7.4. A team of workers has to carve $n$ ice sculptures for an ice town over two days. On the first day, they carved 11 sculptures, with each worker completing an equal share of the work. On the second day, only 7 workers showed up because some were transferred to another project. As a result, each worker had to do three times more work than on the first day, and the ice town was completed on time. How many ice sculptures could there have been in the town? List all possible options and explain why there are no others.
Answer: 12, 14, 18, 22, or 32.

Solution: Let $k$ be the number of workers who started the job on the first day. The amount of work done by each worker on the first day is $\frac{11}{k}$, and on the second day, since each worker did three times more, it is $\frac{33}{k}$. On the other hand, on the second day, 7 workers carved $(n-11)$ sculptures, so the amount of work done by each worker is $\frac{n-11}{7}$. We can set up the equation:

$\frac{n-11}{7}=\frac{33}{k}$, from which $k(n-11)=231$.

Thus, $k$ must be a divisor of 231: 1, 3, 7, 11, 21, 33, 77, or 231.

The options 1, 3, and 7 do not work, as it is known from the problem statement that some workers were transferred to another project, i.e., $k>7$. In the other cases,

| Number of workers $k$ | 11 | 21 | 33 | 77 | 231 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $n-11$ | 21 | 11 | 7 | 3 | 1 |
| Number of sculptures $n$ | 32 | 22 | 18 | 14 | 12 |

## Criteria:

7 points - correct answer and complete correct justification;

correctly and justified found only 4 solutions - 5 points; 3 solutions - 4 points; 2 solutions - 3 points; 1 solution - 2 points;

the equation $k(n-11)=231$ was found in some form, but the solution is not completed or has errors - 2 points

only answer - 0 points.