a little brainteaser

Here’s a little problem that Halyun brought up in group meeting today — a little googling showed that it’s a Putnam prep problem, but I won’t hold that against it. The problem is “Determinant Tic-Tac-Toe.” This is like regular Tic-Tac-Toe except that Player One puts a “1” in the square and Player Two puts a “0.” The grid forms a 3×3 matrix (call it A), and Player Two wants to make \det(A) = 0, whereas Player One wants to make \det(A) \ne 0. Player One gets to move first. Is there a winning strategy for either player? What if both players can place arbitrary real numbers? What about a general n \times n grid?

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