From:

Subject:

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