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A grid with all whites fits on the left because it does not contain black squares, hence requiring 0 paths, an even number.
A grid with all blacks requires 1 path to go through all black squares (such as a zigzag through every row), an odd number.
The checkerboard grid on the left contains 12 black squares that cannot be joined with a path, hence it requires 12 paths, an even number. | 
