There are many ways to associate a spectrum of numbers to a surface: Two of the most classically studied are the eigenvalues of the Laplacian and the lengths of closed geodesics. People often ask whether two different surfaces can have the same spectrum of numbers, and there’s a long and beautiful story attached to that question. Here’s a twist on the setup: Now consider a polygon in the plane and label its sides with letters. Follow a billiard ball trajectory around the surface and record the “bounce sequence,” or the sequence of labels hit by the ball as it moves. Is it possible for two different billiard tables to have all the same bounces?