After the close of the soccer season, the final standings for the four teams named A, B, C and D were:
| Team
|
Won
|
Draw
|
Lost
|
GF
|
GA
|
Pts
|
| 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
|
Each of the four teams played against one other one time.
Team A beat team B, 3 - 0.
Can you deduce the results of the other five games?
In the standings,
GF stands for the number of goals that team scored
GA stands for the number of goals that team allowed
Pts stands for Points,and is calculated by mutiplying each win by 3 and each draw by 1. No points are awarded for a loss.
Solution:
Team B beat team D, 2- 0.
Team B drew with team C, 0 - 0.
Team A beat C, 2 - 1.
Team A beat D, 2 - 0.
Team C beat D, 2 - 1.
We see at once from the table that team B beat team D and drew with team C.
As B scored 2 goals to 0 in these games, they must have won 2 - 0 and drawn 0 - 0.
This disposes of B, and leaves three games, C x D, A x D, and A x C to be determined.
Now, A had only 1 goal scored against them (C or D).
D scored only 1 goal, and that must have been against A or C.
Assume that it was against A. In that case, C did not score against A. But C scored 3 goals altogether; therefore these must have been scored against D.
We find D had 6 goals against them : 2 scored by B, as shown, 3 by C (if we assume that D scored versus A), and the remaining goal was scored by A.
But, as we have just assumed, D scored 1 goal against A, the match would have been drawn.
It was won by A, and therefore D could not have scored against A.
Thus, the goal against A must have been scored by C.
And as C scored 3 goals, the other 2 must have been versus D, who must have scored their only goal against C.
Thus, A beat C by 2 - 1 and D by 2 - 0, while C beat D by 2 - 1.