In attempting to understand the "meat ax" of finite group theory, Diaconis has formulated an urn model. In the simplest case, balls numbered 0 and 1 are placed in an urn. At times $n = 1,2,...,$ two balls are drawn with replacement. Those balls are replaced in the urn, and a new ball that contains the sum mod $2$ of the numbers on the drawn balls is added to the urn. A conjecture is that the fraction of balls numbered $1$ converges to $1/2$. This conjecture and some generalizations are proved as a two-fold application of the almost supermartingale convergence theorem of Robbins and Siegmund $(1972)$. This is joint research with Benny Yakir.