Question

[World cup fun problem!]

\(\large 7\) countries are participating in a tournament.

In the first round, each country plays a match against every other country and
the lowest scoring \(3\) teams will be eliminated.

Points are awarded as below :
```
win : 2
draw : 1
loss : 0
```

Find
1) Minimum points required by a team that guarantee surviving the first round.
2) All the possible permutations of points possible after first round.

Answer 1

\(\large \color{black}{\begin{align} \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
\text{teams} & \text{ A} & \text{B}& \text{ C} & \text{D}& \text{E} & \text{F}& \text{G} &\text{total}\\ \hline
\text{A} & -& 0& 1 & 1& 1 & 1& 1 & 5\\ \hline
\text{B} & 2& -&0 & 0& 1 & 1& 1& 5 \\ \hline
\text{C} & 1& 2&- & 1& 0 & 0& 0&4 \\ \hline
\text{D} & 1& 2&1 & -& 1 & 1& 1&7 \\ \hline
\text{E} & 1& 1&2 & 1& -& 1& 1& 7\\ \hline
\text{F} & 1& 1&2 & 1& 1&-& 1& 7\\ \hline
\text{G} & 1& 1&2 & 1& 1&1& - &7\\ \hline
\end{array}

\end{align}}\)

Answer 2

from this table teams \(\large A,B\quad \text{and}\quad C\) are eliminated and
the minimum point seems \(\large 7\)

Answer 3

your table actually shows that \(6\) points will guarantee survival because the 3 eliminated teams have points less than \(6\)

Answer 4

look at the last column for points
http://gyazo.com/f9a654a36f3c59f2ed76858e55ebcb1e

If your table is correct, it seems india and south africa have cleared the first round ?

Answer 5

the question then is : can a team get eliminated even after scoring 6 points ?

Answer 6

yes it is \(6\) but i just selected an arbitrary case, where the 4 teams had \(7\) points

Answer 7

then why india and south africa are not celebrating yet lol

Answer 8

i honestly don't watch cricket , and they have already celebrated i think, they are in last six , so way ahead to go, do or die every match

Answer 9

Okay but it seems india and south africa have already scored the minimum required points(6). So they will survive even if they lose all remaining matches

Answer 10

yes they will survive guaranteed.

Answer 11

(again thats according to your table) I haven't seen any news channels mentioning about these countries clearing the first round yet..

Answer 12

but the run rate matters in case they tie in top six , so every run\win counts

Answer 13

i have read this in newspaper i think

Answer 14

is there a chance for more than four teams to score >= 6 points ?

Answer 15

in world cup it is not single round robin in first phase i think , but in your question it is right

Answer 16

the maximum matches a team can play in first phase is \(4\) , i think

Answer 17

The tournament will feature 14 teams, the same number as the 2011 World Cup.

The format is the same as the 2011 edition: 14 teams will take part in the initial stages, divided into two groups of seven; the seven teams play each other once before the top four teams from each group qualify for the quarter-finals.

Answer 18

The format ensures that each team gets to play a `minimum of six matches` even if they exit in the group stage.

Answer 19

oh i see , it is single round robin

Answer 20

yes

Answer 21

so it all boils down to proving below :

There is no chance for more than four teams to score \(\ge 6\) points

Answer 22

that will prove 6 is the minimum required points for guaranteeing survival

Answer 23

yep six!

Answer 24

chess tournament has also the similar scoring system

Answer 25

but in case of tie , the two players play a duel match

Answer 26

*tie for the first place

Answer 27

http://en.chessbase.com/post/zurich-rapid-and-playoff-nakamura-clinches

Answer 28

yeah rapid round

Answer 29

so how do we prove 6 is the minimum ?

Answer 30

draw a table where 4 teams have 6 points

Answer 31

is this possible ?

\[\large \color{black}{\begin{align} \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
\text{teams} & \text{ A} & \text{B}& \text{ C} & \text{D}& \text{E} & \text{F}& \text{G} &\text{total}\\ \hline
\text{A} & -& 0& 1 & 1& 1 & 1& 1 & 5\\ \hline
\text{B} & 2& -&2 & 0& 1 & 1& 1& 7 \\ \hline
\text{C} & 1& 0&- & 1& 0 & 0& 0&2 \\ \hline
\text{D} & 1& 2&1 & -& 1 & 1& 1&7 \\ \hline
\text{E} & 1& 1&2 & 1& -& 1& 1& 7\\ \hline
\text{F} & 1& 1&2 & 1& 1&-& 1& 7\\ \hline
\text{G} & 1& 1&2 & 1& 1&1& - &7\\ \hline
\end{array}

\end{align}}\]

Answer 32

\(\large \color{black}{\begin{align} \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
\text{teams} & \text{ A} & \text{B}& \text{ C} & \text{D}& \text{E} & \text{F}& \text{G} &\text{total}\\ \hline
\text{A} & -& 0& 1 & 1& 1 & 1& 1 &\color{red}{ 5}\\ \hline
\text{B} & 2& -&0 & 0& 1 & 1& 1& \color{red}{ 5} \\ \hline
\text{C} & 1& 2&- & 2& 0 & 0& 0&\color{red}{ 5} \\ \hline
\text{D} & 1& 2&0 & -& 1 & 1& 1&\bf\color{black}{ 6} \\ \hline
\text{E} & 1& 1&2 & 1& -& 1& 1& \color{green}{ 7}\\ \hline
\text{F} & 1& 1&2 & 1& 1&-& 1& \color{green}{ 7}\\ \hline
\text{G} & 1& 1&2 & 1& 1&1& - &\color{green }{ 7}\\ \hline
\end{array}
\end{align}}\)

this case shows the minimum is \(\bf \Large 6\)

Answer 33

But why u didn't consider each game in first round as only lose or win game ? And how are the countrties gonna play like how many games in.first round allowed ?

Answer 34

Oh wait my b.ad huh got it now

Answer 35

it can be possible to consider ,where a country either losses or win , but the their are many combinations but i chose maximum draws to make it simpler

Answer 36

there are total of seven countries and each country will play total of 6 games