Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Optimal Token Allocations in Solitaire Knock ’m Down. | Optimal Token Allocations in Solitaire Knock m Down Arthur T. Benjamin Matthew T. Fluet Department of Mathematics Department of Computer Science Harvey Mudd College Cornell University Claremont CA 91711 Ithaca NY 14853 benjamin@ fluet@ Mark L. Huber Department of Statistics Stanford University Stanford CA 94305 mhuber@ Submitted February 28 2000 Accepted August 14 2000 Abstract 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. MR Subject Classifications primary 91A60 1 Introduction Knock m Down is a simple game to play that proves surprisingly difficult to analyze. At the beginning of the game the player allocates t tokens to N bins. An allocation can be described by a vector X x-1 x2 . XN of non-negative integers with 2 xi t. At each step of the game a bin is chosen at random with the probability of choosing bin i fixed at pi 0. If at least one token remains in bin i when it is chosen then one token is removed from the bin. The game continues until all of the tokens have been removed. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 no. 2 2001 R2 1 In Solitaire Knock m Down the goal is to remove all the tokens in the shortest time possible. Given t tokens N bins and a hxed probability vector P p1 p2 . pN we will say that an initial token allocation X x x . x N is optimal if it minimizes the expected time needed to remove all of the tokens. Intuitively one would expect the optimal token placement to resemble the shape of the probability histogram so that roughly tpi tokens .
