Problem
COM-B1-M09-P019 Tournament of \(7\) Players
#19
★★★☆☆ Level 3 of 5
In a tournament of \(7\) players, everyone played everyone exactly once, with no draws. Prove that some player won at least \(3\) games and some player lost at least \(3\) games.
There were \(\binom{7}{2}\) games.
There are \(\binom{7}{2}=21\) games, hence \(21\) wins in total. The average number of wins per player is \(21/7=3\), so someone won at least \(3\) games. Similarly, there are \(21\) losses in total, the average number of losses is \(3\), so someone lost at least \(3\) games.
Gentle introduction to directed graphs via tournaments.