4. On a $7 \times 7$ board, we are playing a game of battleships. There is one ship of size $2 \times 3$ on it. We can ask about any square on the board, and if we hit the ship, the game ends. If not, we ask again. Determine the smallest number of questions we need to ask to be sure to hit the ship.
SOLUTION. According to Fig. 2, we can place 8 disjoint rectangles $2 \times 3$ on the board (the middle cell of the board will remain empty). To definitely hit the ship, we need to ask about at least one cell in each of the eight marked rectangles, so the necessary number of questions is at least 8.

In Fig. 3, an example of selecting eight cells is given, asking about which is enough to ensure that no ship $2 \times 3$ can be placed on the board outside of them. Therefore, these 8 questions are always sufficient to hit the ship.

![](https://cdn.mathpix.com/cropped/2024_04_17_5d278118f968b4b8dba7g-5.jpg?height=434&width=437&top_left_y=1913&top_left_x=478)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_04_17_5d278118f968b4b8dba7g-5.jpg?height=431&width=429&top_left_y=1915&top_left_x=1156)

Fig. 3

From both of the above considerations, the following conclusion follows.

Answer. The smallest number of questions we need to definitely hit the ship is exactly 8.

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