Since we have this mailing list I am tempted to put on record an
interesting property of the cube that some of you might not be aware
The cube has a degree of freedom that is rarely considered. The six
center faces which are the pivots about which rotations are performed
do not always return to their original orentation. In other words, if
you were to paint little arrows on the center square of each face of a
solved cube, randomized the cube, and then solved it, you might well
find that the arrows no longer pointed in the same directions as they
did before. (Now you have to get them back. And you thought you had
already solved the damn thing!)
This extended cube problem has been solved independently twice to my
knowledge (not by me, I cheated and learned someone else's solution).
The property was first shown to me by Spencer Love. Does anybody know
if anybody else ever discovered it independently?
Without considering this property we already knew that there were 43252003274489856000 = (8! * 3^8 * 12! * 2^12)/12 different rearrangements of the cube (all those numbers are obvious except the 12 in the denominator). Considering this property raises that number to 88580102706155225088000 = (8! * 3^8 * 12! * 2^12 * 4^6)/24 (the 24 being just as hard to explain as the 12 was).