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Suppose you have a group of 6 football teams. The same as the World Cup, but with 6 teams instead of 4.

Suppose each team plays each other twice, meaning each team plays 10 games.

A win grants 3 points, a draw grants 1 point and a loss grants 0 points.

How many points would one team have to win in order to guarantee 1st place in the group?

How many points would it take to win in order to guarantee 2nd place or better (please note I mean 2nd place OR BETTER, meaning either 2nd or 1st place) in the group?

How many points guarantee 3rd place or better?

I tried to work it out with permutations, but since this theoretical group would have 30 matches total, and 3 possible results for each (team A wins, draw, team B wins) that means there are over 200 trillion permutations. Is there a simpler way to do this? Along with the answer, could you explain how you come to that answer?

Thank you and warm regards, Max

Max Jarl
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Partial answer, showing that the right method is to consider the extreme cases, not to worry about all the permutations.

If the group is very unbalanced then it's possible to have two teams each of which beats all the others and splits their two matches. So 27 points guarantees only a tie for first.

I think 28 will suffice - that's 9 wins and a draw. If the second best team was involved in that draw they might have 25. I think 25 might guarantee second place at least.

Can you finish?

Ethan Bolker
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Suppose team A wins both games against teams C, D, E. Say they win one and draw one against team B. And say that team B wins both of its games against teams C, D, and E.

That means that team B wins 8 games, draws one, and loses one. That comes out to 25 points, so clearly 25 points does not guarantee first place.

Okay, so suppose team B has 26 points instead. There's only one way to get 26 points: 8 wins and two draws. There are two cases here: either both draws were against the same team, or they were against two different teams. If they were against the same team, then that team also had two draws so it had at most 26 points. If they were against different teams, then each team that team B tied also lost a game to team B. As such, the teams that played team B to a draw could have at most 8 wins, 1 draw, and 1 loss, for a total of 25 points. So 26 points guarantees at least being tied for first place.

Okay, so suppose team B has 27 points. That can only be 9 wins and one loss. The team that team B lost to also has a loss to team B, so they have at most 9 wins and one loss. So 27 points also guarantees at least being tied for first. (It's impossible for one team to have 26 and another to have 27.)

Okay, so suppose team B has 28 points. That's 9 wins and a draw. Since team B beat every other team at least once, no other team has more than 27 points. So 27 points guarantees first place.

Your best bet is likely to continue reasoning in this manner. It will presumably get increasingly complicated, but I think it will be manageable especially if you're happy using a computer to check multiple cases.

Daniel McLaury
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