Individual-pairs tournament design for 16 players

Julian D. A. Wiseman, June 2003

This 16-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).

Available formats:
PDF (A4)	Schedule, in score-sheet, with running totals; Schedule by player; Blank score sheet, with running totals
PDF (A3)	Schedule, in score-sheet, with running totals; Schedule by player; Blank score sheet, with running totals
PDF (USL)	Schedule, in score-sheet, with running totals; Schedule by player; Blank score sheet, with running totals
Text	Human-readable schedule, machine-readable schedule
Also see the individual-pairs explanation and the links to designs for other numbers of players.

There is another design of a 16-player individual pairs tournament, in which groups of four players play a mini-tournament amongst themselves, before changing venues and groups after three games. In contrast, this design, in which no set of three players meet together more than once, is slower but more sociable.

Properties of this tournament design:


       i       ii       iii      iv    
 1  D+I:F+B  L+A:C+G  K+O:M+H  N+J:E+P 
 2  A+J:N+L  D+K:O+I  P+G:C+E  M+F:B+H 
 3  E+H:B+C  F+G:P+M  N+O:I+L  D+A:J+K 
 4  P+J:K+M  C+L:I+B  D+F:G+A  O+H:E+N 
 5  D+O:H+F  E+G:A+N  J+L:C+P  B+M:K+I 
 6  G+M:B+E  D+J:L+O  I+N:A+K  C+H:F+P 
 7  K+P:F+A  H+N:I+C  B+L:O+E  D+G:M+J 
 8  I+M:J+C  A+E:O+F  D+H:N+G  L+P:K+B 
 9  D+L:P+H  K+N:G+B  M+E:A+I  F+C:J+O 
10  N+C:F+K  D+M:E+L  O+B:G+J  A+P:H+I 
11  J+I:H+G  P+B:O+A  F+E:L+K  D+N:C+M 
12  O+C:M+A  G+K:L+H  D+P:B+N  E+I:J+F 
13  D+E:I+P  J+B:N+F  C+K:G+O  H+A:M+L 
14  B+A:H+J  D+C:K+E  L+F:N+M  G+I:P+O 
15  M+O:P+N  I+F:L+G  H+K:E+J  D+B:A+C

This is an individual pairs for 16 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.
If games are played on a rectangular table with play clockwise and left players on the leading diagonal (as in tiddlywinks), each player opposes each other once from the same end of the table and once from the opposite end.
Players play on the venues with distributions as follows: 6 players 4:4:4:3; 4 players 5:4:3:3; 4 players 5:5:3:2; 2 players 6:5:2:2.
No player plays at the same venue in two consecutive rounds.
If players are ranked, from A the best to P the worst, this tournament has an unfairness measure of 7705190.5596.
This design is constructed in an unusual way, specific to powers of 2: index the players by the elements of a four-dimensional vector space V over the field of two elements, and the rounds by the non-zero elements of that vector space. Find a bijection f:V→V such that the function x→x+f(x) is also a bijection; then, in round x, player y partners y+x against f(y) and f(y)+x.