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Tournament Designs: All-Play-All

Julian D. A. Wiseman, January 2007

Abstract: Twelve teams wish to play each other in an all-play-all format at six venues over eleven rounds. How can this be arranged?

Contents: Introduction; Presentations and file formats; Technical notes (permutation score; left-right asymmetry measure).

Publication history: Earlier versions of most of the designs here have previously been made available in paper form by Dr Nicholas F. J. Inglis, for the use of ETwA and CUTwC tournament organisers. This is believed to be the only publication on the web. Usual disclaimer and copyright terms apply.


      i   ii   iii  iv    v   vi  
  1  F:C  L:E  G:I  K:D  H:A  B:J 
  2  D:E  J:H  K:C  B:I  L:F  G:A 
  3  H:I  K:F  D:L  A:J  G:B  E:C 
  4  G:J  I:A  H:B  C:L  E:K  F:D 
  5  B:F  C:J  I:D  H:E  A:L  K:G 
  6  C:G  F:I  A:K  L:B  J:E  D:H 
  7  J:D  B:K  L:G  F:A  C:H  I:E 
  8  K:H  A:D  B:E  I:C  F:G  J:L 
  9  E:A  H:L  F:J  D:G  K:I  C:B 
 10  I:L  E:G  C:A  J:K  B:D  H:F 
 11  A:B  D:C  E:F  G:H  I:J  L:K 

Twelve teams are all to play all the others. Each team has to play eleven games, so this tournament requires at least eleven rounds. With six games happening simultaneously, there must be at least six venues. On the right can be found an example 12-player all-play-all at six venues over eleven rounds. All-play-all tournament designs are published for all numbers of players from twenty-six down to the (trivial) one: 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 and 1.

There are ‘excess-venue’ designs (defined below) for 8 on 8, 8 on 7, 8 on 6, 8 on 5 (as well as the minimal-venue 8 on 4), 7 on 6, 7 on 5, 7 on 4 (and minimal-venue 7 on 3), 6 on 6, 6 on 5, 6 on 4 (6 on 3), and 5 on 3 (5 on 2).

Also see the complete list of all-play-all links.

The precise meaning of some of the terms can vary according to the game being played.

Presentations and file formats

Each of these all-play-all tournaments is published in a number of different presentations and file formats.

Technical notes

There can be many ways to set out an all-play-all tournament of any particular size. A particular one is chosen using some constraints and two optimisations.

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