GK 2020 Q3
3. There are 4 teams in a football tournament. Each team plays every other team exactly once. In each match, the winner gets 3 points and the loser gets 0 points. In the case of a draw, both teams get 1 point. After all matches have been played, which of the following total number of points is it impossible for any team to have obtained?
- A 4
- B 5
- C 6
- D 7
- E 8
Grey Kangaroo 2020, Question 3. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 8
Each team plays exactly three matches. Hence the maximum number of points any team can obtain is 3 × 3 = 9. A draw only gets 1 point. Hence the next highest total number of points possible, from two wins and a draw, is 2 × 3 + 1 = 7. Therefore it is impossible to obtain 8 points.
(Note: totals of 4, 5 and 6 points can be obtained by one win, one draw and one loss, one win and two draws and two wins and a loss respectively.)
Solution © UK Mathematics Trust, from the GK 2020 solutions.