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Permutation of \(1,\ldots,2n\) in the sum of rows and columns

Source: Pan-American Girls&rsquo; Mathematical Olympiad 2023 P2

August 8, 2023

square gridcombinatorics

Problem Statement

In each cell of an

n \times n

grid, one of the numbers

0

1,

2

must be written. Determine all positive integers

n

for which there exists a way to fill the

n \times n

grid such that, when calculating the sum of the numbers in each row and each column, the numbers

1, 2, \ldots, 2n

are obtained in some order.