lew (01/26/83)
I have thought about the weighted dice problem before and I offer this analysis: The energy of a die is its kinetic energy plus its potential energy and its potential energy is proportional to the height of its center of mass above some horizontal surface. For each orientation of the die there is some minimum height that the center of mass must have. Two orientations are equivalent if one can be obtained from the other by a rotation about a vertical axis. Therefore every distinct orientation can be obtained by a rotation about some horizontal axis. These can be mapped to the surface of a sphere by marking the final position of the zenith of the original fixed position. The minimum height as a function of orientation then forms a scalar function on the surface of the sphere. As the total energy decays, the region of the sphere accesible to the die diminishes in area. This region will split into disconnected subregions, and these will split, and so on until the only regions left are the stable points. The implicit assumption here is that all accessible orientations are equally probable. This is best approximated by a hard bouncy surface, which gives a slow decay and hence time to wander. I think that the probability of the die ending up in a given subregion is given by the area of that region AT THE SPLITTING POINT. For complex shapes, the probability of ending at each stable point must be calculated by an iterative process. Lew Mammel, Jr. ihuxr!lew