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In a tournament with $7$ teams, each team plays one match with every other team. For each match, the team earns two points if it wins, one point if it ties, and no points if it loses. At the end of all matches, the teams are ordered in the descending order of their total points (the order among the teams with the same total are determined by a whimsical tournament referee). The first three teams in this ordering are then chosen to play in the next round. What is the minimum total number of points a team must earn in order to be guaranteed a place in the next round?

  1. $13$
  2. $12$
  3. $11$
  4. $10$
  5. $9$

My attempt:

We have $7$ team and each team will be play $6$ matches. Possible scenario for the maximum marks of third team:

First team won $4$ matches with last $4$ teams and two ties with second and third team then total possible marks of first $=4\times2+1+1=10$

Second team won $4$ matches with last $4$ teams and two ties with first and third team then total possible marks of second $=4\times2+1+1=10$

Third team won $4$ matches with last $4$ teams and two ties with first and second team then total possible marks of first $=4\times2+1+1=10$

Now any of lost $4$ teams can be upto $=3\times2=6$ marks, since last four teams lost three matches each with first, second and third team.

So, maximum marks can be $10$ for third team.

Can you explain in formal way? please.

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  • $\begingroup$ If a team wins all mathes then max marks are $12$ i didnt get you exactly $\endgroup$ – Archis Welankar Apr 10 '16 at 7:31
  • $\begingroup$ @ArchisWelankar , question ask maximum of third team not for first team. and in that way, second teams can be max marks $10$ and third team $8$ or both second and third can get maximum marks $9$ each. Third team will never get maximum marks $10$. Am I right? $\endgroup$ – 1 0 Apr 10 '16 at 8:45
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You are correct that the maximum points for the third place team is 10 points. However the question asked for the minimum number of points that would guarantee at least a third place finish.

For example, if a team would get 9 points, are they a guaranteed a top-3 finish?

The answer is that 9 points is not a guarantee since it is possible that the top 4 teams all finish with 9 points. This would happen if each of the top 4 had one win, one loss, and one tie against the other 3 teams in the top-4, while all beating the bottom 3 teams.

However 10 points would guarantee a top-3 finish. To prove this, consider that there are 21 total games, which means there are 42 total points awarded. The top 4 teams can combine for a maximum 36 points since the other 3 teams will get 6 points in the games played amongst themselves. If 4 teams split 36 points, it is impossible for all 4 teams to get 10 points or more, therefore 10 points guarantees being in the top-3.

In conclusion, 10 is the minimum number of points that would guarantee a team at least a third place finish.

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