I'm not quite satisfied with the numbers that have been quoted
for various cube groups. I believe they are probably correct,
but I haven't seen anything resembling a satisfactory sketch
of a proof.
Number of Reachable Positions:
The standard calculation runs like : we have 8 corners that can
be in all 8! arrangements, and 12 sides that can be in all 12!
arrangements, except that the total permutation must be even.
( I'll talk about orientations later)
I can accept this figure as an upper bound, but can anyone demonstrate
that all the positions not ruled out can actually be reached? The only
way I can think of is an actual construction, which means finding
a generator for the group (for instance of corner cube positions) and
showing that its order is 8!. This is somewhat less than elegant,
and requires an unspecified bit of magic to 'find' a generator
for the group.
I also don't like the recourse to geometric arguments ( the corners
and sides can't be interchanged ) but I am willing to accept it.
Number of Reachable orientations:
A similar line of reasoning is used to argue the number of reachable
orientations : the amount of twist on all corners and (independantly)
on all sides is a multiple of 360 degrees. I haven't seen a demonstration
that all the not-forbidden orientations are actually reachable.
Finally, I haven't seen a demonstration that the orientation and permutation
subgroups are indepentant, that is, that you can get to an arbitrarily
selected location and an arbitrarily selected orientation at the same
time. This assumption is the basis for Singmasters claim that there
are 12 orbits of cubes : ( even-spacial-permutation X even-side-orientation
X one-of-three-corner-orientation = 2 X 2 X 3 = 12 )