Four Proofs of the Ballot Theorem 1

One of the great pleasures in mathematics occurs when one considers several different proofs of a single result. In fact, when one considers the myriad proofs of the Pythagorean theorem and the irrationality of √ 2 constructed over the centuries, it seems we humans can never be satisfied with just one proof. Why do we continue to devise new approaches to known results? There is something in the reasoning itself that brings insight to the problem beyond what the result tells us, like looking at a sculpture from many different perspectives to appreciate it as fully as possible. In this article we present four proofs of the ballot theorem, describe some of the history surrounding each of the proofs, and consider the different perspectives that each brings to the problem.