AT-OEMO-REG 2019 Problem 3

Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$ . (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$ .