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coloring with 2 colours a mxn chessboard iff m \cdot n is even

Source: INAMO Shortlist 2014 C3

July 13, 2019
perimeterColoringChessboardcombinatorics

Problem Statement

Let nn be a natural number. Given a chessboard sized m×nm \times n. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if m⋅nm \cdot n is even.
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