2012 CMO 第 4 题

Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$ . Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement *good* if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?

设 $p$ 为素数。我们将 ${\{1,2,\ldots ,p^2} \}$ 中的数字排列为 $p \times p$ 矩阵 $A = ( a_{ij} )$ 。接下来，我们可以选择任何行或列，并将 $1$ 添加到其中的每个数字，或从其中的每个数字减去 $1$。如果我们能够在有限次数的此类移动中将矩阵的每个数字更改为 $0$，我们称这种排列为*好*。有多少好的安排？