0 All Around
I filled up a 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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1 solution
Moderator note:
Great solution.
How does this generalize to a larger grid?
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!