Mathematics (HMC), Mathematics (CMC)
In the game Knock 'm Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution.
© 2001 The Electronic Journal of Combinatorics
Benjamin, Arthur T., Mark L. Huber, and Matthew T. Fluet. "Optimal Token Allocations in Solitaire Knock'm Down." The Electronic Journal of Combinatorics 8.2 (2001): 1-8.