Date: Wed, 05 Jun 96 18:54:44 -0400 (EDT)
From: Dan Hoey <hoey@aic.nrl.navy.mil>
To: Cube Lovers
Subject: Re: A essay on the NxNxN Cube : counting positions and solving it

  All the 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
different.

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.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil