Requires an odd number of paths to cover all black squares vs requires an even number of paths to cover all black squares

A path is formally defined as a sequence of black squares such that no two squares are the same and every pair of consecutive black squares are orthogonally adjacent to each other.

Every grid on the left requires a minimum of an even number of paths such that every black square is a part of exactly one path, while every grid on the right requires a minimum of an odd number of paths to do so.

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.

Matt Nguyen