International Mathematical Olympiad 2016 Problem 2

Find all positive integers $n$ for which each cell of an $n \times n$ table can be filled with one of the letters $I$, $M$ and $O$ in such a way that:

• in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and
• in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.

Note: The rows and columns of an $n \times n$ table are each labelled 1 to $n$ in a natural order. Thus each cell corresponds to a pair of positive integers $(i,j)$ with $1 \leq i,j \leq n$. For $n > 1$, the table has $4n - 2$ diagonals of two types. A diagonal of the first type consists of all cells $(i,j)$ for which $i + j$ is a constant, and a diagonal of the second type consists of all cells $(i,j)$ for which $i - j$ is a constant.