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sum of -1 and 1 in a nxn grid , when is zero

Source: Austrian Regional Competition For Advanced Students 2019, p3

March 2, 2020
combinatorics

Problem Statement

Let n≥2n\ge 2 be a natural number. An n×nn \times n grid is drawn on a blackboard and each field with one of the numbers −1-1 or +1+1 labeled. Then the nn row and also the nn column sums calculated and the sum SnS_n of all these 2n2n sums determined. (a) Show that for no odd number nn there is a label with Sn=0S_n = 0. (b) Show that if nn is an even number, there are at least six different labels with Sn=0S_n = 0.
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