1
$\begingroup$

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!

$\endgroup$
3
$\begingroup$

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 $10$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.