From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 7 21:38:40 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 VAA27186 for ; Wed, 7 Apr 1999 21:38:39 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu From: Jerry Bryan To: Cube Lovers Subject: Re : Re: Inventing your own techniques In-Reply-To: <14088.45800.718995.311244@cipr.no_spam.rpi.edu> Message-Id: Date: Tue, 6 Apr 1999 20:49:37 -0400 (Eastern Daylight Time) On Mon, 05 Apr 1999 08:56:08 -0400 (EDT) "Frederick W. Wheeler" wrote: > Wei-Hwa Huang sent me this teaser about conjugation. > > whuang@ugcs.caltech.edu (Wei-Hwa Huang) wrote: > > After I understood conjugation well enough, I have never invented a > > move that I can in all honesty call "new" -- although they may > > appear "new" to others. The only new part is just applying it to > > different types of moves and seeing what the result is. > > Later, at my request, Wei-Hwa Huang was kind enough to elaborate on > conjugation. > > whuang@ugcs.caltech.edu (Wei-Hwa Huang) wrote: > > I keep on meaning to write a more detailed explanation but can never > > seem to find the time. > > > > Essentially, by conjugation I mean taking two routines (call 'em A > > and B), consider their reverses (a and b), and juxtapose them (do > > the move ABab). When A and B have a small intersection the results > > of the conjugation is a simple permutation. And pretty much more > > cube puzzles can be solved if you have the simplest permutations. > > > > Eg, to rotate two corner pieces, let > > A = R'DRFDF' (rotate one corner in the top face without affecting > > the rest of the top face) > > B = U (rotate the top face) > > > > As A and B have a small intersection (one corner cubie), the move > > ABA'B' is quite useful. > > > > Note that A is itself a move arrived at by conjugation. There are two separate ideas here. A process of the form XYX'Y' is called a commutator rather than a conjugate. As you say, a commutator which moves very few cubies can be a very useful process. In fact, the number of cubies moved by XYX'Y' can be used as a sort of informal measure of how close X and Y come to commuting. In the extreme case where X and Y do commute, we have XYX'Y'=YXX'Y'=YY'=I so that no cubies are moved. And conversely, two processes X and Y which "nearly" commute and/or which intersect in very few cubies are good candidates for forming a useful commutator. A process of the form Y'XY is called a conjugate, and in particular is called the conjugate of X by Y. Note that YXY' is also a conjugate, and in particular is called the conjugate of X by Y'. This can be a little confusing because a few books (incorrectly in my opinion) call YXY' the conjugate of X by Y. Of Y and Y', which is the "real" process and which is the inverse is totally arbitrary. For example, if Z=Y', then Z'=Y. So we could write a conjugate as YXZ (the conjugate of X by Z) and another conjugate as ZXY (the conjugate of X by Y) if we know that Y and Z are respectively the inverses of each other. It is sometimes said that the conjugate Y'XY results in X shifted by Y, which is the real utility of using conjugates to solve a cube. Use a process you know, but shift it to apply to a slightly different set of cubies. Your process A=R'DRFDF' consists of the conjugates R'DR (the conjugate of D by R) and FDF' (the conjugate of D by F'). It is often the case that useful processes can be formed from both commutators and conjugates. I am perversely proud that my own personal solution technique for solving the cube consists of only two processes -- one for the corners and one for the edges -- plus conjugates of those two processes. I think it is indicative of the power of conjugates that a cube can be solved with so few processes provided only that they are combined with conjugation. I am "perversely proud" because my "two processes" technique probably yields one of the slowest solution times of anybody on Cube-Lovers. I am always embarrassed to read about those people who can do it in under 30 seconds. I have taught myself some of the faster techniques, but I always find that after a few months the only technique my hands can remember is the old, slow one which I invented myself many years ago. ---------------------------------------- Jerry Bryan jbryan@pstcc.cc.tn.us