From hoey@aic.nrl.navy.mil Wed Nov 29 12:18:52 1995 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23715; Wed, 29 Nov 95 12:18:52 EST Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA26702; Wed, 29 Nov 95 12:14:41 EST Return-Path: Received: by sun13.aic.nrl.navy.mil; Wed, 29 Nov 95 12:14:40 EST Date: Wed, 29 Nov 95 12:14:40 EST From: hoey@aic.nrl.navy.mil To: mschoene@math.rwth-aachen.de (Martin Schoenert), frb6006@cs.rit.edu (Frank R Bernhart), Cube-Lovers@life.ai.mit.edu Newsgroups: sci.math Subject: Generating Rubik's Cube Message-Id: <9511291210.Hoey@AIC.NRL.Navy.Mil> References: <1995Nov29.054118.9651@cs.rit.edu> Distribution: About generating the cube's group with arbitrary elements of that group, mschoene@Math.RWTH-Aachen.DE (Martin Schoenert) writes: > ... Rubik's cube can be generated by 2 elements. > Moreover almost any random pair of elements will do the trick.... Actually, I think it's more accurate to say that a random pair of elements has nearly a 75% probability of generating the cube. At least, I'm pretty sure that's an upper bound, and I don't see any reason why it shouldn't be fairly tight. That's for the group where the whole cube's spatial orientation is irrelevant. I think it's more like 56% (9/16) if you also need to generate the 24 possible permutations of face centers. About the minimal presentation of the cube group on the usual generators, frb6006@cs.rit.edu (Frank R Bernhart) writes: > The answers may be in SINGMASTER, et.al. > "Handbook of Cubic Math" or BANDEMEISTER (sp?) "Beyond R. Cube" I recall Singmaster wanted to know if anyone found a reasonably-sized presentation; I don't know if any have been found in the intervening fifteen years. The best I know of is a few thousand relations, some of them several thousand letters long. I've been meaning to try chopping that down a bit. Dan posted and e-mailed Hoey@AIC.NRL.Navy.Mil