Competition problem — Combinatorics

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Combinatorics Difficulty 4.8 AIME Prove it Soviet Union

Problem:
Given an $n \times n$ array of numbers. $n$ is odd and each number in the array is $1$ or $-1$ . Prove that the number of rows and columns containing an odd number of $-1$ s cannot total $n$ .

Solution

Solution:
If we change a $-1$ to $1$ , we affect the total number of rows and columns (containing an odd number of $-1$ s) by $0$ , $2$ or $-2$ . After changing all the $-1$ s we have total of $0$ . Hence the starting total must be even. So it cannot be $n$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.