**Problem**: At any party of 1000 people, must there always exist two people at the party who have the same number of friends at the party? For the sake of this problem, one cannot be friends with oneself, and friendship is bidirectional.

**Solution**: This must always happen. Suppose to the contrary, that every person at the party has a different number of friends at the party. The minimum number of friends one could have is 0, while 999 is the maximum. Thus, we have 1000 different numbers to pick from, and 1000 people to assign numbers, where no two people have the same number. So somebody at the party, say Bill, has no friends at the party, while another, say Julie, is friends with everybody. But this means Julie is friends with Bill, which is a contradiction, because Bill has no friends.

The same method obviously works for any party of size $ n \geq 2$.

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What if nobody at the party knows each other, and just turned up due to a public advertisement? Then each person can have an arbitrary number of friends right?

The proof still holds. And if nobody knows each other, then everybody has zero friends, so there are two people who have zero friends. The point is that we only count friends who are

atthe party.The only way it can be arbitrary is if the concept of friendship is directed.