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Matching Problem: Set out the Fixtures for Teams in a Tournament |
Problem:
Given a set of n teams, where n is an even number, draw up a list of fixtures of matches between the teams.
Each team must play against every other team and each team must play at each round.
Analysis:
There are n teams, n is an even number.
Total number of rounds: n-1. e.g. For 6 teams there will be 5 rounds.
Total number of matches: (n-1)n/2. For 6 teams there will be 15 matches in total.
Number of matches during each round: n/2. For 6 teams there will be 3 matches during each round.
Example:
For 6 teams a possible list of fixtures appears as follows:
|
Teams |
A, B |
C, D |
E, F |
|
Round 1 |
AvB |
CvD |
EvF |
|
Round 2 |
AvC |
FvB |
DvE |
|
Round 3 |
AvF |
EvC |
BvD |
|
Round 4 |
AvE |
DvF |
CvB |
|
Round 5 |
AvD |
BvE |
FvC |
Problem restated: Find a method by which the fixture list may be constructed given any even number of teams.
Further analysis: To highlight the problem, observe what may happen when the fixtures are made without planning ahead.
The fixture list for the 6 teams could begin like this:
|
Teams |
A, B |
C, D |
E, F |
|
Round 1 |
AvB |
CvD |
EvF |
|
Round 2 |
AvC |
DvB |
|
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Round 3 |
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Round 4 |
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Round 5 |
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Teams E and F cannot play in Round 2 since they have already played each other in Round 1. Obviously, DvB was a poor choice for the second match of Round 2.
For this set of 6 teams the correct choices are not too difficult to make. Poor choices are easily identified and better ones made.
Before you look at the solution you may wish to experiment with sets of 8, 10, or 12 teams or even higher numbers of teams. See a Solution for 6 teams. See a Solution for 20 teams.