NIM

Nim is a simple game of strategy that can be played with any type of object, but matches are the most common. The goal of the game is to avoid taking the last object.

The rules of Nim are as follows:

• The game starts with two or more players and several piles of objects.
• On each turn, a player must remove at least one object from a pile.
• A player can remove any number of objects from a pile, as long as they all come from the same pile.
• A player can also empty a pile, effectively removing it from the game.
• The game ends when one player takes the last object. That player loses the game.

The winning strategy for Nim is to always make a move that leaves the Nim sum of the piles equal to 0. The Nim sum of a set of piles is the remainder when the sum of the number of objects in each pile is divided by the number of piles. For example, if there are 2 piles of 3 objects each, the Nim sum is 0, because 3 + 3 = 6, which is divisible by 2.

If a player leaves the Nim sum of the piles equal to 0, then the other player has no choice but to take the last object and lose the game.

Here is an example of a game of Nim:

• The game starts with 3 piles of 3 objects each.
• Player 1 removes 2 objects from the first pile.
• Player 2 removes 1 object from the second pile.
• Player 1 removes 1 object from the third pile.
• The game ends because there are no more objects left in any of the piles. Player 2 loses the game.

Nim is a simple game, but it can be very challenging to win. The key to winning is to always be thinking about the Nim sum of the piles and making moves that will leave the Nim sum equal to 0.