Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 1995/11/29
Subject: Generating Rubik's Cube

About generating the cube's group with arbitrary elements of that
group, mscho...@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,
frb6...@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