Question 30
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 the centre (when $n$ is odd). Below shows the arrays for $n=4$.
$$
A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix} \qquad B = \begin{bmatrix} 1 & 3 & 6 & 10 \\ 2 & 5 & 9 & 13 \\ 4 & 8 & 12 & 15 \\ 7 & 11 & 14 & 16 \end{bmatrix}
$$
$$
A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix} \qquad B = \begin{bmatrix} 1 & 3 & 6 & 10 \\ 2 & 5 & 9 & 13 \\ 4 & 8 & 12 & 15 \\ 7 & 11 & 14 & 16 \end{bmatrix}
$$