Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of their own card. Prove that the table can be rotated so that at least 2 guests are seated in front of their own name card.
The extreme case is having swapped names in pairs i.e.
1 sits where 2 should sit and vice versa, 3 sits where 4 should sit and vice versa,...
Because no rotation will get two correctly seated, unless their number is odd from pigeon. Missing something? Is this really the extreme case?
Consider rotating the table through all possible positions. Then every person has every card in front of them once, and so the total number of correct matches is $15$. Since it is given that in one position there are no correct matches, the $15$ correct matches must be distributed over $14$ positions of the table. By the pigeonhole principle, there must be a position of the table which affords at least two correct matches.