5.5. After the hockey game, Anton said that he scored 3 goals, while Ilya only scored one. Ilya said that he scored 4 goals, and Seryozha scored as many as 5. Seryozha said that he scored 6 goals, and Anton only two. Could it be that together they scored 10 goals, given that each of them told the truth once and lied once? Explain your answer.
Answer: No, it could not.

Solution. The first method. There are two possible cases.

1) If Anton told the truth about himself, then he scored 3 goals. Then Seryozha lied about Anton, so Seryozha told the truth about himself, meaning he really scored 6 goals. Consequently, Ilya lied about Seryozha and told the truth about himself, meaning he scored 4 goals. In this case, the boys scored a total of $3+6+4=13$ goals.
2) If Anton lied about himself, then he told the truth about Ilya, meaning Ilya scored 1 goal. Then Ilya also lied about himself and told the truth about Seryozha, meaning Seryozha scored 5 goals. Consequently, Seryozha lied about himself and told the truth about Anton, meaning Anton scored 2 goals. In this case, the boys scored a total of $1+5+2=8$ goals.

The second method. A total of six statements were made, three of which are true. Each boy made two different statements, one of which must be true and the other false. Therefore, Anton scored either 3 goals or 2, Ilya scored either 1 or 4 goals, and Seryozha scored either 5 or 6 goals. Note that the sum of 10 can only be achieved in one way: if Anton scored 3 goals, Ilya scored 1, and Seryozha scored 6. However, the statement "Anton scored 3 goals, and Ilya scored 1" was made by the same boy, so this situation is impossible.

## Grading criteria:

+ The correct answer and a fully justified solution are provided

$\pm$ The correct answer and a generally correct solution are provided, but there are gaps and deficiencies in the justification

Ғ The correct answer is provided, but only one of the two possible cases is considered (see the first method)

$\mp$ The correct idea of the solution is present (see the second method), but there are errors in calculation or reasoning

- Only the answer is provided
- The problem is not solved or is solved incorrectly