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Define a valid ``coloring'' to occur when no two faces with a common Edge share the same color. Given two colors, there is a single way to color an Octahedron. Given three colors, there is one way to color a Cube and 144 ways to color an Icosahedron. Given four colors, there are two distinct ways to color a Tetrahedron and 4 ways to color a Dodecahedron. Given five colors, there are four ways to color an Icosahedron.
See also Coloring, Polyhedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 238-242, 1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 82-83, 1989.