will that be changed by any turn of a central slice to bring them back to an even permutation. I the corners are correct (which I guess is the normal situation when the problem with the swapped edge pieces shows up) then, though I say so with some hesitation, I do not think Jerry Bryan is right in saying that the pair of swapped edge pieces will be matched by a pair of swapped centre pieces. For example, the process I quoted switches edge pieces, and though it has no visible effect on the centre pieces, it does in fact change the positions of the centre pieces on the front face (if I have correctly identified the results of a bit of hasty work with little Post-it stickers). However, the whole block of four rotates through 180 degrees, which is two 2-cycles and thus of even parity. Edwin Saesen could mark the centre pieces, get them back to their original position and still find the edge pieces swapped, but that will not prevent his correcting the edge pieces, and then, if he wants to, correcting the centre pieces with even-parity processes. Luckily, for the 4x4x4, we do not have to worry about twists for the edge pieces or the centre pieces, since that is fixed geometrically for each position they can occupy. When an edge piece is in its home position it must be the right way round. When it moves to its next-door position it must flip. I imagine this is the point behind Allan Wechsler's charming square-dancing analogy. The centre pieces always present the same corner to the central intersection of the face. Roger Broadie