Knock 'm Down

• ejc.pdf (with Arthur T. Benjamin and Mark L. Huber; Appeared in the Electronic Journal of Combinatorics)

  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.

• amm.pdf (with Arthur T. Benjamin; Appeared in the American Mathematical Monthly)

• thesis.pdf (Harvey Mudd College Senior Thesis)

  Knock ’m Down is a game of dice that is so easy to learn that it is being played in classrooms around the world as a way to develop students’ intuition about probability. However, as analysis has shown, lurking underneath this deceptively simple game are many surprising and highly unintuitive results. In the original description of the game, two players are each given one six sided die, 12 tokens and a board labeled with the values 2, 3, ..., 12. Each player distributes his/her tokens among the values on his/her board. Now, the players roll their dice together and each removes a token from his/her board on the value equal to the sum of the dice (if he/she has one there). Turns continue in this fashion. The winner is the first player to remove all twelve tokens. The problem posed by this game is to determine which allocation of tokens will maximize a player’s chances of winning. Results will demonstrate that the answer to this question depends on many factors, and small variations in the rules of the game can lead to markedly different answers. In addition to the major theoretical results, the principal computational challenges of this problem will be discussed.