(Solved): A dodecahedron is a regular convex polyhedron made up of 12 regular pentagons. (a) Suppose you color ...

A dodecahedron is a regular convex polyhedron made up of 12 regular pentagons. (a) Suppose you color each pentagon with one of three colors. Prove that there must be two adjacent pentagons colored identically. (b) What if you use four colors? (c) What if instead of a dodecahedron you colored the faces of a cube?