19 chess boards
There are 19 chess boards, each with 64 removable squares.
Two players take turns removing an \(n\times n\) group of squares from any single chess board they like, until all the squares are gone. The winner is the one who takes the last square (or last group of squares).
Can either player guarantee a win?
Details and Assumptions:
- In a given turn, players only remove from one chess board, and must remove a complete \(n\times n\) array of squares. That is, if they remove a 3 \(\times\) 3 array of squares, they should remove exactly nine squares.
- Ignore the borders of the chessboards, and consider only the 64 squares in them.
- Both players play optimally.