5. There is a $10 \times 10$ grid for playing "Battleship". In this case, the game follows unusual rules, and only ships of size $1 \times 2$ are allowed, which can be placed both horizontally and vertically. What is the maximum number of these ships that can be placed on such a field, if the ships must not extend beyond the boundaries of the square, and they cannot touch each other even at a point?

For a complete solution to each problem, 7 points are awarded

The maximum total score is 35
5. Answer: 13.

One of the examples of placing 13 ships is shown in the figure:

|  |  |  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  |
|  |  |  |  |  |  |  |  |  |  |
|  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  |
|  |  |  |  |  |  |  |  |  |  |
|  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  | $\mathrm{x}$ | $\mathrm{x}$ |  |
|  |  |  |  |  |  |  |  |  |  |
|  | $\mathrm{x}$ |  | $\mathrm{x}$ |  | $\mathrm{x}$ |  | $\mathrm{x}$ |  |  |
| $\mathrm{x}$ |  | $\mathrm{x}$ |  | $\mathrm{x}$ |  | $\mathrm{x}$ |  |  |  |
|  |  |  |  |  |  |  |  |  |  |

We will prove that it is impossible to place more ships. For each rectangle $1 \times 2$, consider two pairs of parallel lines, located outside its contour at a distance of $1 / 2$. They define a rectangle $2 \times 3$ with an area of 6. Also, for each side of the square $10 \times 10$, consider lines at a distance of $1 / 2$ from the contour of the square, located inside it.

We will say that the framing has been done. From the condition of the problem, it follows that the framings of the rectangles and the square cannot overlap each other. Therefore, all the framings of the ships are contained within the inner square $9 \times 9$. If there were $k$ ships, then their total area together with the framings is no more than the area of the inner square: $6 k \leq 81$. From this, it is clear that $k \leq 13$.