Question 147
On a board with $2024$ rows and $2023$ columns, Turbo the snail tries to move from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then moves one step at a time to an adjacent cell sharing a common side. He wins if he reaches any cell in the last row. However, there are $2022$ predetermined, hidden monsters in $2022$ of the cells, one in each row except the first and last rows, such that no two monsters share the same column. If Turbo unfortunately 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.
Suppose Turbo is allowed to take $n$ attempts. Determine the minimum value of $n$ for which he has a strategy that guarantees reaching the last row, regardless of the locations of the monsters.
(Hong Kong)
Comment. One of the main difficulties of solving this question is in determining the correct expression for $n$. Students may spend a long time attempting to prove bounds for the wrong value for $n$ before finding better strategies.
Students may incorrectly assume that Turbo is not allowed to backtrack to squares he has already visited within a single attempt. Fortunately, making this assumption does not change the answer to the problem, though it may make it slightly harder to find a winning strategy.
Suppose Turbo is allowed to take $n$ attempts. Determine the minimum value of $n$ for which he has a strategy that guarantees reaching the last row, regardless of the locations of the monsters.
(Hong Kong)
Comment. One of the main difficulties of solving this question is in determining the correct expression for $n$. Students may spend a long time attempting to prove bounds for the wrong value for $n$ before finding better strategies.
Students may incorrectly assume that Turbo is not allowed to backtrack to squares he has already visited within a single attempt. Fortunately, making this assumption does not change the answer to the problem, though it may make it slightly harder to find a winning strategy.