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Sporting Team Draw Matrix
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Calculating The Draw
In days gone by allocating team or individual sporting draws into individual rounds of a competition, require people to sit down with pen an paper an draw crisscross lines across the page to map drawn teams to each other. One major problem facing any sporting team coordinator or organisation is allocating the correct team draw combinations to complete one round in any given competition. For example if you have 30 plus individuals members or teams in any given competition, there would be over 435 individual draw combinations alone. How can the draw coordinator be absolutely sure they have allocated all teams to play every other team once only throughout the competition season? The following document will show you how to calculate any size team draw easily an accurately, furthermore you will not need to know anything about mathematics to be able to accomplish this daunting task... |
Yes indeed you will be able to
map any size team draw from the lowest possible combination of 2 teams
right up to infinite unknown teams. There is no limit to the accurate
combinations you can derive from within this sporting draw matrix
system. And you can do it simply an easily with a pen an a piece of
paper. With a little insight you may soon realise that this matrix
system can be used for non-sporting applications as well. For now we
will stay within the context of sporting events.
Pre-Draw OrganisationBefore any team draw can be calculated you need to know two items of information.
You will need to know unconditionally how many teams are going to be in the final draw. If your overall sporting clubs do not forward their nominated teams into the central coordinator, then there is no accurate method for calculating any final team draw. The draw coordinator must know before hand how many teams are in the final competition before you can calculate the draw table for those teams. The draw coordinator must also know the duration of the competition season so as to be able to determine how many rounds he/she can place into the final overall competition. One Round One round of competition is where every team has played against all other teams once only. Depending on the sporting event, draw coordinators may allocated multiple rounds throughout the season. Once a matrix draw is accurately calculated the draw coordinator need only multiply these values with the same team names to have multiple rounds. The draw coordinator can then work out the duration of the competition for the coming season. Once the duration of competition is known the draw coordinator may then decide to add or remove competition rounds to have the entire season of competition end on a specific calendar date. |
Building The Draw MatrixI'll assume you have determine the amount of teams in the competition and the approximate duration the competition will run till the end of the season. Ok let's get started with a simple competition using 2 teams in one single round of competition. A 2x2 cell matrix for a two team competition would look something like this :
figure 2 In the above 2x2 draw matrix of figure 2 there are two teams : Team 'A' and Team 'B'. The draw coordinator would simply replace the letters 'A' and 'B' with the corresponding team names making them relevant to their own competition. It should be obvious to people that Team 'A' can not play against Team 'A' ,likewise team 'B cannot play against itself, so this null combination is given an 'X' in that cell to denote that this draw combination is impossible. (also see 4x4 matrix figure 2) We then simply increment the count while moving down the columns for the given number of teams minus 1. We start counting from 1 on up to the matrix root base minus one (e.g. 2x2 matrix root = 2-1). Because each team cannot play against itself there will always be one less in the weekly games count. Example, there are two teams in the matrix of figure 2, so we start incrementing the count moving downwards from the first available column (aqua shaded cell) under the heading Team 'A' using the number '1' . We continue incrementing the count while going down the column until we run off outside the matrix . |
We then move across from left
to right to the top of the next column under Team 'B' and then increment
the count moving down the columns again. (also see
4x4 matrix in figure 2) It is vitally important
that you understand this incremented count while moving down the columns
principle. Incremented counting while moving down the columns allows you
to understand and build the draw matrix correctly. In the next 6 x 6
matrix example below I will explain the important column counting an
projection principle a bit further. In the 2x2 matrix of figure 2 there
is no more cells to be able to fill so we have confirmed that there is
only one round with one game play off in this two team competition. Team
draw matrices are always even value matrices; never odd. How to
calculate odd team draws will be explained further as we go...
