Chess Tournament Pigeonhole Principle
A chess tournament has participants, and any two players play one game against each other. Then it is true that in any given point of time, there are two players who have finished the same number of games.
There are players. If there is a player who has finished all of his games, then all other players must have played at least one game (against player ).
So, the set of number of games played by players cannot have the values and in it simultaneously.
So, the number of games played has only possible values. The number of players is . By the Pigeonhole Principle, there must be two players who have played the same number of games.