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30 soccer teams participate in a tournament. Each team plays against every other team exactly once. For a victory a team gets 2 points, for a draw 1 point and for a defeat 0 points. At the end of the tournament all teams have a different amount of points. The team that finishes in second place has the same amount of points as the last eight teams together.

How did the game end between the teams on the 22nd and 24th place?

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    $\begingroup$ 2 points for a win?! What year is this - 1980?! ;-) $\endgroup$
    – Stiv
    Feb 7, 2020 at 14:54
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    $\begingroup$ It seems strange to have so many numbers on a lateral-thinking question...are you sure it is so? $\endgroup$
    – George Menoutis
    Feb 7, 2020 at 14:58
  • $\begingroup$ this means they played 465 games? $\endgroup$
    – jmbmage
    Feb 7, 2020 at 15:07
  • $\begingroup$ @abbaf33f 435, surely? (29 rounds of 15 matches) $\endgroup$
    – Stiv
    Feb 7, 2020 at 15:10
  • $\begingroup$ @Stiv, right it was either 1954 or 1974, unless you prefer 1966! $\endgroup$
    – ThomasL
    Feb 7, 2020 at 16:26

1 Answer 1

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Team 22 beat team 24

I decided that the easiest way to start this problem was

to assume the 1st place team beat all other teams, the 2nd place beat all teams except first place, etc.

Then I counted up the points for the last 8 teams and

They added up to 56, which is equal to the 2nd place team.

Most importantly,

Because 56 is the maximum score the 2nd place team can have (57 would force a tie for 1st) and 56 is the minimum combined score for the last 8 teams (they lost every game to every team 22nd place or better), we know that team 22 must have beaten team 24. A tie or a loss means that the combined score for the last 8 teams is greater than 56, which is more than the maximum for 2nd place.

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  • $\begingroup$ Are you sure this satisfies: "At the end of the tournament all teams have a different amount of points." $\endgroup$
    – jmbmage
    Feb 7, 2020 at 17:42
  • $\begingroup$ In the one scenario I set up before trying to prove the overall system, every team had a different amount of points. So if even if that is the only scenario where each team has a different amount of points, it still works. $\endgroup$
    – Joel Rondeau
    Feb 7, 2020 at 18:02

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