The second observation I would like to make regarding the
Pons Asinorum involves the Supergroup (also known as "the extended
problem") in which the orientation of face centers is considered.
The process UUDDFFBBLLRR turns each of the face centers 180
degrees, so Pons Asinorum is symmetric in the Supergroup as well.
(Turning each face center 180 degrees is the M-symmetric position
Big Ben Squared, which I will call Noon.) There is another optimal
way of making a (pseudo-) Pons Asinorum, (UD'FB')^3, which differs
from the true Pons only in the face center orientations. According
to an exhaustive search I carried out by hand, this is the only
pseudo-Pons (up to M-conjugacy) that can be obtained with six slice
moves. I would be very interested in hearing about any other
twelve-qtw positions which differ from the Pons Asinorum only in
I have found a 16-qtw process for Pons Asinorum composed
with Noon, (UD FB FB UD)(FB UD UD FB), which looks like Pons
Asinorum, but does not rotate the face centers. This in turn gives
a 20-qtw process for Noon itself:
LLRR UUDD (UD FB FB UD) (FB UD UD FB) FFBB
= LLRR (U'D' FB FB UD) (FB UD UD F'B').
Of course, there's no assurance of optimality here.
It occurs to me that many readers of this list may find
details of the Supergroup uninteresting. I have more on this
subject, so if you would or wouldn't like to know more about the
Supergroup, send a vote to Hoey@CMU-10A and we'll see what to do.