If every tile has an integer side, so does the whole rectangle

Question

A large rectangle is divided into finitely many smaller rectangles, with sides parallel to those of the large one. Each small rectangle has at least one side whose length is a whole number (the other side may be anything).

Prove that the large rectangle also has at least one side of whole-number length.

There are several proofs; the most elegant uses a colouring.

Accepted Answer

Colour the plane in a checkerboard of squares of side 1/2, aligned so that a corner of the big rectangle sits at a corner of the pattern. A rectangle with an integer side is balanced. Take any small rectangle whose width is a whole number w (the argument for an integer height is identical). Slide a vertical line across it: at each height the rectangle spans a horizontal segment of length w. Along a horizontal line the colours alternate every 1/2, so any segment of whole-number length contains equal lengths of black and white, wherever it starts. Integrating over all heights, the rectangle con…

If every tile has an integer side, so does the whole rectangle

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