Chess Tournament Pigeonhole Principle

Book: A Walk through Combinatorics


A chess tournament has nn 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 nn players. If there is a player AA who has finished all of his n1n - 1 games, then all other players must have played at least one game (against player AA).

So, the set of number of games played by players cannot have the values 00 and n1n - 1 in it simultaneously.

So, the number of games played has only n1n - 1 possible values. The number of players is nn. By the Pigeonhole Principle, there must be two players who have played the same number of games.