At a dinner party there are 8 guests. The dinner takes place a table shaped like a regular octagon. Each edge has a place setting labeled with the name of a different guest. Originally each person sits in the wrong place. Explain why the table can be rotated so that at least two persons are sitting in the right place.
Let the eight guests be pigeons and the eight possible positions the pigeonholes.
There are no pigeons in the first hole, because no guest is correctly seated in the first position.
This leaves seven pigeonholes and eight pigeons, so two of them must go in the same hole. That is, two guest must be correctly seated in one of the seven rotations.