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filling a 3xn table with numbers 1 to 3n, sum of rows and columns related

Source: Austria Beginners' Competition 2018 p3

February 24, 2020
combinatoricsSum

Problem Statement

For a given integer n≥4n \ge 4 we examine whether there exists a table with three rows and nn columns which can be filled by the numbers 1,2,...,,3n1, 2,...,, 3n such that ∙\bullet each row totals to the same sum zz and ∙\bullet each column totals to the same sum ss. Prove: (a) If nn is even, such a table does not exist. (b) If n=5n = 5, such a table does exist.
(Gerhard J. Woeginger)
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