Complete Solutions for a Combinatorial Puzzle in Linear Time

In this paper we study a single player game consisting of n black checkers and m white checkers, called shifting the checkers. We have proved that the minimum number of steps needed to play the game for general n and m is nm+n+m. We have also presented an optimal algorithm to generate an optimal move sequence of the game consisting of n black checkers and m white checkers, and finally, we present an explicit solution for the general game.