Problem 5.8. In the large family of Ivanovs, there are no twins. A reporter came to the Ivanovs to interview them.

During the interview, each of the children said: "I have an older brother." After thinking for a moment, the reporter was very surprised. But the head of the family explained that some of the children were joking, and only six of them told the truth. How many children can there be in this family, given that there are 4 more boys than girls? List all possible options.
Answer: 8, 10.

Solution. Let the Ivanov family have $x$ boys. Then the oldest boy joked, and the remaining $x-1$ boys told the truth.

If $x \geqslant 8$, then at least 7 children told the truth, which contradicts the problem's condition.

If $x \leqslant 5$, then there are no more than 1 girl in the family. But then no more than $4+1$ children could have told the truth, which contradicts the problem's condition.

If $x=6$, then there are 2 girls in the family. This situation is possible if the oldest and the youngest child in the family are girls. The total number of children in the family is $6+2=8$.

If $x=7$, then there are 3 girls in the family. This situation is possible if the three oldest children in the family are girls. The total number of children in the family is $7+3=10$.