8.4 Workers were laying a square floor of size $n \times n$ with tiles of size $2 \times 2$ and $3 \times 1$. They managed to completely cover the floor, using the same number of each type of tile. For which $n$ is this possible? Justify your answer.
Solution. Let $a$ tiles of each type be used. Then, from the equality of areas, we have $4a + 3a = n^2$. Hence, the number $n^2$, and therefore the number $n$, must be divisible by 7. We will show that any $n$ divisible by 7 will work. Indeed, in this case, the square floor can be divided into non-overlapping $7 \times 7$ squares, so it is sufficient to show how to cover a $7 \times 7$ floor using an equal number of tiles of both sizes (7 tiles of each type). This can be done as shown in the figure.

Answer: For all $n$ divisible by 7.

![](https://cdn.mathpix.com/cropped/2024_05_06_4f1964101946872669e0g-6.jpg?height=303&width=323&top_left_y=1299&top_left_x=1483)

To the solution of problem 8.4

| INCLUDED IN THE SOLUTION | SCORE |
| :--- | :---: |
| Correct and justified answer | 7 points |
| Justification that $n$ must be divisible by 7 and correct examples for specific $n$ divisible by 7 | 5 points |
| Only justification that $n$ must be divisible by 7 | 4 points |
| Correct proof that all numbers divisible by 7 satisfy the condition | 3 points |
| Correct tiling examples for specific $n$ divisible by 7 | 1 point |
| Answer without justification or with incorrect justification | 0 points |