Soccer

[absoluzation]

Four soccer teams from the cities Icetown, Frostville, Glacierhampton and Coldbury have participated in the Xmas soccer tournament. During this tournament, each team played exactly one match against each of the other three teams. A win yields three points, a draw one point, and a loss zero points. No two of these six matches ended with the same result. (For example: if some match ended in 3:4 then none of the other five matches ended with 4 goals for one team and 3 goals for the other team.) Here is the final of table of the tournament:

Icetown - 2 wins, 0 draws, 1 loss - 5 Goals and 1 Goals against - 6 points
Frostville - 2 wins, 0 draws, 1 loss - 3 Goals and 5 Goals against - 6 points
Glacierhampton - 1 win, 0 draws, 2 losses - 5 Goals and 6 Goals against - 3 points
Coldbury - 1 win, 0 draws, 2 losses - 4 Goals and 5 Goals against - 3 points

Which of the following statements is true?
1. Cold. won its match against Glac. 1:0
2. Cold. lost its match against Glac. 0:1
3. Cold. won its match against Glac. 2:0
4. Cold. lost its match against Glac. 0:2
5. Cold. won its match against Glac. 3:0
6. Cold. lost its match against Glac. 0:3
7. Cold. won its match against Glac. 2:1
8. Cold. lost its match against Glac. 1:2
9. Cold. won its match against Glac. 3:1
10. Cold. lost its match against Glac. 1:3

So basically what I did was this:

Icetown has to lose (0,1) as they only lose one time with one goal being allowed.
They can't win (4,0) as that would make the other match (1,0) which is not allowed (No match ends with the same result).
So Icetown wins (2,0) and (3,0) and loses (0,1)

The only way Frostville can obtain 3 points in 2 matches is for the winning scores to be 2 and 1.
Since Icetown already had a (2,0) match, Frostville wins (2,1) and (1,0). In total, Frostville allowed 5 goals,
so that means they lost a match (0,4)
With this information, you can conclude that Frostville beats Icetown (1,0).

So now I'm wondering whether Glacierhampton or Coldbury wins the (4,0) match against Frostville.
If Coldbury won the match, that would mean their other matches would be (0,0)/(0,5) or (0,1)/(0,4) or (0,2)/(0,3)
(0,0) is impossible because there are no draws.
(0,4) is impossible because no match ends with the same result.
However, I don't know if (0,2)/(0,3) is possible. If it isn't, then Glacierhampton definitely wins the (4,0) match.

Can anyone help me with this? This might be really easy but I just can't see it.

 

[Mr. Bland]

This is an entertaining logic puzzle, but I'm not sure I'd call it a math problem. In what context was this presented to you?

Ultimately it boils down to process of elimination. One could hypothetically brute force every possible combination of scores, but logic helps us narrow down and rule out several without spending fifteen minutes writing them all out. It all works out very cleanly, so props to the puzzle designer.

Icetown lost 1 game and had 1 goal scored against them, so that game must therefore have been 0:1. Icetown's own 5 goals could be distributed across its 2 wins as either (4:0, 1:0) or (3:0, 2:0). Since the 1:0 combination is already accounted for by the loss, we know that Icetown's games ended in 3:0, 2:0 and 0:1.

Frostville won 2 games with 3 goals. The ways this could have happened are (2:0, 1:0) and (2:1, 1:0), both of which contain a 1:0 victory over Icetown. Since the 2:0 scenario is already accounted for by one of Icetown's victories, we know that the results of Frostville's games were 2:1, 1:0 and 0:4.

Glacierhampton and Coldbury both suffered two losses. They each lost to Icetown, meaning one had a loss of 0:2 and the other had a loss of 0:3. One of the two also lost to Frostville. Since Frostville won two games and one of them was against Icetown with 1:0, their other winning score of 2:1 must have been against either Glacierhampton or Coldbury. Therefore, of these two teams, the one who lost to Frostville would have had losing scores of either (0:2, 1:2) or (0:3, 1:2).

Suppose Glacierhampton lost to Frostville. Since Glacierhampton had 5 goals total and one of them is accounted for in this hypothetical loss to Frostville, the other 4 must therefore have been in a victory against Coldbury. However, Coldbury had 5 goals total scored against them, with as few as 2 coming from Icetown. Therefore, Glacierhampton could have scored at most 3 goals against Coldbury, rendering this scenario impossible. Glacierhampton did not lose to Frostville, but instead lost to Coldbury.

Coldbury lost to Icetown either 0:2 or 0:3 and lost to Frostville 1:2. If it lost to Icetown 0:3, then it won against Glacierhampton 3:0, but this is a duplicate result. Therefore, we know that the results of Coldbury's games were 3:1 (Glacierhampton), 0:2 (Icetown) and 1:2 (Frostville).

Filling in the blanks, we know that Glacierhampton's games were 4:0 (Frostville), 0:3 (Icetown) and 1:3 (Coldbury).