TITLE: Square Achievement Game on a Grid
QUESTION [8 upvotes]: Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid.
The first player to occupy the vertices of a square with horizontal and vertical sides
is the winner.  What is the smallest n such that the first player has a winning strategy?
Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides.  So the game must result in a winner (the first player) when n=15.

REPLY [3 votes]: I believe the first player has a winning strategy for N=7.
In the configuration below (with "o" having the next move), "o" can force a win by making a "cross".  Note that the "cross" can also be applied to two non-adjacent "o"s in the same row or column.

The first player plays in the center.  The second player has 9 unique responses.  I'll consider the 3 most interesting responses here.
Below, the first player places the second "o" next to the first, threatening to make a cross.  So, the second player has 2 unique moves available.  However, both moves allow the first player to force a few moves and achieve the "cross", as illustrated.  (The two starting pieces for the "cross" are marked with a square around them.)

Another possible position for the initial "x".  This is also a win for "o".

Another possible position for the initial "x".  This is also a win for "o".  (I omitted 2 branches to save space, but they follow the same pattern).

I didn't consider the other 6 initial moves for "x", but I think the same techniques can be used to solve them.
FWIW, I'm pretty sure N=5 is a draw.  I haven't considered N=6, yet.