The last two transforms you describe sound similar to the two I
learned. Mine are also rather repititous. Perhaps it is the case
that the two configurations are very distant using the obvious metric:
smallest number of twists from one to the other.
I wonder if anyone knows very much about the nature of that metric
anyway? I understand that it is known that no two points are more
than 94 (or is it 93?) twists apart (disregarding the extended
problem). I don't know if that number is actually attained, or if it
is only the currently known upper bound based on the best algorithm.
(Or perhaps there isn't an algorithm that good yet, just a proof of
I believe that you and ACW and I once did the math to show that
whatever that longest distance is, it has to be greater than something
around 30, and for the extended cube problem it must be even bigger
than that, so since I can do the transformations you speek of in about
28 (I think) moves, those must not be most distant points.