Let a $3 \times 3$ matrix have the elements $1,2,\dots,9$. What is the maximum value the determinant may have?
I have found the desired value and an intuitive feeling/approach as to why that must be optimal. I struggle to really prove that claim, though.
At the Online Encyclopedia of Integer Sequences the maximal value is tabulated for $n\times n$ matrices for $n\le7$. Bounds are given for $8\le n\le10$, and an upper bound good for all $n$ is given. A reference and a link is given to Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.
412 is correct for $n=3$.