Problem

The game of NIM. Determine the best strategy for each player in the following two-player game. There
are three piles, each of which contains some number of coins. Players alternate turns,
each turn consisting of removing any (non-zero) number of coins from a single pile.
The goal is to be the person to remove the last coin.

Solution

The game of NIM is a solved game, meaning that there is a known optimal strategy for each player. The strategy is as follows:

If the number of coins in a pile is a multiple of 4, the player should remove enough coins to leave a multiple of 4.

If the number of coins in a pile is not a multiple of 4, the player should remove enough coins to leave a pile with 3 coins.

The reason this strategy is optimal is because it always leaves the player with a position that is either a win or a forced loss. A position is a forced loss if, no matter what move the player makes, the opponent can always respond in a way that leads to the player losing.