Let M be a compact surface and let N be the surface of one of the Platonic solids (a "Platonic surface"). A function f : M —> N is an "n-fold branched covering of N, branched over its vertices" if both 1) and 2) hold: ----- 1) For every point p of M such that f(p) is *not* a vertex of N, p has a neighborhood U that f takes homeomorphically onto the subset f(U) of N. Also, f^(-1)(f(p)) contains exactly n points of M. —and— 2) If p is a point of M such that f(p) *is* a vertex of N, then p has a neighborhood U that f takes to f(U) by wrapping around f(p) n times (just like the mapping z |—> z^n near the origin of the complex plane). In this case, f^(-1)(f(p)) contains only the point p of M. ----- Puzzle: ------- For each Platonic surface N and for each integer n ≥ 2, construct such a mapping f : M —> N for some compact surface M. —Dan