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An Algorithmic Analysis of a Combinatorial Game

Daxin Zhu, Lei Wang, Jun Tian, Xiaodong Wang

Abstract


In this paper, we study an interesting shifting checkers game consisting of n black checkers and 1 white checkers, called Moving Checkers Game, as a means to elucidate the process of analyzing a problem and synthesizing observations into an algorithm that, in turn, can be used to generate a programming solution. While not directly presented as such, the material discussed could be easily adapted to allow for small-group cooperative learning. Some properties of the optimal solutions of the problem are also interesting. We have proved that the minimum number of steps needed to play the game for general n is 2n+1.
We have also presented an optimal algorithm to generate all of the optimal solutions in linear time for very large size. The number of solutions for the game of size n is the (n+2)th Fibonacci number. We present also an explicit solution for the general game. This explicit solution will give the exact actions for each number of moves in constant time. The step by step process used to solve the problem is reflective of the process an expert goes through when solving the problem.

Keywords


Computer games, optimal algorithms, explicit solutions.

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