Question 93
Let $m$ and $n$ be natural numbers with $mn$ even. Jetze tiles an $m\times n$ board, consisting of $m$ rows and $n$ columns, with dominoes so that each domino covers exactly two squares, no domino extends outside the board or overlaps another, and every square is covered. Merlijn then colors every domino red or blue. Determine the smallest nonnegative integer $V$, depending on $m$ and $n$, such that Merlijn can always arrange that, in each row, the number of squares covered by red dominoes and the number covered by blue dominoes differ by at most $V$, regardless of Jetze’s tiling.