Together with Kloosterman's result for their third and fourth phase (which
together form Kociemba's second phase) the upperbound on God's algorithm
is now 37.

well, at least i had the record for a couple of days! ;-)

          Below follows the set of distances for the first phase:
0:          1
1:          4
2:         74

but i don't understand how we can get 74 positions at distance 2 from
only 4 positions at distance 1. the 4 positions at distance 1 are
easy to see: they're the positions obtained from START by the turns
B, F, L and R. with only 18 different face turns, each should
extend to at most 18 positions at distance 2. am i missing something
obvious here? (the numbers do seem to add up, though.)

I conjecture that the maximal distance in phase 2 is at most 16. There is a
lower bound on it of 14.

the pattern (written in permutation notation) (FR, FL) (UFL, DFR) is
at distance 15, so that's (also) a lower bound. however, if the whole
cube is turned so that the F face becomes the U face, then the new
pattern is still in the subgroup of stage 2, but is now at distance 14.

mike