2023 USAMO 第 5 题

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

设 $n\geq3$ 为整数。如果每行中的数字可以排列形成等差数列，则我们说 $1$、$2$、$\dots$、$n^2$ 在 $n \times n$ 表中的排列是行有效的，如果每列中的数字可以排列形成等差数列，则我们说是列有效的。对于什么 $n$ 值，可以通过排列每行中的数字来将任何行有效排列转换为列有效排列？