# Champions League Football (group stage)

24en1KDexd29
Posers and Puzzles 22 Feb '13 16:56
1. 22 Feb '13 16:56
Two questions.

(a) What is the most points a team can get and still not qualify for the knockout stage?

(b) What is the least points a team can get and still qualify?

For those not into European football:
In the group stage there are four teams in each group. Every team plays every other, once only. 3 points for a win. 1 point for a draw. 0 points for a loss. The top two teams go through to the knockout stage. If two teams are on equal points there is a formula (irrelevant to this problem) to decide who is placed the higher.
2. talzamir
Art, not a Toil
22 Feb '13 17:45
I can see a team failing to advance with six points, and two points being enough to make the cut?
3. 23 Feb '13 01:07
For the most it must be 6, there are 6 matches at stake, and at most 3 points per match so 18 points maximum in total.

To fail to qualify with X, two teams must get equal to or better than X, which proves 6 points is the max (as 3*6 = 18, so two other teams must get 6 points too)

This can be achieved, given teams A, B, C and D
AvB = 3 pts to A
AvC = 3 pts to C
AvD = 3 pts to A
BvC = 3 pts to B
BvD = 3 pts to B
CvD = 3 pts to C

now teams A,B and C are all on 6 points and one must go out

For min, the worst case is if one team beats all the others and the others fight it out with draws e.g
AvB = 3 pts to A
AvC = 3 pts to A
AvD = 3 pts to A
BvC = 1 pt to B, 1 pt to C
BvD = 1 pt to B, 1 pt to D
CvD = 1 pt to C, 1 pt to D

now A goes through with 9 pts, but B,C and D each have 2 points and only one more can go through.
4. 24 Feb '13 21:20
actually, the problem as described is for the world (or euro) cup only. i think it has happened several times that 3 nations were tied with 6 points and one got out.
in champions league every team plays twice vs all other team, i guess the logical answer would be 12 then. i don't think case has ever happened though, but i might be wrong.
5. 06 Mar '13 16:52
Originally posted by iamatiger
For the most it must be 6, there are 6 matches at stake, and at most 3 points per match so 18 points maximum in total.

To fail to qualify with X, two teams must get equal to or better than X, which proves 6 points is the max (as 3*6 = 18, so two other teams must get 6 points too)

This can be achieved, given teams A, B, C and D
AvB = 3 pts to A
AvC ...[text shortened]... ow A goes through with 9 pts, but B,C and D each have 2 points and only one more can go through.
If every match finished in a home win, except one which is drawn, the team who drew at home would have seven points and not qualify as all the others would have nine points(in fact one would have ten)
6. 07 Mar '13 22:18
Originally posted by deenny
If every match finished in a home win, except one which is drawn, the team who drew at home would have seven points and not qualify as all the others would have nine points(in fact one would have ten)
I think you are saying that all teams have 3 home games and 3 away games (like i described the CL above)? In any case, you shouldn't focus on the 4th ranked team, but on the 3rd ranked. So the maximum without advancing can be achieved if all have 2 wins against the last ranked team (which gets 0 points) plus 2 wins and 2 losses against the other 2 teams. That would make it 12 points for 1st - 3rd ranked with one having to go out.
7. 07 Mar '13 23:00
Originally posted by deenny
If every match finished in a home win, except one which is drawn, the team who drew at home would have seven points and not qualify as all the others would have nine points(in fact one would have ten)
The question said each team plays each other once and once only. So you don't play the same team home and away.