A closed volume is bounded by conducting surfaces that are the n sides of a regular polyhedron (n = 4, 6, 8, 12, 20). The n surfaces are at different potentials V_i, i = 1, 2,..., n. Prove in the simplest way you can that the potential at the center of the polyhedron is the average of the potential on the n sides. This problem bears on Problem 2.23b, and has an interesting similarity to the result of Problem 1.10.