Coloring a Cube

  1. Coloring each face a solid color and using each of N colors (N=2,3,4,5,6) at least once, in how many ways can you color a cube, such that a duplicate cannot be found simply by rotating the cube?
  2. Using each of N colors at least once (N=2,3,4), in how many ways can you color a tetrahedron?
The "Using each of N colors at least once" in the problem statements helps to limit the number of solutions. Can you solve them if that phrase is changed to "Using N colors"?

Source: Several sources.


Solution
Mail to Ken