Ним

Nim is a mathematical strategy game of perfect information. Players take turns removing objects from distinct piles. The player who takes the last object wins (normal play) or loses (misere play). Despite simple rules, Nim has deep mathematical structure.

Take the last object(s) to win (normal play), or avoid taking the last object (misere variant).

Objects (traditionally matchsticks or stones) are arranged in several piles. Classic setup: rows of 1, 3, 5, and 7 objects (16 total). Many variations exist with different starting configurations.

1. Players alternate turns.
2. On your turn, remove any number of objects from a single pile (at least one, up to all of them).
3. You can only take from one pile per turn.
4. Play continues until all objects are removed.
5. In normal play: the player who takes the last object wins.
6. In misere play: the player forced to take the last object loses.

Win by taking the last object (normal) or forcing your opponent to take it (misere). The winning strategy involves binary XOR calculations of pile sizes.

• *Learn the Nim-sum: XOR all pile sizes in binary. If Nim-sum is 0 before your move, you're in a losing position (with perfect opponent play).
• *Always aim to leave a Nim-sum of 0 after your move.
• *With two equal piles, mirror your opponent's moves in the other pile.
• *In misere play, the strategy differs only at the endgame - aim for an odd number of 1-piles.
• *Recognize losing positions and play to maximize opponent mistakes.
• *Practice calculating Nim-sums quickly to find winning moves.

Nim was comprehensively analyzed by Charles Bouton in 1901, who proved the winning strategy. The name may come from German 'nimm' (take) or archaic English 'nim' (steal).