If 1-9 is filled in the $3 \times 3$ determinant, and each number appears once,then the maximum value of the determinant is $412$.
For example, the following determinant can take the maximum value of $412$: $$\left| \begin{array}{ccc} 1 & 4 & 8 \\ 7 & 2 & 6 \\ 5 & 9 & 3 \\ \end{array} \right|=412.$$
Question: if 1-16 is filled in the $4\times4$ determinant, and each number appears once, what is the maximum value of the determinant? Is it necessarily less than $16 \times 15 \times 14 \times 13= 43680$?
The largest known value is $$\left| \begin{array}{cccc} 12 & 13 & 6 &2 \\ 3 & 8 & 16 &7\\ 14 & 1 & 9 &10 \\ 5 & 11 &4 &15 \end{array} \right|=40800.$$
See this paper and the OEIS sequence A085000 as a reference.