Summary: Investigating A Single-Elimination Tournament

Part A: Investigating a single-elimination tournament
1. Given that there are 16 teams in the round of 16, how many teams will go forward to the
quarter-finals 8
semi-finals 4
The final? 2
2. Describe how you found the number of teams at each stage.
Here is an example, if two teams play against each other it equals 1 winning team, so if we begin with 16 teams, we follow the same rule. Divide 16 into 2 or half the number to give us 8 teams into the next round.
The winning team of each match goes onto the quarter rounds, while the team that lost is eliminated.
Similarly with the semi-finals, to go through it again though it would be easier just to divide the number (in this case 4) in 2, which the answer is 2.
3. Could a single-elimination tournament begin with any number of teams? Explain your answer.

Yes, it can, for instace It would seem no at the beginning, but multiples of 16 co-inerce perfectly with single elimination tournament chart tables.

I guess that another reason for yes is because, even with numbers that aren't in the power of two tournaments can begin and end with any number of teams, there are only disadvantages such as some teams having to play more matches, while other teams have their first match in the second round.
4. A single-elimination tournament begins with n teams.
What can you say about the value of n?
Using algebra describe how to find the number of teams going into each round of the tournament.
The value of n can be virtually any number. It is in the power of two and if it is not a multiple of 16 then some of the teams may have to go through some 'bye' rounds, which, as described earlier, means that some teams have to miss out on one round.
n teams will go into the first round.
h teams will go into the quarter finals.
d teams will go into the semi

finals.
b teams will go into the finals.

5. In a single-elimination tournament with 16 teams, how many matches in total will be played in deciding who the winner is?
Explain how you found your answer to this question.
The answer is 15. Similar to question 2, the basic step is counting the matches of each round (you get this by dividing the number of teams by two) and adding them all together.
So 8 (16 divided into 2) + 4 (8 divided into 2) + 2 (4 divided into 2)+ 1 (2 divided into 2) equals 15.
6. If instead of there being 16 teams at the start there were 32 teams, how many matches would now be played in deciding who the winner is?
All the steps are taken, so really all we need to do is follow the table I have made below and add
2 Teams = 1 Match
4 Teams = 2 Matchs
8 Teams = 4 Matchs
16 Teams = 8 Matchs
32 Teams = 16 Matchs

1 + 2 + 4+ 8 + 16 = 31 matches

7. Describe any patterns you notice in the calculations you have done.
Looking down the table I can see that the number of matches, such as the number of teams double themselves everytime we go up a multiple of 16
8. By looking at other starting number of teams, investigate possible connections between the number of starting teams and the number of matches needed in deciding a winner. Can you write a mathematical rule that describes the connection you found?
Hint: It may help you to draw up, and continue, a table like the one shown below – is there a connection between the total number of matches played and the power of two (n)?
Yes, there is a large connection between the total number of matches that are played and the powers of 2, in fact single elimination tournaments are based on powers of 2. The simple mathematic rule is simply to divide the number of teams, as in the starting number for the total number of matches played. If there are two teams playing against eachother it equals one match or game.
Starting number - Total number of matches played
2 - 1
4 - 2
8 - 4
16 - 8
32 - 16
64 - 32
Part B: Group Matches
In next year's World Cup, before the single-elimination tournament can take place all the group matches need to be played first to decide who the 16 teams are.
In the 2010 FIFA World Cup there are 8 groups (A – H), each with 4 teams.
Consider Group A with teams A1, A2, A3 and A4. Each team plays all the other teams in the group.
1. How many games will be played altogether in each group? There were 6 games played in each group.
This explanation is most clearly understood with examples - A1, A2, A3, and A4.
Team A1 will play Team A2, Team A3 and Team A4
Team A2 will play Team A3 and Team A4 (as Team A1 their team have already played against each other)
Team A3 will play with Team A4 ( they have already played against Team A1 and A2)
And Team A4 has played with all the other teams.
3 + 2 + 1 = 6

2. If instead of only four teams in each group there were six teams. How many games would be played altogether with six teams in a group? Altogether there would be 15 matches played in each group if there were 6 teams.
9 + 6 = 15

3. How many games would be played altogether if there were 8 teams in a group?
Describe any patterns you notice in what you have done this far.
If we change the nuber of teams in a group to 8 it would equal 28 matches played altogether in each group. The pattern so far is stated already in Q1 Part B, Where Team 1 would play Team 2, 3 and 4, Team 2 would verse Team 3 and 4 and Team 3 versing Team 4 and Team 4 already having matches with everyone.
To simplify it further it can be seen as this: with each number of teams the number one less down ( so for instance if it was 9 teams it would be 8). That umber then adds to all the numbrs before it, so -
For example - 9 teams
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36 matches played all together
If it were an single elimination tournament this process would be different, but since each team is versing every single other team in its group this is the persific rule for group matches.

