Date: Sun, 03 Dec 95 14:46:30 -0500 (EST)
From: Dan Hoey <hoey@aic.nrl.navy.mil>
To: Cube Lovers
Subject: Re: Generating Rubik's Cube

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