11.3. Each cell of a square board of size $n$ by $n$ is colored blue or red. A row or column is called bluish if it contains more blue cells than red ones. Conversely, a row or column is called reddish if it contains more red cells than blue ones. What is the maximum value that the sum of the number of reddish rows and the number of bluish columns can take for some coloring of the board depending on $n$?
Answer. The maximum value of the sum of the number of reddish rows and the number of bluish columns is: a) if $n$ is odd, then $2 n-2$, b) if $n$ is even and $n \geq 8$, then $2 n-4$, c) for $n=2,4,8$ respectively $2,5,9$.

Solution. Consider an arbitrary coloring of the board in two colors, denote by $x$ the number of reddish rows and by $y$ - the number of bluish columns in it.