4. Investigate the number of games played for other numbers of teams and record your data in a table.
EXAMPLE
22 Teams would equal in group matches:
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21 = 231
EXAMPLE
18 Teams equals to 153 in group matches:
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 + 17= 153
5. Write a mathematical rule which will give you the number of matches played when you know the number of starting teams in a group - remember to test your rule to see if it works.
Try to explain why your rule works.
My rule workes for group matches perspifically, you add every number leading up to the actual number, but not the actual number itself =
Example = 7 teams in a group, 4 groups:
The matches equal 1, 2, 3, 4, 5, 6, but not 7 itself.
We then add those numbers together =
1+2+3+4+5+6 = 21
And if you chose to go further to the equation=
21 x 4 (groups) = 84
You can see it being proven that it works and it is a faster and easier way then working out bylabelling which team played which other.
Example = 13 teams in each group, 3 groups
1+2+3+4+5+6+7+8+9+10+11+12+13 = 91
91 x 3 = 273
Part C: Your Proposal
Having completed Parts A and B you should now have a much better idea about how to organise the hockey tournament that the PE department will run in November.
How do you think the PE department should organise their hockey tournament?
My schedule of which the hockey tournament should be organised will make sure that there will be a smooth, unhurried schedule with extra hours to spare. I think that the hockey tournament should follow my schedule because not only is there time allocated but the whole tournament is cut down to 3 days.
The whole hockey tournament is a single elimination tournament, single elimination tournament's have a lot of advantages, but there are also some disadvantages due to the fact that competition is strained as a team must win all to go onto the next round.
For instance, in a single elimination tournament we are forced to have 'bye' rounds, which are rounds where, for instance, two teams play a game, the winner of that game then versus the one team that was left out. This has another disadvantage as one team's first match will be in the second round.
The order in which 'byes'are determined is by taking the number of tournament teams and taking the next highest number which is in the power of two, so 2, 4, 6, 8, etc. The way this works is all using the power of two, as an example, a single elimination tournament has 18 teams, so the highest power of two next is 20. Another example;e is 14 teams, 16 is the next highest PO 2 (Power of two).
The reason why having extra time is essential is because there is no strict determination of how long a match will be, the time allocated for a match may be not enough, counting in-between and pre-game breaks and with other matters, such as team transitions and announcements that can use valuable time .
As my research shows, genuine field hockey matches take on average about 3 hours, with small breaks each 20 minutes and / or half hours. However, on some occasions, a match can take up to 3 hours and a half, so it would be necessary for more time to make sure that the hockey tournament doesn't go over our 12 hours a day.
There will be 18 or 20? games played altogether. I point worth going over is the fact that we have access to 4 hockey fields, so if wanted, we can play 4 matches at a time easily.
Our tournament will be able to manage 4 - 6 games in one hockey field each day, depending on whether there are smooth team transitions or problems. However, we do not have to use every match at once, it depends on the crowd number.
I think that each day there will be at least 2 hockey fields being played on, if not all where possible.
On the first day of 20 teams play on 4 pitchers equals 3 games of 6 teams play on one pitch, 3 games of 6 teams on pitch number 2, 4 teams play 2 games and 4 teams play 2 games on field number 4.
Each game averages 3 hours, therefore, 10 matches times 3 hours per match equals 30 hours divided by four pitchers.
The pitches with 2 games equal 6 hours per pitch. The pitchers with 3 games equal 9 hours.
Day 2 - there is 10 teams remaining. We use 3 pitchers and have one game per pitch and the fourth pitch has 2 games
Day 3 - There are 6 teams remaining, there is one game per pitch played on 3 pitches, the two teams with the lowest points are eliminated. The remaining 4 teams go into the quarter finals 2 teams are eliminated and the remaining two teams play in the semi-finals.

Part D: Reflection
Having completed the task and made your presentation you now need to reflect on what you have done. Look at the rubric for Criterion D, this asks you to:
Explain in some detail how you were able to use your findings from parts A and B to help you organise the hockey tournament.
Explain in some detail how what you have done in this task could be useful in a real life situation
Having completed the task, look back at what you did to consider how appropriate the methods you used were – if you could do it all again would you do it the same way?
Part A and Part B were essential to learn and complete before Part C, and studying all the subjects made sure that I was more prepared for Part C. From the first two parts I learnt and worked outmathematicle rules by myself, thus gaining a skill and making sure that I was able to connect my own single elimination tornament schedual.