  Consider the following game between a random player R and a deterministic
player D. There is a pile of n elements at the beginning. The rules for playing
are as follows: In each turn of R, if the pile contains exactly m elements, R
removes k elements from the pile, where k is independently identically
distributed from {1, . . . , m}. In each turn of D, D removes only one element.
The winner is the player that, at the end of its round, has no elements
remaining. R starts first to play. This short paper shows that Dn, which is
defined as the probability of D winning the game (when is initialized with n
elements), approaches 1/e when n increases; and more specifically, Dn = dn/n!,
where dn is the n-th derangement number.
