2023 IMO Shortlist C4

Let $n \geq 2$ be a positive integer. Paul has a $1 \times n^{2}$ rectangular strip consisting of $n^{2}$ unit squares, where the $i^{\text {th }}$ square is labelled with $i$ for all $1 \leq i \leq n^{2}$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an $n \times n$ square satisfying the following property: if the unit square in the $i^{\text {th }}$ row and $j^{\text {th }}$ column is labelled with $a_{i j}$, then $a_{i j}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

令 $n \geq 2$ 为正整数。 Paul 有一个由 $n^{2}$ 个单位方块组成的 $1 \times n^{2}$ 矩形条，其中 $i^{\text {th }}$ 方块对于所有 $1 \leq i \leq n^{2}$ 都标有 $i$。他希望将条带切成几块，每块由许多连续的单位方块组成，然后平移(不旋转或翻转)这些块以获得满足以下性质的 $n \times n$ 方块：如果 $i^{\text {th }}$ 行和 $j^{\text {th }}$ 列中的单位方块被标记为 $a_{i j}$，则 $a_{i j}-(i+j-1)$ 是可整除的由$n$。确定保罗为了完成这个任务需要制作的最少零件数量。