On Rolling Loaded Dice

When an unfair die is tossed, what are the factors that determine the side upon which it lands? Sir Hermann Bondi (see European Journal of Physics 14, pp. 136-140) asked a related theoretical question in 1993 with the intention of determining the theoretical probability of a coin landing on its edge. He notes that the center of mass, the coefficients of restitution and friction, and the radius of gyration all play a role, perhaps. A simple model assumes that the probability of landing on a particular side is proportional to the solid angle subtended from the center of mass, but this model predicts too few base landings for tall cylinders, and too many rolling landings for squatty cylinders. Here we propose a thermodynamic modification of this model which qualitatively improves the match between experiment and theory by introducing an effective ``temperature'' parameter. We apply the model to several different geometrical shapes where the landing odds are not even, including right circular cylinders, rectangular prisms, hemispheres and semi-cylinders. We obtain, perhaps unreasonably, somewhat promising results.