  Let $x_1,\ldots,x_n$ be a fixed sequence of real numbers. At each stage, pick
two indices $I$ and $J$ uniformly at random and replace $x_I$, $x_J$ by
$(x_I+x_J)/2$, $(x_I+x_J)/2$. Clearly all the coordinates converge to
$(x_1+\cdots+x_n)/n$. We determine the rate of convergence, establishing a
sharp "cutoff" transition, answering a question of Jean Bourgain.
