Some Two-Player Achievement Games

Note: These are open problems: we don't have solutions. If you make any progress on them, please let us know. Send email to Joe Hemmeter at hemmeter@provide.net.

Problem 1. 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 four 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?

Problem 2. 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 a set of n cells with no two cells in the same row or column is the winner. What is the outcome of the game given best possible play by both players?

Problem 3. Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is unimportant). The last player who can move wins. Who should win?