From:

~~~ Subject:

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