# Number of game won in a round-robin tournament

Sixteen players participated in a round-robin tennis tournament. Each of them won a different number of games. How many games did the player finishing sixth win?

I got some idea here:Round Robin Tournament

The post here suggests that if there are $$2^{n}$$ players, then at least one player won $$\dfrac{2^{n}(2^{n}-1)/2}{2^{n}}$$ (and round up) games.

I believe this can also be applied any $$n$$ players.

So for us, at least one player won 8 games. However, by hypothesis that each player cannot win the same number of games. We can conclude that there must be a unique player who won 8 games.

I do not think this attempt has any help to this question, and I don't know how to do the next.

Also, how could I use the fact that "player finishing sixth"?

Thank you!

Define a player's score as the number of games won by that player.

By hypothesis, there are $$16$$ distinct scores.

Consider the maximum score.

Each player plays exactly $$15$$ games, so the maximum score can't be more than $$15$$.

But the maximum score can't be less than $$15$$ either, else there would be at most $$15$$ distinct scores.

Thus, the maximum score is $$15$$.

It follows that the $$16$$ scores are just $$0,1,2,...,15$$ hence the player who finished in $$6$$-th place had a score of $$10$$.

• Thank you so much!! Brilliant Answer – JacobsonRadical Oct 21 '18 at 6:19