This engaging lecture takes the audience on a foray through a number of mathematical areas in search of connections between two very different problems. The first problem is a theorem of G. A. Miller from 1910 which comes from the structure theory of finite groups. The second problem is a theorem of von Neumann from 1935 concerning spectra of Hermitian operators on a Hilbert space. Demonstrating his gift for lucid exposition, Halmos shows how these two seemingly unrelated problems are actually linked to the well-known marriage problem. The unity of mathematics emerges as the dominant theme as Halmos skillfully ties together a number of mathematical threads involving group theory, topology, operator theory, finite combinatorics, analysis, and infinite set theory. The lecture would be accessible to undergraduates with a basic background in group theory and point set topology.