THE RELATIVE DOWNTOWN REGION IN N-GONS AND THE PLATONIC SOLIDS
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The following problem appeared on the Fiftieth William Lowell Putnam Mathematical Competition in 1989: A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge [LKA91]. Similar problems in two and three dimensions were considered by Baeth et. al. in the paper “The Downtown Problem: Variations on a Putnam Problem” in 2017 [BLM17]. We generalize these results to all regular n-gons as well as the Platonic solids.