# Can we arrange $\{1,…,16\}$ in $4\times 4$-grid so {products of rows} = {products of columns}?

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.

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