I filled up a $3 \times 3$ grid with numbers, such that if we pick any 2 (not necessarily consecutive) rows and columns, the sum of the 4 numbers in their intersection is equal to 0.

Which is the most specific/restrictive sentence that we say about the numbers in the grid?

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Label the grid:

Fix two rows and consider the three equations in them:

We can add them together to obtain $2(a+d) + 2(b+e) + 2(c+f) = 0$ . Divide by two and subtract each of the original three equations separately to obtain $a+d = 0, b+e = 0, c+f = 0$ .

We can do the same for the other two pairs. We obtain $a+g = 0$ and $d+g = 0$ among the six equations produced. Now take this and $a+d = 0$ , and do the same thing as above: sum, divide by two, subtract each. We have $a = 0, d = 0, g = 0$ .

We can get the same with the others, giving all zeroes.

Thus we solved the system, obtaining the solution of all zeroes; this must be the answer.

Note: We have 9 linear equations in 9 variables. Since there is only one solution (all zeroes), this shows that none of the linear equations are redundant!