From hoey@aic.nrl.navy.mil Fri Nov 9 15:01:52 1990 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA16350; Fri, 9 Nov 90 15:01:52 EST Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA24929; Fri, 9 Nov 90 14:57:26 EST Return-Path: Received: by sun13.aic.nrl.navy.mil; Fri, 9 Nov 90 15:02:48 EST Date: Fri, 9 Nov 90 15:02:48 EST From: hoey@aic.nrl.navy.mil Message-Id: <9011092002.AA00993@sun13.aic.nrl.navy.mil> To: Cube-Lovers@life.ai.mit.edu Subject: Rubik's Cube reassembly problem and solution References: <3924@idunno.Princeton.EDU> <1990Nov8.182534.18625@agate.berkeley.edu> Reply-To: Hoey@aic.nrl.navy.mil (Dan Hoey) In rec.puzzles article <1990Nov8.182534.18625@agate.berkeley.edu>, greg@math.berkeley.edu (Greg Kuperberg) writes: >Consider a standard Rubik's cube. Disassemble it and put it back >together at random. Find, with proof, the probability that it can be >solved. It depends on how you take it apart. If you just pull out the corner and edge pieces then put them back in without respect to color, the probability is one in 12 that you will put it back into the right orbit. I won't bore you with yet another proof of this; if you spent the last decade in a box see the archives, Singmaster's NOTES ON RUBIK'S MAGIC CUBE, J. A. Eidswick's article in the March 1986 Math Monthly, or even Hofstadter's METAMAGICAL THEMAS. Now if you take the face centers off and scramble them, then there is only one chance in 60 of getting it right. Of the 720 permutations of the six face centers, only 24 can be generated by rigid motions of the cube. But half of these 24 permutations are odd, and leaving the cube in an unsolvable orbit. If you put the face centers on in the ``standard'' configuration with opposite faces ``differing by yellow'' (i.e., white opposite yellow, red opposite orange, and blue opposite green), your chances go up to one in four--half the time you will get an odd permutation, and half the time you will get a mirror-reversed configuration. But wait, if you took the face centers off you probably noticed that the corners and edges don't stay on very well. So, say you scrambled all three kinds of pieces. You will be able to solve the resulting cube if you could solve the corner/edge permutation and the face- center permutation. But if the only thing keeping you from solving the corner/edge permutation and the face-center permutation is that both permutation parities were odd, then you will be able to solve the two of them together. Therefore your chances of success are one in 360 (= (1/12)*(1/60)*2), or one in 24 if you preserved opposite pairs of face centers. Now suppose you peeled off the 54 colored stickers and stuck them back on at random (carefully keeping them out of the reach of children, as there are rumors the paint contains lead, especially on the cheap Taiwanese knockoffs), what is the probability of getting a solvable cube? This question was posed years ago (in Singmaster?) but I believe it is still open. Dan Hoey Hoey@AIC.NRL.Navy.Mil