Question 33
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 $1 \le i, j \le n$. For $n \gt 1$, 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.
* 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 $1 \le i, j \le n$. For $n \gt 1$, 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.