Problem 2

Find all 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 on that diagonal are

I

, one third are

M

, and one third are

O

. (Note that an

n\times n

table has

4n-2

diagonals in total, in both directions.)

Step 2 of 5: Necessity: mark the good cells

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In plain words

The row and column conditions force $3\mid n$ . The cells with both coordinates congruent to $2$ modulo $3$ are the centers that can be counted from two directions.

n=3k,\quad G=\{(3r+2,3s+2):0\le r,s<k\}

Detailed analysis

Each row contains one third of each letter, so $3\mid n$ ; write $n=3k$ . Call a cell $(3r+2,3s+2)$ good. Let $a$ be the number of good cells containing $I$ . Every good cell lies on one diagonal of each direction, and both diagonal lengths are multiples of $3$ .