Reading information from the Matrix To decode the draw information from the 2x2 matrix of figure 2 you simply read off the identical grouping of numbers. In the case of the 2x2 matrix you would read the grouping of '1's' from the column and row intersection. Do not read the diagonal impossible 'X' combinations. After you have read the matrix you would end up with the following formatted draw table of figure 3.
After you have completed a few team draw matrices of your own, you will begin to realise that you only need to read off the top right triangle section of any draw matrix, as the lower left triangle section of the matrix is a mirror image opposite of it's top right section. So in reality we will never really use all of the matrix information presented before us. |
1. Blank Matrix
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2. Starting Top Row
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3. Counting Down
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4. First Projection
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5. Second Projection
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6. Third Projection
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7. Forth Projection
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8. Completed Matrix
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figure 4
Step-By-Step Six Team Matrix.Remember you can never have a truly odd value matrix because we will always add in an imaginary team called 'Bye' to make up for any odd team draw. To explain further, a 6x6 matrix is used for both an odd value 5 team, and an even value 6 team competition draw. Similarly a 100x100 draw matrix is used for an odd value 99 team draw and an even value 100 draw competition. All sporting team draw matrices are even numbered regardless. Any size even matrix can be used for a next size down odd team draw. That is, if you have an odd number of teams in the competition draw then use a matrix that is one increment or order bigger and place the name 'Bye' as a team name to even up the matrix. You then calculate out the matrix as I'm about to show you... Step-by-Step Remember when counting, the maximum number of weeks to each round equals the matrix root/order minus 1. E.g. if the matrix is a 6x6 then the maximum number of weeks in the round is 6-1 = 5 . When you reach the maximum incremented count of 5 in any one column you must start the loop over again from number 1. You must continue incrementing while moving down the column until all cells of that column are filled, or when crossing the diagonal 'X' line, the value is projected to the last column and row. |
The importance of column
incrementing correctly :
The trick in getting the matrix to work correctly is the incremented count while moving down the columns and projecting at each 'X' cell or the diagonal separation line. Reiterating :You must always increment the count to the maximum matrix root/order -1. Example, in the matrix of figure 4 the root value = 6, minus 1, which equals a maximum incremented count of 5. You must always increment and complete each column to the maximum count. You may then move to the top of the next column and start from the next incremented number corresponding in the top row of the column you are working. Important : when you encounter a cell with an 'X' in it you must project the current number both straight down an on a 90 deg angle along the row to the very last column an row of the matrix. I can not stress the importance of understanding this projection principle. You must understand that you increment the count while moving down the columns firstly, then when you encounter an 'X' you project the current number along the column and row to both the very last bottom row and the far right column in the matrix. When you finally understand this principle you may develop ways of completing the matrix via different counting techniques. It may be obvious to some that transferring the diagonal values may be easier an simpler. I leave that up to you... See the step-by-step matrix draw in figure 4. A handy hint is to treat the 'X' cell like a stove hot plate. Whenever you touch the 'X' cell hot plate the numbers jump to both the last column and row. |
| Translating
From The Completed Matrix :
6x6 Team Draw Matrix
figure 5 Once the matrix is filled out and checked for correctness we can then read off the accurate draw information from within the completed matrix. (see figure 4 & 5.) Using the matrix in figure 5 we can quickly see that the numbers in the matrix are grouped into groups of 3, i.e. 3x1's, 3x2's, 3x3's etc... The highest number in the matrix is also five. Therefore we will have 5 weeks duration to one single round of this competition. The matrix further tells us we will require a grouping of 3 games per week over a total of 5 weeks for every team to play all other teams once. Therefore the derived information tells us we need a minimum 3 x 5 column table to tabulate the results onto a printable draw format. Remember we do not read from the intersection of 'X' cells null diagonal. Furthermore, we need only read from the top right triangle section of the matrix (see yellow shading in figure 5), as the lower left triangle section is a mirror image opposite of the upper right section. By reading both the upper and lower triangle halves of the matrix we would be duplicating the same draw values but only in reverse. I will now show you step-by-step