DK-GEORGMOHR 2022 Problem 4

Georg plays the following game. He chooses two positive integers $n$ and $k$ . On an $n \times n$ - board where all the tiles are white, Georg paints k of the tiles black. Then he counts the number of black tiles in each row, forms the square of each of these n numbers and adds up the squares. He calls the result $S$ . In the same way he counts the number of white tiles in each row, forms the square of each of these n numbers and adds up those squares. He calls the result $H$ . Georg would like to achieve $S - H = 49$ . Determine all possible values of n and k for which this is possible.

Example: If Georg chooses $n = 5$ and $k = 14$ , he could for example paint the board as shown. Then $S = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 = 1 + 4 + 9 + 9 + 25 = 48$ , $H = 4^2 + 3^2 + 2^2 + 2^2 + 0^2 = 16 + 9 + 4 + 4 + 0 = 33$ , so in this case $S - H = 48 - 33 = 15$ .