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.