Let
$$M_n := \left\{ |A| : A \in \Bbb R^{n \times n}, a_{ij} \in \{ 1, 2, 3, \dots, n^2\}, a_{ij} \neq a_{kl}, \forall (i,j)\neq(k,l) \right\}$$
be the set of determinants of $n \times n$ matrices whose entries are taken from $\{ 1, 2, 3, \dots, n^2\}$ without repetition. Then what is $\max M_n$?
Some remark: the case $n=3, 4$ was asked here where it is stated without proof that $$\max M_3 = 412, \ \ \ \max M_4 = 40800. $$ $\max M_n$ can also be found in OEIS, where one can find more information, which includes an upper bound proved in here (which is also linked in the answer here).
Also, one may also ask what is $\min M_n$, but indeed $\min M_n = - \max M_n$, which can be seen by switching two neighboring rows.