Question 5
Let $A$ and $B$ be two $n\times n$ square arrays. The cells of $A$ are labelled by the numbers from $1$ to $n^2$ from left to right starting from the top row; whereas the cells of $B$ are labelled by the numbers from $1$ to $n^2$ along rising north-easterly diagonals starting with the upper-left-hand corner. Stack the array $B$ on top of the array $A$. If two overlapping cells have the same number, they are coloured red. Determine those $n$ for which there is at least one red cell other than the cells at top left corner, bottom right corner and at the centre (when $n$ is odd). Below shows the arrays for $n = 4$.
$$A=\begin{bmatrix}1&2&3&4\\5&6&7&8\\9&10&11&12\\13&14&15&16\end{bmatrix},\qquad B=\begin{bmatrix}1&3&6&10\\2&5&9&13\\4&8&12&15\\7&11&14&16\end{bmatrix}.$$
$$A=\begin{bmatrix}1&2&3&4\\5&6&7&8\\9&10&11&12\\13&14&15&16\end{bmatrix},\qquad B=\begin{bmatrix}1&3&6&10\\2&5&9&13\\4&8&12&15\\7&11&14&16\end{bmatrix}.$$