The Problem of finding fixtures for Teams in a Tournament

A Solution

Example for 6 Teams

1. Label the teams alphabetically in positions 1 to 6.

Position      1  2  3  4  5  6

Team           A B C D E  F

2. Keep Team A in position 1 at all times.
3. Teams will play against the team in the next position.

Thus, for Round 1, the games will be:   AvB;  CvD;  EvF.

4. Let Team B vary its position for each new round.  All other teams, except Team A, will follow the pattern established by the movement of Team B through the positions and each team (other than Team A) will have a different position for each round.
5. In a set of 6 teams, each team will have to play 5 games.  That is, there will be 5 rounds to play.  The games for Round 1 are already decided (at stage 3 above) so that there are 4 rounds to be decided.

6. Form pairs of numbers from the list of numbers 1 to (n-2):i.e. 1, 2, 3, 4 such that each pair adds to 5.

In this case form the pairs (1,4) and (2,3).   [The numbers within pairs may be switched to give (4,1) and (2,3) etc.]

7. Match this list of pairs against the list of positions, starting in Position 3.  The pairs may be placed in any order.

   Configuration 1                  or                           Configuration 2                               

Position  3  4       5  6                                                    Position  3  4       5  6

Pairs       (1, 4)    (2, 3)                                                   Pairs      (2, 3)    (1, 4)

Round     2  5      3   4                                                    Round    3  4       2  5

(Notice that the Round number is 1 greater than the corresponding number within the pair.)

(Notice that  8 different configurations are possible from this initial starting position.  That is, by setting up all combinations of the pairs and within the pairs such as (3, 2) (1, 4) etc. although only 2 essentially different fixture lists are produced.  Similar results may be found by letting Team B be the team fixed in Position 1 and allowing Team A to occupy the Positions of Team B while the fixtures are being made.)

8. The configuration decides the possible positions for Team B during each of the rounds.

Using the first possible configuration (Configuration 1):

Team B

Round 1 :  Position 2

Round 2:   Position 3

Round 3:  Position 5

Round 4: Position 6

Round 5: Position 4

9. The position of each team may be identified from the Position rotation list for Team B which (for Configuration 1) is

- 2 – 3 – 5 – 6 – 4 –

Team B starts in Position 2 and uses positions          2,      3,      5,       6,      4

Team C starts in Position 3 and uses positions          3,      5,      6,       4,      2

Team D starts in Position 4 and uses positions         4,      2,       3,       5,      6

Team E starts in Position 5 and uses positions          5,      6,      4,        2,      3

Team F starts in Position 6 and uses positions          6,      4,       2,       3,      5

                                                during Round number     1       2        3        4       5

10. The teams can then be placed in the relevant positions for each round with Team A in Position 1 each time.  The fixtures are then made by having each team play against the team in the adjacent relevant position.

 The fixtures then for each round for 6 teams are as follows

11. If Configuration 2 is used then Team B would occupy positions

-2 – 5 – 3 – 4 – 6

so that the team positions would be as follows

Team B               2, 5, 3, 4, 6

Team C               3, 4, 6, 2, 5

Team D               4, 6, 2, 5, 3

Team E                5, 3, 4, 6, 2

Team F                6, 2, 5, 3, 4

for Rounds         1  2  3  4  5

and the fixture list would be

12. The same process may be used to set up the fixture lists for any number of teams.

13. Note: There may be possible fixture list configurations which this method cannot produce in the cases when the number of teams is greater than 6.

14. In order to deal with an odd number of teams simply allow an extra ‘dummy’ team to play while the fixture list is being constructed.  The team that plays the ‘dummy’ team in each round has a ‘bye’.

15. The method may be extended to any number of teams.  See the Example dealing with 20 teams.

15. There is an alternative method of solving this problem which is given by Math Forum – Ask Dr. Math.  The method involves using points on polygons to represent the teams and lines linking the points to represent match fixtures.  By using an asymmetrical system of lines and rotating the points the complete set of fixtures may be constructed.

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  Page created by Neil Hallinan   29 October, 2001