11.3. Workers were laying a floor of size $n \times n(10<n<20)$ using two types of tiles: $2 \times 2$ and $5 \times 1$. It turned out that they managed to completely cover the floor such that the same number of tiles of each type was used. For which $n$ could this have happened? (Cutting tiles or overlapping them is not allowed.)
Answer: $12, 15, 18$.

Solution. Let the workers use $x$ tiles of each type. Then the area occupied by the tiles is $4x + 5x = 9x = n^2$. Therefore, $n^2$ must be divisible by 9, which means $n$ must be divisible by 3. Thus, the possible values for $n$ are $n=12, n=15$, and $n=18$.

We will show how to tile the required squares. A $6 \times 6$ square can be tiled using an equal number of both types of tiles (see Fig. 10). Since the squares $12 \times 12$ and $18 \times 18$ can be divided into $6 \times 6$ squares, they can also be tiled in the required manner.

![](https://cdn.mathpix.com/cropped/2024_05_06_2be4d8e018291d9e4bcfg-2.jpg?height=254&width=250&top_left_y=809&top_left_x=351)

Fig. 10

![](https://cdn.mathpix.com/cropped/2024_05_06_2be4d8e018291d9e4bcfg-2.jpg?height=389&width=389&top_left_y=741&top_left_x=757)

Fig. 11

Fig. 11 shows how to tile a $15 \times 15$ square using 25 tiles of each type.

Remark 1. A $18 \times 18$ square can be divided into $2 \times 9$ rectangles, each of which can be tiled using 2 tiles of each type.

Remark 2. It can be shown that any square $3m \times 3m (m > 3)$ can be tiled in the required manner using the tiles.

Comment. It is shown that $n^2$ is divisible by 9 and an incorrect conclusion is drawn that $n$ is divisible by 9 - 0 points.

It is shown that $n$ is divisible by 3 - 2 points.

It is shown that the floor can be tiled when $n=12$ - 2 points.

It is shown that the floor can be tiled when $n=15$ - 2 points.

It is shown that the floor can be tiled when $n=18$ - 1 point.