Question 48
Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_{1}, a_{2}, \ldots, a_{2 r}$ of $2 r \geqslant 4$ pairwise distinct red squares is called a bishop circuit if for every $k \in\{1, \ldots, 2 r\}$, the squares $a_{k}$ and $a_{k+1}$ lie on a diagonal, but the squares $a_{k}$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2 r+1}=a_{1}$ and $a_{2 r+2}=a_{2}$).
In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.
(Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^{\circ}$.)
In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.
(Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^{\circ}$.)