The paper's aim is to demonstrate the utility of the well-known Nim game when it is used as a tool in teaching the binary system. A way to increase student comprehension, it also forms a basis for student projects and extracurricular activities on the topic. The paper offers descriptions of two versions of the game and the underlying mathematical theory, the winning algorithm, an elementary introduction to the binary system, structured proof of the winning strategy, and data structures and algorithm for computer implementation. All materials are offered in a form oriented towards presentation and discussion in the classroom. (Contains 2 tables, 1 figure, and 4 online resources.)