how I derived the valid draw information from the 6x6 matrix in figure 5. Again the aqua coloured cells are the current column I'm working on. In reality the draw coordinator would simply substitute the letters 'A' through 'F' as the real team names in the competition. Also in the case of an odd team draw, the draw coordinator may substitute the letter 'F' for the team named 'Bye' to even out an odd 5 team competition draw. Every team would play the team named 'Bye' which in reality means that team steps forward to the next week of games. The completed draw table signifies one round of the competition. The draw coordinator need only multiply the matrix draw table to make a simple multiple round competition. You may also change the team names 'A' through 'F' for each new round in the competition without having to change the data mapped into the matrix. I therefore recommend you use single letters to denote team names in the matrix as I have done. You then substitute the matrix letter for the real team name when deriving the final draw table. It is much quicker an easier to use single letter team name variable values inside a draw matrix. Also to eliminate favouritism or be accused of bias to a competition draw, the draw coordinator may like to place the letters 'A' through 'F' in one hat, and the actual real team names in another hat. The coordinator simply draws a matrix letter an a corresponding team name out of each hat to define a true an totally random draw. |
An again you need only
translate the letters of the matrix to the actual team names when
completing the final draw table. If you are clever you may be able to
use a 12x12 matrix to map a 6x6 team draw into two full rounds of
competition. You will really need to know what you are doing if you are
to complete this type of matrix.
Step-By-Step Matrix Translation
Reading the grouped 1's intersection in the yellow shaded section of figure 5
Reading the grouped 2's intersection in the yellow shaded section of figure 5
Reading the grouped 3's intersection in the yellow shaded section of figure 5
Reading the grouped 4's intersection in the yellow shaded section of figure 5
Reading the grouped 5's intersection in the yellow shaded section of figure 5
(The abbreviation V's in the above table denotes 'verses or plays against'.) |
Draw Matrix
figure 6 |
Translation From Draw Matrix
(The 'V' in the above table denotes verses.) |
| The above 10 x 10 matrix for figure 6 has 45 combinations to complete for one round of the competition. The competition would require 5 games per week over a period of 9 weeks. The above matrix can also be used in a pool type competition where all teams may be divided into separate individual competition pools. | A pooled competition would see that the top teams of each separate pool combine for a play off at the end of the season. Using separate pools in a competition draw reduces the combinations and duration needed to complete the season on time. The drawback with using pools is that not all teams get to test their true ability against all other teams. |
Sample Golf Team Competition Matrix
(4 x 4 Team and 12 x 12 Individual )
3 or 4 Golf Team Matrix
figure 7 |
Translation
From Draw 4 x 4 Matrix
(The 'V' in the above table denotes verses.) figure 7 |
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The 4 x 4 translation draw table can be used for an odd or even individual or team draw. The number of players in each team can be any number of players you desire. If you wish to set a handy cap for Golf teams, then you may wish to place more players in one team with respect to all the other teams. A handy cap example might be 3 players in team 'A', 5 players in team 'B', two players in team 'C' and 4 players in team 'D'. To use the matrix as an odd team draw you simply replace one of the column alphabet headings with the team name "BYE". Example if the team named 'D' is not present for the competition then you simple replace the label 'D' with the corresponding name "BYE". All teams will then be required to miss playing in one day of each round of the competition. |
You can duplicate the total matrix translation into as many total rounds as you wish. Example, you can duplicate the above matrix translation table in figure 7 by 3 times to have a total of 3 rounds to the entire competition. The 4 x 4 matrix of figure 7 can be used along with the 12 x 12 matrix of figure 8. The 4 x 4 team draw matrix can be separated down into 12 individual players that can then play off against every other individual player once over in a single round with in the entire competition. (See figure 8 matrix below.) |
12 x 12 Individual Matrix
12 x 12
Individual Golf Matrix
figure 8 |
The 12 x 12
matrix of figure 8 can be used in a golf competition as an individual or
team draw in an 11 day or round robin type of competition. The Matrix is
coloured to show people how the draw table is translated from the 12 x 12
matrix. We only use the top triangle half of the matrix as the bottom
triangle half of any team draw matrix is a mirror image opposite to the top
half. See how the colour of the columns in translation table of figure 9 corresponds to the colour of the groups of numbers in the 12 x 12 matrix in figure 8. The actual number in each number grouping then translates to the placement of each match, day or week value. This all depends on how you play out each match of the competition, and/or whether you further totally randomise each draw.
