Game Of Stones
Chris and Agnishom are playing a game of stones. There are two piles of stones, each with a positive number of stones. Each player takes turns taking any positive amount of stones from only one of the two piles. The player who takes the last stone wins.
Provided that the situation allows, what strategy should Chris follow to win the game?
A. Always take 1 stone from the larger pile.
B. Always take a positive amount of stones from the larger pile until it is 1 more than the smaller pile.
C. Always take a positive amount of stones from the larger pile until it's the same as the smaller pile.
D. Always take a positive amount of stones from the larger pile until it is 1 less than the smaller pile.
E. Always take 1 stone from the smaller pile.
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1 solution
The player who takes the last stone wins, so the goal is to force the other player to leave you one ball.
If there is more than one ball in one of the piles, the other player can take all the balls in that pile except for one, forcing you to leave them with only one ball no matter which one you take. If you leave a pile with no balls, your opponent can take them all and win. Therefore, in order to win you must (assuming everyone is playing to the best of their ability) leave your opponent with one ball in each pile.
To ensure you get the opportunity to leave your opponent with one in each pile, you must always even out the piles on your turn. You cannot add stones, so to do this you must take stones from the larger pile. Your opponent then has no choice but to take stones from one of the even piles, so you will always be presented with a larger pile again to continue the method. Each time you're presented with a slightly lower piles, until you even them out to one and one, and you've won the game.