zermelo's theorem

Finding winning strategies

This time I want to show how to solve a game using a perfect play which means being able to predict the outcome of the game from any position. The perfect play is the strategy that leads to the best possible outcome for that player regardless of the response by the opponent. For the illustration I have selected two children’s games with stones. The rules are very simple. There is a pile of 99 stones. Two player take n stones out of the pile in turn. The one who takes the last stone is the winner. In each game n is different: it is in the closed interval between 1 and 3 power of 2 Let’s find out how to always win these kind of games.