Soccer

Dec 2019
16
2
tik tok on the clock
(Can someone also give me the calculations? These kinds of questions are really hard to me. It would mean a lot! :) )
Four soccer teams from the cities Icetown, Frostville, Glacierhampton, and Coldbury have participated in the Christmas 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 3:2, then none of the other five matches ended with 2 goals for one team and 3 goals for the other team.) Here is the final table of the tournament:


Icetown - 2 wins - 0 draws - 1 loss - 5 goals and 1 goal against - 6 points
Frostville - 2 wins - 0 draws - 1 loss - 3 goals and 5 goals against - 6 points
Glacierhamption - 1 win - 0 draws - 2 losses - 5 goals and 5 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. Coldbury has won its match against Glacierhampton 1:0.
  2. Coldbury has lost its match against Glacierhampton 0:1.
  3. Coldbury has won its match against Glacierhampton 2:0.
  4. Coldbury has lost its match against Glacierhampton 0:2.
  5. Coldbury has won its match against Glacierhampton 3:0.
  6. Coldbury has lost its match against Glacierhampton 0:3.
  7. Coldbury has won its match against Glacierhampton 2:1.
  8. Coldbury has lost its match against Glacierhampton 1:2.
  9. Coldbury has won its match against Glacierhampton 3:1.
  10. Coldbury has lost its match against Glacierhampton 1:3.

 
Jan 2009
448
127
There must be a typo.
Sum of goals for: 5+3+5+4= 17
Sum of goals against: 1+5+5+5 = 16
Supposing for Coldbury, it's 4 goals for and 6 goals against.


Reasoning Below


Shortening the names of teams to their first letter.

For I,

It must lose (0,1). Team loses exactly one game, but only allows 1 point.
It cannot win (4,0) as this would force the other game to be (1,0) but this contradicts no score being the same ignoring order.
It hence wins (2,0) (3,0)

For F,
It must win (2,1) and (1,0) as F scores 3 points and the only way to win twice scoring 3 points is for the winning scores to be 2 and 1 but (2,0) was already won by I.
Since it allowed 5 total goals, it lost (0,4) in a game.
Hence F beat I (1,0) by symmetry.

For G,
It must win the (4,0) game as C winning the (4,0) game implies that the remaining games only allowed goals totaling 6, of pairs (0,0)/(0,6) (0,1)/(0,5) (0,2)/(0,4) (0,3)(0,3) but these result in contradictions.
* There were no draws, so no (0,0)
* (0,1) was already played between F and I
* Can’t have two (0,4) games
* Can’t have two (0,3) games
Hence G beats F (4,0)
This leaves one game where it lost but scored 1 point. It can’t be (1,2) as this would mean G played F twice. This leaves G to lose (1,3) as losing (1,4) would mean it also lost (0,1) but that game was played between I and F.
Since it played games (4,0) and (1,3), that leaves (0,2) as it allowed 5 points.
Hence it lost to I in the (2,0) game, and lost to C in the (1,3) game.
The only remaining game is the (3,0) game. I beat C (3,0).

Summary

Team I 5/1
W(3,0) vs C
W(2,0) vs G
L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/5
L(0,2) vs I
L(1,3) vs C
W(4,0) vs F

Team C 4/6
L(0,3) vs I
L(1,2) vs F
W(3,1) vs G

Hence the answer is (9) only.
 
Last edited:
Dec 2019
16
2
tik tok on the clock
GUYS im so sorry! Glacierhampton actually has 6 goals against. I'm so so sorry. The rest is correct tho. :/
 
Jan 2009
448
127
In that case, I think my previous answer can be modified except that the reasoning to get there will vary slightly. Should generally be the same approach though. (using partitions)
Without looking deeply, reasoning should be similar as the case above as to why G must win (4,0) vs F, leaving the loss of (0,3) vs I as the remainder instead

Team I 5/1
W(3,0) vs G
W(2,0) vs C

L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/6
L(0,3) vs I

L(1,3) vs C
W(4,0) vs F

Team C 4/5
L(0,2) vs I

L(1,2) vs F
W(3,1) vs G
 
Dec 2019
16
2
tik tok on the clock
There must be a typo.
Sum of goals for: 5+3+5+4= 17
Sum of goals against: 1+5+5+5 = 16
Supposing for Coldbury, it's 4 goals for and 6 goals against.


Reasoning Below


Shortening the names of teams to their first letter.

For I,

It must lose (0,1). Team loses exactly one game, but only allows 1 point.
It cannot win (4,0) as this would force the other game to be (1,0) but this contradicts no score being the same ignoring order.
It hence wins (2,0) (3,0)

For F,
It must win (2,1) and (1,0) as F scores 3 points and the only way to win twice scoring 3 points is for the winning scores to be 2 and 1 but (2,0) was already won by I.
Since it allowed 5 total goals, it lost (0,4) in a game.
Hence F beat I (1,0) by symmetry.

For G,
It must win the (4,0) game as C winning the (4,0) game implies that the remaining games only allowed goals totaling 6, of pairs (0,0)/(0,6) (0,1)/(0,5) (0,2)/(0,4) (0,3)(0,3) but these result in contradictions.
* There were no draws, so no (0,0)
* (0,1) was already played between F and I
* Can’t have two (0,4) games
* Can’t have two (0,3) games
Hence G beats F (4,0)
This leaves one game where it lost but scored 1 point. It can’t be (1,2) as this would mean G played F twice. This leaves G to lose (1,3) as losing (1,4) would mean it also lost (0,1) but that game was played between I and F.
Since it played games (4,0) and (1,3), that leaves (0,2) as it allowed 5 points.
Hence it lost to I in the (2,0) game, and lost to C in the (1,3) game.
The only remaining game is the (3,0) game. I beat C (3,0).

