Polyhedron coloring assigns colors to the faces of a polyhedron. A proper coloring requires that no two faces with a common polyhedron edge share the same color. In the counts below, the colors are distinguished and colorings related by a rotation of the regular polyhedron are identified. Mirror image colorings are counted separately unless a rotation identifies them.
Given two colors, there is a single proper coloring of an octahedron (Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one proper coloring of a cube (Ball and Coxeter 1987, pp. 238-239) and 144 of an icosahedron (Ball and Coxeter 1987, pp. 239-242). Given four colors, there are two proper colorings of a tetrahedron (Ball and Coxeter 1987, p. 238) and four of a dodecahedron, consisting of two enantiomorphous pairs (Ball and Coxeter 1987, p. 238; Steinhaus 1999, pp. 196-198). There are four colorings of an icosahedron with five colors under the stronger requirement that the five faces incident to each vertex all have different colors (Makai and Tarnai 2021). There are 30 colorings of a cube in which each of its six faces has a different color (Steinhaus 1999, p. 167). The chromatic polynomial of the dual graph of a polyhedron's skeleton counts proper colorings before identifying those related by rotations. Additional restrictions, such as requiring all faces at a vertex to have different colors, are not imposed by this chromatic polynomial.
The following table gives the numbers of ways to color the faces of the Platonic solids using at most 
colors, with no restriction on the colors of adjacent faces.
The colors remain distinguished and colorings related by rotations
are identified. The counts follow from the group action
of the rotational symmetry group on the faces
using the Pólya enumeration theorem.
| solid | polynomial | OEIS | colorings for  , 2, ... |
| cube |  | A047780 | 1, 10, 57, 240, 800, 2226, 5390, ... |
| dodecahedron |  | A000545 | 1, 96, 9099, 280832, 4073375, 36292320, ... |
| icosahedron |  | A054472 | 1, 17824, 58130055, 18325477888, 1589459765875, ... |
| octahedron |  | A000543 | 1, 23, 333, 2916, 16725, 70911, 241913, ... |
| tetrahedron |  | A006008 | 1, 5, 15, 36, 75, 141, 245, 400, 621, ... |
