Is Notakto Solved? Yes for 3×3, and an Open Question Beyond

Notakto on the standard 3×3 board is solved. The mathematicians Thane Plambeck and Greg Whitehead worked out its complete misère theory in 2013, and for the two- and three-board game we play here, perfect play is a settled fact: a lookup, not a search. But push the board to 4×4 and the certainty runs out: nobody yet knows whether that game can be solved the same way at all.

New here? Start with what is Notakto.

What “Solved” Means

A game is solved when the result under perfect play is known for certain, together with a way to achieve it. It does not mean memorising every position; it means the outcome is settled and a winning method exists. For multi-board Notakto on 3×3 grids, the outcome of every position is known, which is the strongest sense of “solved” there is.

How the 3×3 Game Was Solved

Plambeck and Whitehead’s 2013 paper, “The Secrets of Notakto: Winning at X-only Tic-Tac-Toe” (arXiv:1301.1672), gives the single board a complete misère analysis. The key object is its misère quotient, a small algebraic structure that captures everything strategically relevant about a position, and for the 3×3 board that quotient has exactly 18 elements. Eighteen values are enough to describe how any number of 3×3 boards combine. That is the precise sense in which Notakto is a disguised game of Nim: each board reduces to one of a handful of values, and a fixed table says how those values combine. The intuition behind that is in Notakto is secretly Nim.

What Perfect Play Looks Like Here

Because the theory is settled, the result of every position in the shipped game is fixed in advance. Two facts make it concrete.

First, the opening is decided by parity: from empty boards, the player to move wins exactly when the number of boards is odd:

One board → mover wins. Two boards → mover loses. Three boards → mover wins.

Second, the openings differ sharply in how forgiving they are. From a single empty board exactly one of nine first moves wins (the centre, then a knight’s move). From two empty boards, none of the eighteen openings win for the mover. From three, all twenty-seven do.

Counting positions up to symmetry shows how compact the game is, which is why a single table holds the whole solution:

Those numbers are exactly what our computer opponent reads from at the board, with no live searching. At the Perfect level that means perfect play every time; Casual and Sharp slip on purpose.

The Open Question: 4×4

Here is the part that keeps Notakto interesting to mathematicians. The clean solution on the 3×3 board rests on its misère quotient being finite: only 18 elements. For the 4×4 board, it is not even known whether a finite misère quotient exists. So while the game we ship is fully solved, the natural next size up sits at the edge of what anyone has proved. Notakto is that rare thing: a children’s game with an honest open problem one step away.

Why a Solved Game Is Still Worth Playing

Knowing the answer exists is not the same as finding it at the board. Against a fallible human the win still has to be executed, the losing side still has chances when the opponent slips, and the inverted, misère objective makes positions hard to read. That gap between theory and practice is the whole game; see how to win at Notakto, then play it free and try to hold the winning side against the Perfect level.