real mechanical 3x3x3, 4x4x4, 5x5x5 Cubes that I've seen only have
cubies on the outside, but if you can put back all N^3 cubies in the
one I'm describing then you can certainly do the real ones.
(In Dan Hoey's notation, I believe that this means I treat the Cube as
the G+C group, where G is generated by the outer slice rotations, and
C is the rotations of the entire thing....
Actually, the distinction between G and G+C is that in the latter we
draw a distinction between cubes that differ by a whole-cube move as
When we take account of the internal cubies I call it the "Theoretical
Invisible cube", described in my Invisible Revenge article 9 August
1982. A solution method is given in
Eidswick, J. A., "Cubelike Puzzles -- What Are They
and How Do You Solve Them?", 'American Mathematical
Monthly', Vol. 93, #3, March 1986, pp. 157-176.
that is pretty much like yours, I think.
As for counting the positions, I haven't got around to checking the
numbers in "Groups of the larger cubes", 24 Jun 1987. You might want
to see how they compare to yours.