2022 IMO 第 6 题

Let $n$ be a positive integer. A Nordic square is an $n\times n$ board containing all the integers from 1 to $n^{2}$ so that each cell contains exactly one number. An uphill path is a sequence of one or more cells such that:

a) the first cell in the sequence is a valley, meaning the number written is less than all its orthogonal neighbors,
b) each subsequent cell in the sequence is orthogonally adjacent to the previous cell, and
c) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$ , the smallest possible total number of uphill paths in a Nordic square.

令 $n$ 为正整数。北欧方格是一块 $n\times n$ 的棋盘，包含从 1 到 $n^{2}$ 的所有整数，因此每个方格只包含一个数字。上坡路径是一系列一个或多个像元，使得：

a) 序列中的第一个单元格是一个山谷，这意味着写入的数字小于其所有正交邻居，
b) 序列中的每个后续单元都与前一个单元正交相邻，并且
c) 单元格中所写的数字按顺序递增。

根据 $n$ 的函数求出北欧方形中可能的最小上坡路径总数。