The football problem

Probability Level 3

A football tournament with n teams (n < 9), with each team plays each other exactly once, and gets three points for a win, one point for a draw and zero points for a loss. After the tournament, one team was declared champion for having achieved the highest cumulative points. Interestingly, this was the team with the lowest number of wins. Find n


The answer is 8.

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Anderson Schroeder by Anderson Schroeder , 27, Brazil

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2 solutions

I asked several teachers to do this question and none of them succeeded. This question was taken from a Chinese Olympiad. And this problem was in a math book. I tried to talk to the author of the book and he told me that this was the answer. But do not remember the solution.

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Here are some hints towards the solution.

  1. Transform the scores so that you get +2 for a win, 0 for a draw and -1 for a loss (this makes bookkeeping easier, but otherwise isn't necessary).

  2. The total number of points would then be the total number of win-loss games, which has to be non-zero. The winning team needs to win at least 1 game.

  3. If the winning team only won 1 game, consider the other teams. Show that the number of games lost is more than the number of games won. Hence contradiction.

  4. If the winning team only won 2 games, show that there must be a team which won 3 games, lost 4 games. hence there are at least 8 teams.

  5. Show that it's possible to have 8 teams.

Calvin Lin Staff - 7 years, 1 month ago

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are you able to solve

Rohit Singh - 7 years, 1 month ago

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Yes, the steps that I listed will lead you to a complete solution. It has most of the structure, you just need to fill in a few details.

Calvin Lin Staff - 7 years, 1 month ago

ninnod njan chaya chodichoda kunjirama????

Gokul G Menon - 7 years, 1 month ago
Pranav Singla
Jan 30, 2018

Let winning team's - Number of wins - x - Number of draws - y - Number of draws - z

  • As winning team x min...implies other teams atleast x+1 win...to exceed the extra 3 points of other teams there must be 4×1points
  • 3<y ==> 4<number of teams <9
  • Only 4 cases left 5,6,7 gets easily rejected
  • Therefore 8 is the answer
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