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.
