Problem 6.8. In class 6 "A", there are several boys and girls. It is known that in 6 "A"

- girl Tanya is friends with 12 boys;
- girl Dasha is friends with 12 boys;
- girl Katya is friends with 13 boys;
- any girl will have a friend among any three boys.

How many boys can there be in 6 "A"? List all possible options.
Answer: 13, 14.

Solution. From the condition, it follows that there are at least 13 boys in the class. If there were at least 15, then among them we could choose three who do not befriend Tanya. But then Tanya would not have a friend among them, which is a contradiction. Therefore, there are 13 or 14 boys in total. Let's provide the corresponding examples.

1) Suppose there are 3 girls (Tanya, Dasha, Katya) and 13 boys in the class. Among them, there is a boy Andrey, who does not befriend only Tanya, and a boy Borya, who does not befriend only Dasha (all other pairs of people in the class are friends). It is easy to see that all conditions of the problem are satisfied.
2) Suppose there are 3 girls (Tanya, Dasha, Katya) and 14 boys in the class. Among them, there are boys Vlad and Denis, who do not befriend only Tanya, boys Zhenya and Kirill, who do not befriend only Dasha, and a boy Lёnya, who does not befriend only Katya (all other pairs of people in the class are friends). It is easy to see that all conditions of the problem are satisfied.