May 2023
Problem of the Month
World Cup Soccer
By Paddy Smith in GAMES
In a World Cup soccer group round, four teams in a section play a round robin series of games.
Winning a game is worth 3 points, a tie is worth 1 point, and a loss scores zero points.
In a certain section, the competing teams were Australia, India, Turkey, and the U.S.A.
From the following clues, can you determine each team's point total?
1. Only the U.S.A. had more than one tie.
2. Turkey had one more point than the U.S.A. and one less point than Australia.
Solution
The answer for this group is:
India 7 points
Australia 4 points
Turkey 3 points
U.S.A. 2 points
Turkey defeated U.S.A. and lost its other two games for a total of 3 points.
U.S.A. lost to Turkey and tied both India and Australia for 2 points.
India defeated Turkey and Australia and tied U.S.A. for 7 points.
Australia defeated Turkey, tied U.S.A., and lost to India, for 4 points.
Six games were played in all. From clue 1, at least two were ties. If there were a third tie,
from clue 1, it went to U.S.A. and India. In that case, the other three games were decisive. There
were either three or four decisive games, and so the total number of points earned by all teams was either 15 or 16.
From clue 2, three teams had consecutive point totals. Totals of 0-1-2 are impossible, because that would have required
six wins by the fourth team. Similarly, totals of 1-2-3 are impossible, because that would have required five wins by the fourth team
(as well as one more tie by Turkey). Totals of 3-4-5 would mean that the fourth team had 3 or 4 points; but scores of
3-3-4-5 or 3-4-4-5 would mean that the U.S.A., the only team with two ties, had the highest point total, contrary to clue 2.
Totals of 4-5-6 would mean that the fourth team scored 0 points and therefore had three losses. The teams with 4 and 6
points each had one loss to account for their scores, but this adds up to five losses, which added to the U.S.A.'s two ties
exceeds the number of games played. Totals of 5-6-7 or higher add up to more than the possible point totals of 15 or 16.
By elimination, teams had consecutive point totals of 2-3-4, which means the fourth team (India) had 7 points. The U.S.A. earned
2 points from ties with India and Australia, losing to Turkey. Turkey lost its other two games. India defeated Australia.
To summarize, here are the results of the six matches:
India defeated Australia.
India defeated Turkey.
India tied the U.S.A.
Australia defeated Turkey.
Australia tied the U.S.A.
Turkey defeated the U.S.A.