Question 89
Let $m$ and $n$ be positive integers with $mn$ even. Jetze is going to cover an $m \times n$-board (with $m$ rows and $n$ columns) with domino tiles, in such a way that every domino tile covers exactly two squares, domino tiles do not protrude out of the board or overlap one another, and every square is covered by a domino tile. Merlijn then is going to colour all domino tiles on the board either red or blue. Determine the smallest non-negative integer $V$ (depending on $m$ and $n$) such that Merlijn can always make sure that in each row, the number of squares covered by a red domino tile and the number of squares covered by a blue domino tile differ by at most $V$, no matter in what way Jetze covers the board.