Take n coins where n is a triangular number (i.e., n = 1 + 2 + . . . + k for some positive integer k), and divide them into s ≥ 1 piles, with no restriction on the number of piles or the numbers of coins in the piles. Then perform repeatedly the following operation. Take one coin from each pile and put all of them in a new pile. Show that no matter what the initial partition of n coins is, the process will always arrive at k piles containing 1, 2, . . . , k coins, respectively, after a finite number of iterations. (After reaching this state, the algorithm will obviously remain in it).