2024 Putnam 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 $1,2,\dots,n$?

令 $n$ 和 $k$ 为正整数。 $n$×$n$ 网格的第 $i$ 行和第 $j$ 列中的正方形包含数字 $i+j-k$。对于 $n$ 和 $k$ 来说，可以从网格中选择 $n$ 个方格，不能在同一行或同一列中选择两个方格，这样所选方格中包含的数字恰好是 $1,2,\dots,n$？