Polyhedron Coloring
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 (Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one way to color a cube (Ball and Coxeter 1987, pp. 238-239) and 144 ways to color an icosahedron (Ball and Coxeter 1987, pp. 239-242). Given four colors, there are two distinct ways to color a tetrahedron and four ways to color a dodecahedron, consisting of two enantiomorphous ways (Steinhaus 1999, pp. 196-198, p. 238). Given five colors, there are four ways to color an icosahedron. Given six colors, there are 30 ways to color a cube. These values are related to the chromatic polynomial of the corresponding dual skeleton graph, which however overcounts since it does not take rotational equivalence of colorings in the original solid into account. The following table gives the numbers of ways to color faces of various solids using at most n colors (with no restriction about colors on adjacent faces). This can be computed by finding the graph automorphisms of the skeleton of the polyhedron, removing the symmetries that invert a face (leaving pure rotational symmetries only), then finding the induced symmetry group for the faces and applying the Pólya enumeration theorem. solid | polynomial | OEIS | colorings for n = 1, 2, ... cube | 1/3 n^2 + 1/2 n^3 + 1/8 n^4 + 1/24 n^6 | A047780 | 1, 10, 57, 240, 800, 2226, 5390, ... dodecahedron | 11/15 n^4 + 1/4 n^6 + 1/60 n^12 | A000545 | 1, 96, 9099, 280832, 4073375, 36292320, ... icosahedron | 2/5 n^4 + 1/3 n^8 + 1/4 n^10 + 1/60 n^20 | A054472 | 1, 17824, 58130055, 18325477888, 1589459765875, ... octahedron | 1/4 n^2 + 17/24 n^4 + 1/24 n^8 | A000543 | 1, 23, 333, 2916, 16725, 70911, 241913, ... tetrahedron | 11/12 n^2 + 1/12 n^4 | A006008 | 1, 5, 15, 36, 75, 141, 245, 400, 621, ...
