Number theory begins with down-to-earth questions like the number of solutions to Diophantine equations (polynomials in several variables with integer coefficients). But the methods needed to solve these problems often take us into wild and difficult territory, e.g., automorphic forms. In this talk, we explain the role of the Hecke algebra in giving simpler and very explicit ways of studying certain automorphic forms via algebra and combinatorics. We won't assume prior knowledge of any of these topics, so the talk should be accessible to a wide audience.