Football

Probability Level 3

In a football competition there are eight teams. Every team plays exactly one game with every other team (so there will be $\dfrac{7 \times 8}{2}=28$ matches).

The scoring system is as follows:

In each match

the winner gets 2 points,
the loser doesn't get any point,
if the match ends in a draw, then both team get 1 point.

One of the coaches says the next sentence to his team:

$\quad$ If we score $n$ points, we are in the first four teams for sure!

What is the minimum value of $n$ ?

9 10 8 12 11 13

1 solution

Áron Bán-Szabó
Jun 30, 2017

We will show that the minimum value of $n$ is $11$ .

Suppose there are $5$ teams, so that all of them scored minimum $11$ points. Since in each match excatly $2$ points are given, in the $10$ matches between these $5$ teams, $20$ points were given. From the other three teams they could get maximum $2*3*5=30$ points. So maximum they could get $20+30=50$ points in all, but this contradicts that they achieved $5*11=55$ points together. So $11$ points is enough.

Is $10$ points enough? The answer is no. A counterexample:

Football

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