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,
firstname.lastname@example.org (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