ABSTRACT
This chapter examines averages. Everyone knows how to compute the average of a set of numbers, but the usefulness of averaging is solving problems is often overlooked; sometimes one can even find an answer by starting from an average and, so to speak, reverse-engineering. Eight puzzles allow the reader to use averaging to tip scales, represent numbers as sums of squares, and compute the time it takes to find a jack in a shuffled deck. The theorem presents a surprising and little-known fact from graph theory: that given any graph, and any ordering of its edges, it is possible to find a walk whose edges are in strictly increasing order, and whose length is at least the average degree of the graph.
