I'm given the problem: In a tournament which 18 teams participate, a team being matched with another in a round don’t match again in the follwoing (later) rounds. After 8 rounds prove that there are 3 teams not being matched with each other.
I don't know where to start from. Can anyone help me to aproach to the solution of this problem
The number of matches needed for all teams to have faced each other exactly once is $$\binom{18}2=153$$ Since each team should only be able to play against one team at a time, the number of distinct matches played in total is $$9\cdot 8 = 72$$ This leaves $153-72=81$ unplayed matches.
What does your statement about "$3$ teams not being matched with each other" mean?