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| Translation
From Individual Golf 12 x 12 Matrix
(The 'V' in the above table denotes verses.) figure 9 |
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Organized or Totally Random Draw
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| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | |
| A | X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| B | 1 | X | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 2 |
| C | 2 | 3 | X | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 4 |
| D | 3 | 4 | 5 | X | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 6 |
| E | 4 | 5 | 6 | 7 | X | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 8 |
| F | 5 | 6 | 7 | 8 | 9 | X | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 10 |
| G | 6 | 7 | 8 | 9 | 10 | 11 | X | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 12 |
| H | 7 | 8 | 9 | 10 | 11 | 12 | 13 | X | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 14 |
| I | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | X | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 16 |
| J | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | X | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 18 |
| K | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | X | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 |
| L | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | X | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 3 |
| M | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | X | 6 | 7 | 8 | 9 | 10 | 11 | 5 |
| N | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | X | 8 | 9 | 10 | 11 | 12 | 7 |
| O | 14 | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | X | 10 | 11 | 12 | 13 | 9 |
| P | 15 | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | X | 12 | 13 | 14 | 11 |
| Q | 16 | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | X | 14 | 15 | 13 |
| R | 17 | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | X | 16 | 15 |
| S | 18 | 19 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | X | 17 |
| T | 19 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | X |
Figure 10
| The above 20x20 matrix of figure 10 requires resources to implement 190 total games to enable every team to play all other teams or individuals once. The draw coordinator will need to allocate resources for 10 team games per week over a period of 19 weeks. | Application of this matrix may find viable use in a single person individual darts tournament or the alike. I have not derived the final draw table for this matrix as it will take up a lot of unnecessary page space. You may like to do this exercise for yourself. |
Quick Formula :There is a formula which I use to quickly calculate the number of draws/games required to complete one round of any competition. Matrix/Square Root = Msqrt Games = ( Msqrt2 - Msqrt ) / 2 You may like to use the above formula in your own quick calculations to determine the number of games to be played.... The following key points are a summary of the main points in completing a draw matrix. It highlights the main format you should try to follow when doing any draw matrix. If people are interested I have written a computer program to calculate team draw matrices quickly and easily. You may contact me on the email address shown. |
Key Points :
I hope I have made sorting out your team draw format a little more easier... Enjoy ..... G.D.Mutch [B.I.T] |
[Alternative
Technology Library Home Page. ] |
Alternative
Calculating
a team draw using a matrix. The term team draw is defined from
the days when teams or individual names where written down on a piece of
paper and placed into a hat. The list of names were first divide in half
and placed into two separate hats. One team or individual name was taken
or drawn from hat 'A' and the other team name drawn from hat
'B'. These two teams where then written down on another piece of paper
or chalk board as the first draw; the drawn teams were then required to
play off against each other. All corresponding team names where likewise
drawn from hat 'A' and 'B' until no team names where left. In the case
of an odd number of teams in the competition an imaginary team or
individual name called 'Bye' was placed into the hats to even out the
names drawn. If any team or individual drew the name 'Bye' then that
team automatically 'bye-passed' their current game an where
automatically placed into the next slot of the draw on the competition
table. In a true an accurately balanced odd team draw every team will be
required to have one 'bye', so all rounds and competition points should
always remain normalised.