Feynman gave a famous elementary introduction to quantum theory by discussing the thin-film reflection of light. We make his discussion mathematically rigorous, keeping it elementary, using his other idea. The resulting model leads to accurate quantitative results and allows us to derive a well-known formula from optics. In the process, we get acquainted with mathematical tools such as Smirnov's fermionic observables, transfer matrices, and spectral radii. Quantum walks and the six-vertex model arise as the next step in this direction.