MathNet 2017-part-02 — 176 problems (EN–VI) Question 147

Let $n \geqslant 2$ be an integer. The $n^{2}$ integers $1,2, \ldots, n^{2}$ are placed in the $n^{2}$ cells of an $n \times n$ chessboard as follows: the first row contains the numbers $1,2, \ldots, n$ (from left to right), the second row contains the numbers $n+1, n+2, \ldots, 2 n$, and so on. At each step, one may choose two cells sharing a common side and add the same integer to, or subtract the same integer from, the two numbers in these two cells.

Find all values of $n$ for which it is possible to obtain the configuration consisting only of 0s, and in that case find the minimum number of steps needed to achieve it.