4. [8] A $7 \times 7$ board is either empty or has an invisible $2 \times 2$ ship placed on it "by cells." It is allowed to place detectors in some cells of the board and then turn them on simultaneously. An activated detector signals if its cell is occupied by the ship. What is the smallest number of detectors needed to guarantee that, based on their signals, it can be determined whether there is a ship on the board, and if so, which cells it occupies?

(R. Zhinodarov)
Answer: 16 detectors. Solution: Evaluation. In each $2 \times 3$ rectangle, there should be at least two detectors: the rectangle consists of three $1 \times 2$ dominoes, and if a detector is in the outer domino, we cannot determine whether there is a ship on the other two dominoes, and if a detector is in the middle domino, we cannot determine which of the outer dominoes the ship occupies along with the middle domino. Therefore, there should be no fewer than 16 detectors (see the left image).
![](https://cdn.mathpix.com/cropped/2024_05_06_67a3ec7e241252dcfcabg-2.jpg?height=370&width=966&top_left_y=1408&top_left_x=544)

Example. On the right image, the black squares indicate the placement of 16 detectors. Any ship intersects with exactly one black $2 \times 2$ square in one cell, two adjacent cells, or all four cells. In any case, the position of the ship or its absence is uniquely determined.

Note. 16 detectors can be placed in only three fundamentally different ways. One is shown in the solution, and the other two are shown in the image below.
![](https://cdn.mathpix.com/cropped/2024_05_06_67a3ec7e241252dcfcabg-2.jpg?height=370&width=916&top_left_y=2026&top_left_x=573)