11.4. Workers need to tile a square floor consisting of 36 cells (identical squares). The workers have 12 tiles. 11 strips of three cells and one corner piece of three cells. Will the workers be able to complete the task? Cutting the tiles is not allowed.
Solution. Suppose that a $6 \times 6$ grid can be divided into 11 strips of size $3 \times 1$ and one corner piece consisting of three cells. We place one of the numbers $1, 2, 3$ in each cell as shown in the figure.

| 1 | 1 | 1 | 1 | 1 | 1 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 |

Consider the sum $\mathrm{S}$ of all the numbers in this square. Clearly, $\mathrm{S}$ is divisible by 3. On the other hand, $\mathrm{S}$ is the sum of 12 numbers, which are the sums of three numbers located in the strips and the corner piece. Note that the sum of the numbers in any strip - horizontal or vertical - is a multiple of three. However, the sum of the numbers in the corner piece is not divisible by 3 in any case. Therefore, $\mathrm{S}$, being the sum of 11 numbers that are multiples of 3 and one number that is not divisible by 3, is itself not divisible by 3. This is a contradiction to the assumption of the possibility of such a tiling.

Answer: They will not be able to.