Question 52
Let $N \ge 3$ be an odd positive integer. At the beginning in each square of an $N \times N$ board there is number $0$. In one move one can choose two squares with a common side and increase or decrease by $1$ the numbers in those two squares. If after $K$ moves, the sums of numbers in every row and every column are all equal, show that $K$ is even. (USSR)