After the close of a soccer season, with four teams (named
A,
B,
C and
D),
the final table was:
TEAM Won Drawn Lost Goals Goals Points
For Against
A 3 0 0 7 1 9
B 1 1 1 2 3 4
C 1 1 1 3 3 4
D 0 0 3 1 6 0
Knowing that the four teams played against each other one time, and that team
A beat team
B 3x0, can you deduce the results of the other five games?
There are 6 games played:
A vs B
A vs C
A vs D
B vs C
B vs D
C vs D
A Beats B 3- 0 (Given). Since B allowed only 3 goals for all of its games, it's opponents (C & D) scored 0 goals. B tied C, as they the only teams with ties, so that score was 0 - 0. B scored 2 goals for the tournament, so B beats D 2 - 0. Thus far, we have:
A beats B 3 - 0
B ties C 0 -0
B beats D 2- 0
We know that A beats C, and A beats D, as A is undefeated. A scores 4 goals in these games, (7 total goals less the 3 scored vs. B) and the opponents score 1. A must have at least one goal in each game to win, and two goals in the game where the opponent scores 1. Thus, A scores either 2 in each game or 3 in 1 game and 1 in the other. If A beats C 3-1 and D 1-0, then C has to beat D 2-0 for C to reach its 3 total goals for. However this means that A vs D would be a 1 - 1 tie, as A scores 6 goals agains B & C combined, leaving only 1, and D has no goals against either B or C, leaving 1. That is impossible.
Flipping it over, if A Beats D 3 - 1 and A beats C 1 -0, then C has to score 3 goals in its game against D to reach it's goals for total. This is impossible, as D gives up only 6 goals in all of its games, and has already given up 5 against A and B. Thus, A scored 2 goals each against C and D. This brings us to this point:
A beats B 3- 0
A beats C 2 - ?
A beats D 2 - ?
B ties C 0 - 0
B beats D 2- 0
C beats D ? - ?
C scores 3 for the tournament and gives up 3. D gives up 6 for the tournament. D has given up 2 to A and 2 to B, so C scores 2 vs D. That leaves 1 for C to score against A. A only gives up 1 goal in all of its games, which C scored, so D scores nothing against A. D did score 1 goal in all of its games, which has to be against C, as D was shut out by both A and B. Thus, the final result:
A beats B 3- 0
A beats C 2-1
A beats D 2 - 0
B ties C 0 - 0
B beats D 2- 0
C beats D 2 -1
My Solution
