>It has an interesting article by Herbert Kociemba from Darmstadt, who
>describes his program to solve Rubik's Cube. He states that he has not
>yet encountered a configuration that required more than 21 moves. A short
>description follows:

it would be nice to know how many patterns he has tested.

He does not say how many, but his article gives nine patterns that have been
published earlier in CFF for which he finds shorter answers. Also more are
promised for future issues.

 > > His first stage can loosely be described as working in a three dimensional
 > > coordinate system where the coordinates are resp. twist, flip and permutation.
 > > He searches his way until the coordinate [0,0,0] is reached.
...
 > >                                                           He does also
 > > use tree pruning.

does he describe his method of "tree pruning"? this would seem to
be the "intelligent" part of the program, i.e. recognizing when to
abandon a given approach. if anyone has any insight on the tree
pruning, please let me know.

I can give some information. What he does do is calculate in advance through
the three axis of his space the minimal number of moves needed to get at
[0,0,0]. This is used for tree pruning. It obviously will not prune
everything (e.g. if you are at point [x,y,z] it may very well be that [x,?,?]
for other points requires less moves, and similar across the y and z
direction), but he tells that his pruning is very effective. I do not know
how he prunes in the second case, because he does not completely describes
the coordinates of his second space, but I presume pruning is done in a
similar way.

> it's not likely that this will lead to a proof of an effective upper
> bound. perhaps he can shave a few moves off the 42 obtained by
> kloosterman, but i wouldn't expect him to prove an upper bound
> anywhere near 21.
I think so too. Moreover, it is difficult to take in account what he
found, namely that a minimal solution in the first stage does not guarantee
a minimal overall solution.

> i believe that the diameters of the respective coset spaces are exactly
> those numbers listed in the "Best Possible" line. can anyone confirm
> this? i've finally written a few programs, and those are the diameters
> i get. i'm surprised that thistlethwaite didn't just do an exhaustive
> search on these coset spaces. perhaps it's just a matter of not having
> the technology when he did his work (~12 years ago).
Well, apparently Thistlethwaite did not know that those were the diameters,
otherwise I have no explanation for the question marks as they appear in
Singmaster.

> now how about "superflip," and also "supertwist?"
I will try to contact him to see what he has to say about those.