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Topic: Football Tournament (Read 1394 times) |
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Barukh
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Football Tournament
« on: May 1st, 2007, 12:09am » |
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Mr. H. read an article on a local football tournament in a Sunday newspaper. Six teams A, B, C, D, E, F took place in a round robin tournament. The final table looked as follows:
Team Pts GF GA
-----------------
A 11 8 0
B 11 7 2
C 9 4 0
D 6 2 10
E 3 8 9
F 1 1 9
Mr. H. tried to figure out the result of every single game, but failed to do so. Later, he recalled the score of one of A’s games. After that, he was able to restore all the scores.
Are you?
Notes:
1. “Pts” means Points scored, “GF” – Goals for, “GA” – Goals against.
2. The winner gets 3 pts, the looser – 0; in case of a draw both teams get 1 pt.
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| « Last Edit: May 1st, 2007, 12:10am by Barukh » |
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ThudnBlunder
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Re: Football Tournament
« Reply #1 on: May 1st, 2007, 5:42am » |
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A start:
There were 6x5/2 = 15 games worth a maximum of 45 points.
As the total of points obtained is only 41, there must have been exactly 11 wins/losses and 4 draws.
A dropped 4 points without conceding a goal.
Therefore A drew 0-0 in 2 games and won 3 games.
B dropped 4 points and conceded 2 goals..
Therefore B drew 2 games and won 3 games.
C dropped 6 points without conceding a goal.
Therefore C drew 0-0 in 3 games and won 2 games.
So A,B,C drew 7 games.
So (as there were only 4 draws in total) A,B,C must have drawn 0-0 all the 3 games between themselves.
So (as there were 11 wins and 4 draws) D,E,F shared a total of 3 wins and 1 draw.
This draw must have been 0-0 between C and F as
D wins 2 games, draws none, and
E wins 1 game, draws none.
But D scored only 2 goals.
Therefore D beat both E and F by 1-0
So F scored its only goal against E
and E scored 6 goals against F
and 2 goals against B
A,B,C score all their 19 goals against D,E,F
D,E,F concede a total of 28 goals.
So D,E,F score a total of 9 goals against each other and 2 goals against B
So F has an aggregate score of 0-2 against A and B
So both A and B beat F by 1-0
So A scored 7 goals against D and E
So B scored 6 goals against D and E
So C scored 4 goals against D and E
D conceded 10 goals against A, B,C
E conceded 7 goals against A, B,C
Also, B scores more than 2 goals against E.
There seems to be 6 different solutions, so maybe I am missing something.
------ A ------B ------ C ------ D ------ E ------ F
A ---- * ---- 0-0 ---- 0-0 ----? ------- ? -------1-0
B ---- 0-0---- * ----- 0-0 --- ? ---- ?-2 ------- 1-0
C ---- 0-0 --- 0-0 --- * ------ ? -----? ---------0-0
D ----- ? ------ ? ----- ? ------ * --- 1-0 ------ 1-0
E ----- ? ------2-? ---- ? -----0-1 ---- * --------6-1
F ---- 0-1 ---- 0-1 --- 0-0 --- 0-1 --- 1-6 ----- *
Any predictions for the Champions' League semi-finals?
I say 1-1 to United, and Chelsea after extra time.
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| « Last Edit: May 3rd, 2007, 3:07am by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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ThudnBlunder
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Re: Football Tournament
« Reply #2 on: May 1st, 2007, 7:14am » |
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OK, I have it.
B must have scored 5 goals against E (the game he remembered), as this allows for only 1 solution.
3 goals allows 3 solutions and 4 goals allows 2 solutions.
------ A ------B ------ C ------ D ------ E ------ F
A ----- * ---- 0-0 ---- 0-0 ---- 6-0 --- 1-0 ----1-0
B ---- 0-0---- * ----- 0-0 --- 1-0 ---- 5-2 --- 1-0
C ---- 0-0 --- 0-0 ---- * ------ 3-0---- 1-0-----0-0
D ---- 0-6 --- 0-1 --- 0-3 ----- * ----- 1-0 ---- 1-0
E ----- 0-1 --- 2-5 --- 0-1-----0-1 ---- * ------ 6-1
F ---- 0-1 ---- 0-1 --- 0-0 --- 0-1 --- 1-6 ----- *
Nice puzzle, Barukh!
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| « Last Edit: May 1st, 2007, 8:37am by ThudnBlunder » |
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Barukh
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Re: Football Tournament
« Reply #3 on: May 1st, 2007, 10:15am » |
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on May 1st, 2007, 7:14am, ThudanBlunder wrote:
Uhmm? So what was the A's game score that gave H. the solution?
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ThudnBlunder
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Re: Football Tournament
« Reply #4 on: May 1st, 2007, 10:22am » |
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on May 1st, 2007, 10:15am, Barukh wrote:
Uhmm? So what was the A's game score that gave H. the solution?
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Oh, A's score? I see. Not reading the question again. Many a slip 'twixt cup and lip.
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| « Last Edit: May 1st, 2007, 10:29am by ThudnBlunder » |
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ThudnBlunder
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Re: Football Tournament
« Reply #5 on: May 1st, 2007, 11:04am » |
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on May 1st, 2007, 10:15am, Barukh wrote:
Uhmm? So what was the A's game score that gave H. the solution?
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I don't see how fixing A's score forces only one solution.
A scored 7 goals against D and E
B scored 6 goals against D and E
C scored 4 goals against D and E
D conceded 10 goals against A,B,C
E conceded 7 goals against A,B,C
Fixing d = 5 forces only one solution: b=1, a=6, f=1, e=3, c=1
Fixing a = 4,5,or 6 (or b = 1,2,or 3) does not.
. . . D E
A a b = 7
B c d = 6
C e f = 4
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. . .10 7
where d = 3, 4, or 5
Well, it's 7.30 and time for the real football now, Liverpool-Chelsea!
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| « Last Edit: May 1st, 2007, 11:32am by ThudnBlunder » |
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SMQ
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Re: Football Tournament
« Reply #6 on: May 1st, 2007, 11:52am » |
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on May 1st, 2007, 11:04am, ThudanBlunder wrote:| Fixing a = 4,5,or 6 (or b = 1,2,or 3) does not. |
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What's wrong with fixing a = 4? 
1 < a,b < 6; 3 < d < 5 (since B must defeat E by a score of d to 2); 1 < c < 3 (because d > 3); 1 < g, h < 3.
Fixing a = 4 forces c = e = 3.
Edit: fixed stupid B loses to E mistake with a = 2...
--SMQ
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| « Last Edit: May 1st, 2007, 12:12pm by SMQ » |
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--SMQ
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ThudnBlunder
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Re: Football Tournament
« Reply #7 on: May 1st, 2007, 2:58pm » |
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Yes, you are right.
. . . D E
A a b = 7
B c d = 6
C e f = 4
--------------
. . .10 7
Fixing a = 4, b = 3 forces
. . . D E
A 4 3 = 7
B 3 3 = 6
C 3 1 = 4
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. . .10 7
Fixing c = 1, d = 5 forces
. . . D E
A 6 1 = 7
B 1 5 = 6
C 3 1 = 4
--------------
. . .10 7
Fixing e = 1, f = 3 forces
. . . D E
A 6 1 = 7
B 3 3 = 6
C 1 3 = 4
--------------
. . .10 7
So there is a different table depending on whose result Mr H happens to remember. 
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| « Last Edit: May 1st, 2007, 6:10pm by ThudnBlunder » |
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