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.