The original problem appeared on last year's Putnam exam:
"Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays ﬁrst. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?"
It's not hard to see that Barbara can win this game by reflecting Alan's moves over a vertical line. (In fact, you might say she "wins with multiplicity 1004".) My question is, what if the goals were reversed? That is, suppose Alan (the first player) wants the determinant to be zero, and Barbara wants it to be nonzero. Now who has the winning strategy?
If you expect the result to rely solely on parity, then you should note that Alan wins in the 2×2 case, because he can force a row or column to have only zeroes. Unfortunately, it's not at all clear (to me, anyway) that he can do anything similar to a 4×4 matrix, let alone a 2008×2008 one.