11.1. A literature teacher decided to find out from the students of class 11 "A" how many people from their class are absent. He received the following answers: Peter: "More than one." Victor: "More than two." Tatyana: "More than three." Charles: "More than four." Polina: "Less than four." Shurik: "Less than three." How many people are actually absent from class 11 "A", if exactly three students told the truth? Provide all possible answers and prove that there are no others.
Solution: Note that if Charles is right, then not only he, but also Peter, Victor, and Tatyana told the truth - a total of 4 students. This is a contradiction. Therefore, Charles lied, and no more than four people are absent. Let's assume Tatyana is right. Then exactly four people are absent, and in this case, she, Peter, and Victor told the truth, while the other three students lied. This option works. Let's assume Tatyana also lied, then Polina is right, and among Peter, Victor, and Shurik, exactly one is wrong. This one cannot be Peter: in this case, Victor also lied. The other options work: if Victor is wrong, then Peter and Shurik are right, and then exactly 2 people are absent. If Shurik is wrong, then exactly three people are absent, and this situation also works.

Answer: 2, 3, or 4 people.

Recommendations for checking:

| included in the work | points |
| :--- | :--- |
| Proven that only 2, 3, and 4 people could be absent | 7 points |
| Proven that 2, 3, and 4 people could be absent, but not proven the absence of other options | 3 points |
| Among all possible options, only two are proven | 2 points |
| Among all possible options, only one is proven | 1 point |
| Incorrect solution or its absence | 0 points |