Can We Use Parity?

Consider an \(8 \times 8\) grid of squares. We color every square blue or green such that 3 of the 4 squares on the corners of the grid are blue and the fourth is green.

Will there always exist a \(2 \times 2\) square inside this grid such that there are an odd number of squares of each color inside?

Bonus: Generalize this for the \(n \times n\) grid.

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