Competition problem — Combinatorics

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Combinatorics Difficulty 8.8 Shortlist Find the answer

Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.

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 adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he 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, 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 the game is over.

Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n$ -th attempt or earlier, regardless of the locations of the monsters.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

To solve this problem, we will analyze the board's structure and derive a strategy for Turbo to ensure he reaches the last row in a guaranteed number of attempts. We'll consider the distribution of monsters and Turbo's possible paths.

Given:
- The board has 2024 rows and 2023 columns.
- There is exactly one monster in each row except the first and last, totaling 2022 monsters.
- Each column contains at most one monster.

Objective:
Determine the minimum number $n$ of attempts Turbo requires to guarantee reaching the last row, regardless of monster placement.

### Analysis

1. Board Configuration:
- In total, 2022 monsters are distributed such that each row (except the first and last) contains exactly one monster.
- Since each column has at most one monster, not all columns have a monster.

2. Turbo's Strategy:
- Turbo needs to explore the board in a manner that efficiently identifies safe columns and rows without encountering a monster multiple times unnecessarily.
- Turbo can determine whether a column is safe (contains no monsters) by exploring strategic positions across breadth and depth on the board.

3. Strategy Application:
- First Attempt: Turbo starts by exploring a single path down a column from the first row to the last row.
- If no monster is encountered, Turbo completes the game in the first attempt.
- If a monster is encountered, Turbo records the dangerous columns.
- Second Attempt: Turbo tries an adjacent column next to the previously explored path.
- In this attempt, he checks whether this path leads to a monster-free path.
- Third Attempt: Combining information from the first and second attempts, Turbo systematically explores remaining unchecked paths.

With a systematic exploration strategy, Turbo uses at most three different attempts because:
- Attempt 1: It eliminates either the path as safe or identifies monsters, removing knowledge uncertainties.
- Attempt 2: Validates adjacent safe paths based on new or old information.
- Attempt 3: Finishes off ensuring any unclear pathways are confirmed.

Considering the constraints (2024 rows but only one monster per row, and each column has at most one monster), and considering that Turbo can remember the unsafe paths and adjust his route, the minimum number of guaranteed attempts is 3:
$\boxed{3}$
This ensures that Turbo utilizes a strategic exploration pattern, minimizing redundant moves while guaranteeing reaching the destination row.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.