Math CentralQuandaries & Queries


Question from Peter, a student:

Can you please help with the following questions?

(a) It is known that among any group of the three students in a class two of them are friends. Total number of students is 25. Prove that there is a student who has at least 12 friends.

(b) There are 30 students in a class. They sit at 15 double desks, each one is for two students. Half of the girls sit with boys. Is it possible to make a rearrangement so that half of the boys sit with girls?

Sorry for asking for help on two questions.

Thank you


I can help with b).

If half the girls sit with boys then the number of girls must be even. Now consider the girls who are not sitting with boys. Each of them sits with another girl so this number of girls is even also. Thus the number of girls is divisible by 4. Now suppose they are rearranged and half the boys sit with girls. What can you say about the number of boys?

I hope this helps,

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