How many wins does my team have to get?

Question

I am watching a tournament. 16 teams are participating in the playoff. Every team play against another once. If you win, you score 1 point, if you lose you dont score any point (the outcome of a match can't be draw). To be guaranteed to reach the main event, you have to rank in the top 6. How much win does the team I root for have to secure in order to be sure to reach the main event ?

On a different note, my little cousin who is learning about mutiplication at school told me : "Each teams will play 15 matches thats 16*15 = 240 matches !". I tried to explain to him that he's counting the match twice but he doesn't get it. Is there an easy visual representation of this situation ?

Thanks

Answer 1

How much win does the team I root for have to secure in order to be sure to reach the main event ?

Write down the score of every team :

1. The first one wins against every other : 15 pts.
2. The second one wins against every other except the first one : 14 pts
3. etc...

Therefore (separating the 6 first places) :

Scores : 15 , 14 , 13 , 12 , 11 , **10** | 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    3    4    5      6     7   8   9  10  11  12  13  14  15  16

So, if there is no draw, you have to win at least 10 matches.

Now, what happens if two teams have 11 points ? Well it depends on the rules of your sport, but my guess is that you're no longer in the top 6 (with only 10 points). So let's see how many teams from the right side we can get to the left side.

Suppose that the teams who scored 12 and 13 points loose agains the one who scored 9 :

Scores : 15 , 14 , 12 , 11 , 11 , **10** | 11 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    3    4    4      7      4   8   9  10  11  12  13  14  15  16

And there are 6 teams with a score of at least 11, so that if you win 10 matches, your can't be sure to be in the top 6.

Continuing with this idea, can you be sure with 11 wins ? Let's change the original repartition :

Scores : 15 , 14 , 13 , 12 , **11** , 10 | 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    3    4      5      6   7   8   9  10  11  12  13  14  15  16

to -> (teams 15, 14 and 13 loose against team 9)

Scores : 14 , 13 , 12 , 12 , **11** , 10 | 12 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  

to -> (teams 14 and 13 loose against team 10)

Scores : 13 , 12 , 12 , 12 , **11** , 12 | 12 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    2    2      7      2    2   8   9  10  11  12  13  14  15  16

Therefore, 11 wins in not enough to be in the top 6.

Let's try with 12 :

Scores : 15 , 14 , 13 , **12** , 11 , 10 | 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    3      4      5    6   7   8   9  10  11  12  13  14  15  16

->

Scores : 13 , 13 , 13 , **12** , 13 , 11 | 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    1    1      5      1    6   7   8   9  10  11  12  13  14  15  16

So if you win 12 matches, you're in the top 6.

There is, however a special case :

In case you're draw with the others :

Scores : 15 , 14 , 13 , **12** , 11 , 10 | 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    2    3      4      5    6   7   8   9  10  11  12  13  14  15  16

-> (the three first teams loose against the 9th)

Scores : 14 , 13 , 12 , **12** , 11 , 10 | 12 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  

-> (the two first against the 10th)

Scores : 13 , 12 , 12 , **12** , 11 , 12 | 12 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  

-> (the first against the 11th)

Scores : 12 , 12 , 12 , **12** , 12 , 12 | 12 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
Rank   :  1    1    1      1      1    1    1   8   9  10  11  12  13  14  15  16

In this case, all the 7 first team have the same number of point... See the rules to know what happen.

Answer 2

P - plays

X- does not play/not applicable

For each team count the number of horizontal P + vertical P