Football Fever

Algebra Level pending

This FIFA World Cup, there are 4 teams each group. Teams are awarded +3 points for winning a match , +1 for a draw and 0 points for a loss.Suppose the teams and their points after they have played all the group league matches are as follows: Team A: a points Team B: b points Team C: c points Team D: d points - Find the maximum value of (a+b+c+d)

  • BONUS: Generalize solution with N teams in the group.


The answer is 18.

Richard Feynman by Richard Feynman , 17, India

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1 solution

Zico Quintina
Jun 29, 2018

a + b + c + d a + b + c + d represents the total number of points awarded in all group league matches. To maximize this, there should be no draws (a draw awards one point to each team, i.e. two points in all; a game not ending in a draw awards three points to the winner, i.e three points in all.)

In a league of four teams there will be ( 4 2 ) = 6 \dbinom{4}{2} = 6 matches in total, so the maximum number of points awarded would be 6 × 3 = 18 6 \times 3 = \boxed{18}

In a league of N N teams, there would be ( N 2 ) = N ( N − 1 ) 2 \dbinom{N}{2} = \dfrac{N(N - 1)}{2} matches in total, so the maximum number of points awarded would be 3 N ( N − 1 ) 2 \dfrac{3N(N - 1)}{2}

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