Summary

Team I 5/1
W(3,0) vs C
W(2,0) vs G
L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/5
L(0,2) vs I
L(1,3) vs C
W(4,0) vs F

Team C 4/6
L(0,3) vs I
L(1,2) vs F
W(3,1) vs G

Hence the answer is (9) only.
Can you copy and paste this and adapt the things that vary? I really need help.

I'm so so sorry :(
 
Dec 2019
16
2
tik tok on the clock
In that case, I think my previous answer can be modified except that the reasoning to get there will vary slightly. Should generally be the same approach though. (using partitions)
Without looking deeply, reasoning should be similar as the case above as to why G must win (4,0) vs F, leaving the loss of (0,3) vs I as the remainder instead

Team I 5/1
W(3,0) vs G
W(2,0) vs C

L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/6
L(0,3) vs I

L(1,3) vs C
W(4,0) vs F

Team C 4/5
L(0,2) vs I

L(1,2) vs F
W(3,1) vs G
Basically what you get is this
"G has to win the (4,0) match, 'cause if C did, the remaining goals (which are 5 in total) would be in one of these pairs: (0,0)/(0,5) (0,1)/(0,4) (0,2)/(0,3)
The first pair is impossible as there are no draws so (0,0) can't be a match
The second pair is impossible because a previous match ended with (0,1) already and you can't get the same result (which is(0,4))
I honestly don't know why the third pair is also impossible. If you have time, could you explain it to me?
 
Jan 2009
448
127
Short version: The third pair of (0,2)/(0,3) is impossible because this would imply C was matched against I twice.

Updated Reasoning Below

Shortening the names of teams to their first letter.

For I,

It must lose (0,1). Team loses exactly one game, but only allows 1 point.
It cannot win (4,0) as this would force the other game to be (1,0) but this contradicts no score being the same ignoring order.
It hence wins (2,0) (3,0)

For F,
It must win (2,1) and (1,0) as F scores 3 points and the only way to win twice scoring 3 points is for the winning scores to be 2 and 1 but (2,0) was already won by I.
Since it allowed 5 total goals, it lost (0,4) in a game.
Hence F beat I (1,0) by symmetry.

For G

It must win the (4,0) game as C winning the (4,0) game implies that the remaining games only allowed goals totaling 5, of pairs (0,0)/(0,5) (0,1)/(0,4) (0,2)/(0,3) but these result in contradictions.
* There were no draws, so no (0,0)
* (0,1) was already played between F and I
* (0,2)/(0,3) pair implies C matched against I twice

Hence G beats F (4,0)
This leaves one game where it lost but scored 1 point. It can’t be (1,2) as this would mean G played F twice. This leaves G to lose (1,3) as losing (1,4) would mean it also lost (0,1) but that game was played between I and F.
Since it played games (4,0) and (1,3), that leaves (0,3) as it allowed 6 points.
Hence it lost to I in the (3,0) game, and lost to C in the (1,3) game.
The only remaining game is the (2,0) game. I beat C (2,0).

Summary

Team I 5/1
W(3,0) vs G
W(2,0) vs C

L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/6
L(0,3) vs I
L(1,3) vs C
W(4,0) vs F

Team C 4/5
L(0,2) vs I
L(1,2) vs F
W(3,1) vs G


Hence the answer is (9) only.
 
Dec 2019
16
2
tik tok on the clock
Short version: The third pair of (0,2)/(0,3) is impossible because this would imply C was matched against I twice.

Updated Reasoning Below

Shortening the names of teams to their first letter.

For I,

It must lose (0,1). Team loses exactly one game, but only allows 1 point.
It cannot win (4,0) as this would force the other game to be (1,0) but this contradicts no score being the same ignoring order.
It hence wins (2,0) (3,0)

For F,
It must win (2,1) and (1,0) as F scores 3 points and the only way to win twice scoring 3 points is for the winning scores to be 2 and 1 but (2,0) was already won by I.
Since it allowed 5 total goals, it lost (0,4) in a game.
Hence F beat I (1,0) by symmetry.

For G

It must win the (4,0) game as C winning the (4,0) game implies that the remaining games only allowed goals totaling 5, of pairs (0,0)/(0,5) (0,1)/(0,4) (0,2)/(0,3) but these result in contradictions.
* There were no draws, so no (0,0)
* (0,1) was already played between F and I
* (0,2)/(0,3) pair implies C matched against I twice

Hence G beats F (4,0)
This leaves one game where it lost but scored 1 point. It can’t be (1,2) as this would mean G played F twice. This leaves G to lose (1,3) as losing (1,4) would mean it also lost (0,1) but that game was played between I and F.
Since it played games (4,0) and (1,3), that leaves (0,3) as it allowed 6 points.
Hence it lost to I in the (3,0) game, and lost to C in the (1,3) game.
The only remaining game is the (2,0) game. I beat C (2,0).

Summary

Team I 5/1
W(3,0) vs G
W(2,0) vs C

L(0,1) vs F

Team F 3/5
W(2,1) vs C
W(1,0) vs I
L(0,4) vs G

Team G 5/6
L(0,3) vs I
L(1,3) vs C
W(4,0) vs F

Team C 4/5
L(0,2) vs I
L(1,2) vs F
W(3,1) vs G


Hence the answer is (9) only.
THANK YOU SO SOOOO MUCH !!! I can't express how happy I am!!