Question

The AFL (Australian Football League) has 18 teams. Each team plays one game per week. Assume a team never plays a given other team more than once. Prove that after 8 weeks, there exists at least 3 teams none of which have played each other.

Answer 1

Total number of team=18

Each team plays one game per week

Two teams required for each game.

Therefore, total number of games in one week

Three team can be choosen out of 18 team

Therefore, possible number of ways of choosen 3 teams out of 18 teams

Possible number of ways of choosen 3 teams out of 18 teams=816

If each pair of three teams in one game , then total number of games in 8 weeks=816

But,there are 8 weeks

Therefore, total number of games play in 8 week

Therefore, it is not possible that each pair of three team play atleast one game.

Hence, at-least 3 teams are there none of which have played each other.