This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant X, let t(X) be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true:
1) For any two friends X′ and X′′, we have t(X′)=t(X′′),
2) For every contestant X, the set {t(Y):Y is a friend of X} consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need? graph theorycombinatorics