$\left[\begin{array}{l}\text { Mathematical logic (other) })] \\ {[\quad \text { Case analysis }}\end{array}\right]$

After a hockey game, Anton said that he scored 3 goals, and Ilya only one. Ilya said that he scored 4 goals, and Seryozha 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?
There are two possible cases.

1) Anton told the truth about himself, meaning he scored 3 goals. Then Seryozha lied about Anton, so he told the truth about himself, meaning he 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) 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: Seryozha scored 5 goals. Consequently, Seryozha lied about himself and told the truth about Anton: Anton scored 2 goals. In this case, the boys scored a total of $1+5+2$ = 8 goals.

## Answer

no.