Transversal achievement game on a square grid

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\in\mathbb N$ and let $[n]=\{1,\ldots,n\}$. Two players alternately claim one unoccupied cell of $[n]\times[n]$, starting with player O. A transversal is a set $\{(i,\sigma(i)):i\in[n]\}$ for a permutation $\sigma$ of $[n]$. A player wins immediately upon owning every cell of a transversal; if the board fills without a winner the outcome is a draw. Both players have full information and seek a win, then a draw in preference to a loss.

Statement recorded in the supplied source.

Determine, for every $n\geq1$, the outcome under optimal play: a forced win for O, a forced win for X, or a draw. [S2]

Short English statement

Determine the optimal-play outcome of the alternating square-grid game in which either player wins by occupying one cell in every row and column.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3050 (OPG-37230). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erickson, Martin, Transversal achievement game on a square grid, Open Problem Garden, node 37230. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid.