Nim

The Paper

In 1901, a Harvard mathematician named Charles Bouton published a paper in the Annals of Mathematics titled simply “Nim, a game with a complete mathematical theory.”

The title wasn’t modest. It was accurate. Bouton didn’t just find a good strategy — he completely solved the game. Meaning: given any position, you can calculate the optimal move in seconds. There’s no luck, no gut feel, no experience required. Just arithmetic.

This was the first time anyone had done this for a game. Game theory as a field wouldn’t exist for another forty years — von Neumann and Morgenstern wouldn’t publish their foundational work until 1944. Bouton got there first, in 1901, with Nim.

The game itself is older — variations appear in Chinese and European literature from the 1600s. But Bouton was the first to prove, mathematically, exactly who wins and why.

The Secret

How to beat Hard mode — and why it works.

Step 1: Write each pile in binary

Take the starting position: piles of 5, 4, and 3. Write each as a binary number:

Step 2: XOR them together (the nim-sum)

XOR each column — 1 if an odd number of 1s, 0 if even. This gives the nim-sum.

Step 3: The rule

Try it

Enter any pile sizes and see the nim-sum live. If it’s non-zero, a winning move exists.

Now go back and try Hard mode. The AI knows this secret — but so do you. Set up a position where the nim-sum is already 0 before your first move, and the AI is in a losing position from the start.

More Games from Old Papers

Each of these games was invented or described in an academic paper. Each has a surprising story.