NIM ZERO · 2026.08.01-v1
Take-away 21 strategy: control multiples of four
In Take-away 21, take one token first to leave 20. After that, if the opponent removes x tokens, remove 4−x. Each pair of turns removes four, so you keep returning the pile to a multiple of four and take the last token.
The opening
Twenty-one is one more than a multiple of four, so remove one and leave 20. The opponent can take only 1, 2, or 3, and your reply of 3, 2, or 1 is always legal.
Why the invariant wins
The totals after your turns are 20, 16, 12, 8, 4, and 0. This is a subtraction game with a 1–3 move limit, not unrestricted multi-heap Nim; the complement-to-four rule is the relevant proof.
Worked position
| Turn | Take | Remaining |
|---|---|---|
| Opening | 1 | 21 → 20 |
| Opponent | x (1–3) | 20 − x |
| Your reply | 4 − x | 16 |
| Your checkpoints | 20 → 16 → 12 → 8 → 4 → 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