NIM ZERO · 2026.08.01-v1
Misère Nim strategy: when parity replaces nim-sum
Misère Nim uses the normal nim-sum strategy while at least two heaps contain more than one piece. With exactly one large heap, reduce it so an odd number of single heaps remains for the opponent. When every heap is single, parity alone decides the ending.
The switch point
In 1–1–2, the two-piece heap is the only large heap. Remove one from it, leaving 1–1–1. The opponent receives three single heaps and must eventually take the final piece under perfect play.
Why parity decides
With only single heaps, every legal move deletes exactly one heap. An odd number passed to the opponent becomes even after their move and alternates until they face one final heap. Because the last taker loses, that final forced take is the losing move.
Worked position
| Heap | Binary | 10 |
|---|---|---|
| 1 | 011 | 3 |
| 2 | 101 | 5 |
| 3 | 111 → 110 | 7 → 6 |
| nim-sum | 001 → 000 | 1 → 0 |
Method and source
Method: finite legal moves are exhaustively checked by Nim Zero’s pure rules engine. Mathematical source: C. L. Bouton, “Nim, A Game with a Complete Mathematical Theory,” Annals of Mathematics, Second Series 3, no. 1/4 (1901–1902), 35–39.
DOI: 10.2307/1967631 · 2026-08-01