The Downtown Problem: Variations on a Putnam Problem

Abstract

SummaryThe following problem appeared in the afternoon session of 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. In general, which points are closer to the center of a square (or cube) than to any of the four edges (six faces)? And what do we mean by “closer”? Using a variety of distance functions (metrics), we will investigate these questions and pose several additional open problems for the reader to explore.