Turbo the snail plays a game on a board with rows and columns. There are hidden monsters in of the cells. In each move, Turbo checks a cell: if there is a monster, it is removed and Turbo is told. Otherwise, Turbo is also told. Turbo wins when he knows the exact position of every remaining monster. Determine the minimum value of such that Turbo can guarantee knowing all monster positions after at most moves.
The minimum number of moves is .
The minimum number of moves is .
The minimum number of moves is .
The minimum number of moves is .
Hint 1: Turbo knows there are exactly 2022 monsters. Each check reveals one cell. When can Turbo stop and deduce the rest?
Hint 2: If Turbo has checked enough cells and found some monsters, the remaining monsters are determined by elimination when few unchecked cells remain.
Hint 3: The optimal strategy balances checking cells with deducing from the total count constraint.
Step 1: Turbo needs to determine the positions of all 2022 monsters on a board. Each check reveals whether a cell has a monster.
Step 2 (Lower bound): Turbo needs enough information to distinguish all possible configurations. There are configurations. Each check gives 1 bit, so .
Step 3 (Strategy): An adaptive strategy can eliminate cells systematically. By checking cells in a structured order (e.g., by rows), Turbo can deduce monster positions.
Step 4: The optimal strategy involves checking all but the last row/column, using the constraint that exactly 2022 monsters exist to deduce the remaining positions.
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