2024 IMO 第 5 题

Turbo the snail is in the top row of a grid with 2024 rows and 2023 columns and wants to get to the bottom row. However, there are 2022 hidden monsters, one in every row except the first and last, with no two monsters in the same column.

Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an orthogonal neighbor. (He is allowed to return to a previously visited cell.) If Turbo reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move between attempts, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and Turbo wins.

Find the smallest integer $n$ such that Turbo has a strategy which guarantees being able to reach the bottom row in at most $n$ attempts, regardless of how the monsters are placed.

Turbo 蜗牛位于 2024 行 2023 列的网格的顶行，它想要到达底行。然而，隐藏怪物有2022个，除了第一和最后一行之外，每一行都有一个，同一列中没有两个怪物。

Turbo 进行了一系列尝试，从第一行走到最后一行。每次尝试时，他都会选择从第一行中的任何单元格开始，然后重复移动到正交邻居。 (他被允许返回到之前访问过的牢房。)如果 Turbo 到达有怪物的牢房，他的尝试就会结束，他会被传送回第一排开始新的尝试。怪物在两次尝试之间不会移动，涡轮会记住他访问过的每个单元格是否包含怪物。如果他到达最后一行中的任何单元格，他的尝试就会结束，Turbo 获胜。

找到最小的整数$n$，这样Turbo就有一个策略保证能够在最多$n$次尝试中到达底行，无论怪物如何放置。