Solution
= Solution
This can fail. Color $(x,y)$ by the parity of $x$. If a square has odd side length $q$, then the two endpoints of each horizontal side have first coordinates of opposite parity, so the square cannot be monochromatic.
= Solution
This can fail. Color $(x,y)$ by the parity of $x$. If a square has odd side length $q$, then the two endpoints of each horizontal side have first coordinates of opposite parity, so the square cannot be monochromatic.