From cube-lovers-errors@oolong.camellia.org Thu Jun 5 16:29:27 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 QAA07123; Thu, 5 Jun 1997 16:29:27 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org To: Cube-Lovers@AI.MIT.EDU From: Wei-Hwa Huang Subject: Re: 5x5x5 practical Q's Date: 5 Jun 1997 19:48:39 GMT Organization: California Institute of Technology, Pasadena Message-ID: <5n756n$po3@gap.cco.caltech.edu> References: NNTP-Posting-Host: hedono.ugcs.caltech.edu X-Newsreader: NN version 6.5.0 #3 (NOV) Yes, I was part of the Freecell Project, and it was indeed achieved by many people working in parallel. The rest of this mail will probably be uninteresting to the advanced Cube-lovers on this list. "Cubism for Dummies"? David Barr writes: >I don't understand what ARA'R' means. This is basically the commutator. A and A' represent a sequence of moves and the inverse of said sequence. R and R' represent a sequence of moves and the inverse of said sequence. The general technique is to discover sequences A and R that intersect in as few pieces as possible, then the sequence ARA'R' will only involve those pieces. I.e., suppose sequence A does: (1) an action J on set X and (2) an action K on set Y. Suppose sequence R does: (3) an action L on set Y and (4) an action M on set Z. X, Y, and Z are disjoint; so, Y is defined as the intersection of the two sets that A and R operate on. Ao, the sequence ARA'R' becomes (1) actions J and J' on set X. (2) actions K, L, K', L' on set Y. (3) actions M and M' on set Z. The actions on X and Z cancel out, and you end up with KLK'L' on set Y. If set Y is small, K and L are necessarily simple. Here's an example (take out your cube now): I use this following sequence to flip two edge cubies on the same slice: 1. Position the cube so that the edge cubies are at FL and FR. 2. RF'UR'F 3. Move the center slice to the right. 4. F'RU'FR' 5. Move the center slice to the left. The two edge cubies are now flipped. In this example, A = RF'UR'F, and R (sorry for the redundant notation, hope it doesn't confuse anyone) = Moving the center slice. If you watch the center slice closely during RF'UR'F, you'll notice that all that sequence does is to flip the FR cubie while maintaining the rest of the center slice. It also creates a lot of junk on the top and bottom faces, but since the R sequence only involves the center slice, that junk gets restored in step 4, when we reverse RF'UR'F. To reiterate, our steps are: A : flip an edge cubie, creating some junk R : move the center slice so that another edge is there A' : remove that junk, flipping another edge cubie you just positioned R' : move that center slice back. It is my feeling that a algorithm consisting only of a few basic moves and the ARA'R' technique is the most elegant algorithm of all. (Sorry, God.) Parity of the face pieces aside, the algorithms I use to solve the 3x3x3, the 4x4x4, and the 5x5x5 are almost completely based on this method. They appear very impressive to an onlooker; because of the complexity of A and R, it often appears as if I am messing up the cube, then bringing it back to a slightly more ordered state, then bringing it back to chaos, then ordering it again, etc., etc,. until the cube is solved. The order in which I solve the cube (corners, paired edges, corner faces, edge faces, center edges, corner units, face centers) contributes to this effect. (Similarly, the most impressive way of assembling a jigsaw puzzle is to look at each piece one by one and place each one where it belongs, instead of isolating the edge pieces first, etc.) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ ------------------------------------------------------------------------------- Question everything. Learn something. Answer nothing. -- Engineer's Motto