Question 59
There are $n \gt 1$ guests at Georg's birthday party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. Every guest now proceeds as follows. The guest takes one cup for each of their friends at the party and distributes all the water from their jug evenly in the cups. They then pass a cup to each of their friends. Each guest having received a cup of water from each of their friends pours the water they have received into their jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?