Putnam 2024 B1

Let $n$ and $k$ be positive integers. The square in the $i$th row and $j$th column of an $n$-by-$n$ grid contains the number $i+j-k$. For which $n$ and $k$ is it possible to select $n$ squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly $1,2,\dots,n$?