On the probability that two random elements will generate the entire
cube group, I wrote:
... 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.
I can now answer the spatial orientation part of the question, and
it's much lower. We're talking about C, the 24-element group of
proper motions of the whole cube. If we select two elements at random
with replacement, the probability is only 3/8 that they will generate
the whole group.
The kinds of motions that can take part in a generating pair are a
90-degree rotation about an axis, a 120-degree rotation about a major
diagonal, and a 180-degree rotation about a minor diagonal. Note that
the last kind fixes two major diagonals and an axis. Two motions
generate C iff they are
(48 ways) a 120 and a 180, unless they fix the same major diagonal, (48 ways) a 180 and a 90, unless they fix the same axis, (24 ways) two 90s at right angles, or (96 ways) a 90 and a 120.
The number comes out so even I suspect there's something deeper going
on than the exhaustive analysis I used.
As for generating the (fixed-face) Rubik's group, I still suspect that
two elements almost always generate the entire group unless they are
both even. Anyone who has a Sims's-algorithm implementation handy
could help out with a Monte-carlo approximation to see if this is
approximately right. Or, I wonder, is there a way of getting an exact
number, perhaps with the help of GAP?
Dan posted and e-mailed Hoey@AIC.NRL.Navy.Mil