Question

Beginning with one pile containing n matchsticks, each player must, at his or her turn, separate any existing pile into two smaller subpiles. The first player who cannot move in the loser. For example, if the original pile contains 10 sticks, A might divide it into two piles containing 5 and 5 sticks respectively, or 3 and 7 sticks, etc. In the latter case, B might then subdivide the 3 pile into 2 and 1, leaving piles of 1, 2, and 7. If A then divides the 2 pile, leaving 1, 1, 1, and 7, then B would have to work with the pile containing 7 sticks, and so on. Analyze this game.

Fully shown work and answer please.

Answer 1

Ans). This game can be simplified as putting dividers into the spaces between the sticks. If there are n sticks, there will end up being a total of n-1 dividers to separate them into piles of 1 stick each. Whoever places the last divider wins. Since each player must place 1 divider at each turn, if the number of dividers left to be placed is odd on my turn I am going to win. Thus, if n is even I want to go first. If n is odd I want my opponent to go first.

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