1)I have lots of black and white squares that can be joined together to make cubes. How many distinguishable cubes can I make?

2)Now I try it with triangles and regular octahedrons?

3)Now pentagons and regular dodecahedrons?

4)Triangles again but making regular icosahedrons!?

Note: Distinguishable means rotations are the same, but reflections are not.

Adding a tetrahedron to the problem, the number of sides to the number of variations looks like this:

Solid      Sides      Variations

Tetra:     4             4

Cube      6            10

Octa       8            21

Dodeca  12          116 (assuming I found them all)

While these numbers should have a relationship to factorials, they loosly approximate a cube function.  I am estimating that a icosahedron, with 20 sides will have in the neighborhood of 500 possible variations. Though this may be a gross underestimation.