An Invitation to Proofs Without Words

Abstract

Proofs without words are pictures or diagrams that help the reader see

why a particular mathematical statement may be true, and also see how one might begin to go about proving it true. In some instances a proof without words may include an equation or two to guide the reader, but the emphasis is clearly on providing visual clues to stimulate mathematical thought. While proofs without words can be employed in many areas of mathematics (geometry, number theory, trigonometry, calculus, inequalities, and so on) in our “invitation†we examine only one area: elementary combinatorics. In this article we use combinatorial proof methods based on two simple counting principles (the Fubini principle and the Cantor principle) to wordlessly prove several simple theorems about the natural numbers.