8. How many different ways can you color an icosahedron with one of three colors on each face?

It turns out that every regular polyhedron has a coloring polynomial that can be used to calculate the number of different coloring. They are:

tetrahedron: (11/12)*n^2 + (1/12)*n^4

cube: (1/3)*n^2 + (1/2)*n^3 + (1/8 )*n^4 + (1/24)*n^6

... ...

icosahedron: (2/5)*n^4 + (1/3)*n^8 + (1/4)*n^10 + (1/60)*n^20

So for icosahedrons (20 faced regular polyhedron) and three colors, the answer is 58130055.

These polynomials can be proved using the Polya Enumeration Theorem, which I heard of while in college, but never bothered to find out what it is about.

For a little more detail, see

http://mathworld.wolfram.com/PolyhedronColoring.html