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        riddles >> medium >> Football Tournament
        
(Message started by: Barukh on May 1^st, 2007, 12:09am)

Title: Football Tournament
Post by Barukh on May 1^st, 2007, 12:09am

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.

Title: Re: Football Tournament
Post by ThudanBlunder on May 1^st, 2007, 5:42am

A start:
[hide]
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 ----- *

[/hide]
Any predictions for the Champions' League semi-finals?
I say 1-1 to United, and Chelsea after extra time.

Title: Re: Football Tournament
Post by ThudanBlunder on May 1^st, 2007, 7:14am

OK, I have it.
[hide]
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 ----- *

[/hide]

Nice puzzle, Barukh!

Title: Re: Football Tournament
Post by Barukh on May 1^st, 2007, 10:15am

on 05/01/07 at 07:14:16, ThudanBlunder wrote:

OK, I have it.

Uhmm? So what was the A's game score that gave H. the solution?

???

Title: Re: Football Tournament
Post by ThudanBlunder on May 1^st, 2007, 10:22am

on 05/01/07 at 10:15:39, Barukh wrote:

Uhmm? So what was the A's game score that gave H. the solution?
???

Oh, A's score? I see. Not reading the question again. Many a slip 'twixt cup and lip.

Title: Re: Football Tournament
Post by ThudanBlunder on May 1^st, 2007, 11:04am

on 05/01/07 at 10:15:39, Barukh wrote:

Uhmm? So what was the A's game score that gave H. the solution?
???

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
--------------
. . .10 7
where d = 3, 4, or 5

Well, it's 7.30 and time for the real football now, Liverpool-Chelsea!

Title: Re: Football Tournament
Post by SMQ on May 1^st, 2007, 11:52am

on 05/01/07 at 11:04:05, ThudanBlunder wrote:

Fixing a = 4,5,or 6 (or b = 1,2,or 3) does not.

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

Title: Re: Football Tournament
Post by ThudanBlunder on May 1^st, 2007, 2:58pm

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
--------------
. . .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. :)