Question 46
There is an $n \times n$ chess board. Each of the $n^2$ small boxes can present a number from $0$ to $k$, for some positive integer $k$. In each row and column, there is a button and if we push the button in a row (or column), the number in each of the $n$ small boxes contained in that row (or column, respectively) increases by $1$ and the number $k$ is changed to $0$. We call this a "process". At first, the number $0$ was presented in every one of the $n^2$ small boxes. But there were some processes and the numbers were changed. Show that every number in the $n^2$ small boxes can be changed to $0$ by taking at most $kn$ processes.