Individual-pairs tournament design for 9 players

Julian D. A. Wiseman, June 2003

This 9-player individual-pairs tournament design, last updated in June 2003, is based on an original by Matt Fayers of The Department of Mathematics at Queen Mary, University of London (formerly of The Department of Pure Mathematics and Mathematical Statistics at The University of Cambridge).

There is also a mixed 9-player design, consisting of 12 rounds, each of which has three games of two-versus-one. But in this much faster pure design, each of the 9 rounds consists of two games of two-versus-two and one player having a bye.

Properties of this tournament design:

  Bye    i       ii    
 1 A  B+C:D+G  E+I:F+H 
 2 B  C+A:E+H  F+G:D+I 
 3 C  A+B:F+I  D+H:E+G 
 4 D  E+F:G+A  H+C:I+B 
 5 E  F+D:H+B  I+A:G+C 
 6 F  D+E:I+C  G+B:H+A 
 7 G  H+I:A+D  B+F:C+E 
 8 H  I+G:B+E  C+D:A+F 
 9 I  G+H:C+F  A+E:B+D 

• This is an individual pairs for 9 players.

• Each player partners each of the others exactly once.

• Each player opposes each of the others exactly twice.

• No set of three players meet together more than once.

• Each player opposes each of the others exactly once from the left and exactly once from the right.

• Each player plays immediately before each of the others exactly once, and immediately after exactly once.

• Each player plays four times on each venue.

• Staying at the same venue in consecutive rounds is done by 4 players 4 times, 2 players 1 time, 2 players 3 times, and 1 player 0 times.

• If players are ranked, from A the best to I the worst, this tournament has an unfairness measure of 16413392.28515625.

• This is one of two possible 9-player individual pairs tournaments, and is based on the group C₃×C₃. There is no individual pairs based on the cyclic group of order 9, and it is conjectured that this is the only group of odd order with this property.

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