Turbo the snail is navigating a 2024 x 2023 grid. There are 2022 hidden monsters, with exactly one monster in each row except the first and last, and at most one monster per column. Turbo starts from any cell in the first row and aims to reach any cell in the last row. If Turbo encounters a monster, the attempt ends, and he starts a new attempt from the first row. Turbo remembers the locations of encountered monsters. We need to determine the minimum number of attempts n that guarantees Turbo will reach the last row, regardless of monster placement.
We'll use a combination of graph theory, information theory, and probabilistic analysis to solve this problem. Our strategy will involve:
Lemmas
Lemma 1: In the worst case, Turbo needs to identify the location of all monsters except those in the second-to-last row to guarantee safe passage.
Proof: If Turbo knows the location of all monsters except those in the second-to-last row, he can safely navigate to any cell in the second-to-last row. From there, he has a guaranteed safe move to the last row, as there are no monsters in the last row.
Lemma 2: The entropy of the monster distribution decreases monotonically with each attempt.
Proof: Each attempt either reaches the goal (ending the game) or reveals the location of a previously unknown monster. This new information reduces the uncertainty about the monster distribution, thereby decreasing the entropy.
Main Proof
Theorem: The minimum number of attempts n that guarantees Turbo will reach the last row is 2022.
Proof:
Step 1: Worst-case analysis