Method of Determining Fixtures for Teams in A Round-Robin Tournament

Let n be an even number where n is the number of teams.  Include a ‘dummy’ team if there is an odd number of actual teams.

There will be (n-1) rounds of games where each team plays another team during each round. 

1. Label the teams alphabetically.
2. Set out a list of positions numbered 1 to n.  Each team will occupy a position during each round.  For Round 1, Team A occupies Position 1, Team B occupies Position 2, Team C occupies Position 3, etc.
3. Let Team A occupy Position 1 for each round.
4. Fixtures are set up so that the Team in Position 1 plays the team in Position 2 during each round.

The team in Position 3 plays the team in Position 4 during each round.

The team in Position 5 plays the team in Position 6 during each round.

And so on ….

5. Pair off the numbers from 1 to (n-2) so that each pair adds to (n-1).  The order of the numbers within each pair does not matter.
6. Starting with Position 3, place the number pairs formed at (5) above, successively at Positions 3 and 4;  Positions 5 and 6; and so on.   The pairs do not have to be taken in any particular order.
7. Add 1 to each number within the pairs to give the number of a Round from 2 to n-1.  For each Round number there is now a corresponding Position number.  Thus, the pair (a, b) (where a+b = n-1) may be placed at Positions 3 and 4.  Then Round (a+1) corresponds to Position 3 and Round (b+1) corresponds to Position 4.  

Let these correspondences indicate the positions of Team B for each round. 

8. Beginning with ‘2’ (for Position 2 in Round 1), form a sequence of positions for Team B using the correspondences established in (7) above.  Each position will be occupied by Team B during successive rounds.  This sequence will contain (n-1) numbers.
9. Make a similar sequence of positions for each of the other teams, except Team A.  Start the sequence with the Team Position in Round 1 (e.g. Team C is in Position 3 for Round 1), say m, and continue the sequence by continuing the same sequence as for Team B using  ‘m’ within the sequence for Team B as the starting point and beginning the Team B sequence again to complete the list of (n-1) numbers for the team in question.
10. There are now (n-1) sequences – one sequence for each team other than Team A.  The first number of each sequence gives the positions of each team for Round 1.  The second number of each sequence gives the positions of each team for Round 2, and so on.
11. Place the teams in their correct positions for each round and form the fixtures as in (4) above.

This reference may be useful:

J. Dinitz, E. Lamken, W.D. Wallis, Scheduling a Tournament, The CRC Handbook of Combinatorial Designs, eds. C.J. Colbourn and J. Dinitz, CRC Press, 1995, 578-584.

Home  See Example of Fixtures for 6 Teams;  See Example of Fixtures for 20 Teams;  See Alternative Method.

Page created by Neil Hallinan  29 October, 2001