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My motivation is the following question asked by Jacob Steiner, which he then deleted. For which $n\in\Bbb N$ is it possible to arrange $\{1,\ldots,n^2\}$ in an $n\times n$-grid so that the set of products of columns equals the set of products of rows?

The answer for $n=2$ is clearly No, since the only possibility, up to a transposition and a permutation of rows and columns, is

1 2 
3 4 

Since the set of products of rows are $\{2,12\}$ and the set of products of columns are $\{3,8\}$, this arrangement does not work. So for $n=2$, it is not possible to make such an arrangement.

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  • $\begingroup$ See related discussion here. $\endgroup$ – RobPratt Jan 29 '20 at 5:44
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    $\begingroup$ This is a question from the 2020 PROMYS admissions exam. $\endgroup$ – James Done Mar 2 '20 at 22:02