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Julian D. A. Wiseman, December 2004
This is a 21-player all-play-all tournament design, and 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: | |
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PDF (A4) | Schedule, in score-sheet, with running totals; Blank score sheet, with running totals |
PDF (A3) | Schedule, in score-sheet, with running totals; Blank score sheet, with running totals |
PDF (USL) | Schedule, in score-sheet, with running totals; Blank score sheet, with running totals |
Text | Human-readable schedule, machine-readable schedule |
Also see the all-play-all explanation and the links to designs for other numbers of players. |
Properties of this tournament design:
Bye i ii iii iv v vi vii viii ix x 1 M O:P A:U Q:B K:R F:T E:S I:G L:D H:J N:C 2 Q D:T P:C U:M N:B A:R K:F E:O J:G S:I H:L 3 P F:H I:L G:O A:T J:B C:Q K:S N:E M:R D:U 4 E K:N S:F J:I G:D R:H P:T L:Q B:M U:C O:A 5 F L:J O:D A:P U:Q G:M R:B T:C H:S E:K I:N 6 J G:E T:H C:L O:M K:P N:A U:B I:R F:Q S:D 7 N R:D E:P H:U B:L I:C O:K G:T Q:J A:S F:M 8 U N:Q F:R I:E H:G D:K T:L S:J M:A C:O P:B 9 H J:C D:M P:N F:U E:L B:I O:R S:T K:G A:Q 10 C E:U H:B L:K S:O N:G A:J F:I R:P Q:D M:T 11 A S:M J:N D:F I:P O:Q U:G R:L K:C B:T E:H 12 K C:R M:E N:H L:F U:J I:O D:P T:Q G:A B:S 13 R U:I B:G K:T C:S Q:E J:M H:A P:F D:N L:O 14 B M:L N:O F:A Q:I P:S G:R J:D C:H T:E U:K 15 I Q:S R:J E:D P:H T:N L:U C:M A:K O:B G:F 16 D I:A G:Q T:R E:C S:U M:H B:K F:O N:L J:P 17 O B:F K:I S:G T:J L:A D:C Q:H U:N P:M R:E 18 G H:K L:S O:J D:A C:F Q:P M:N E:B R:U T:I 19 L P:G U:T B:C M:K H:O S:N A:E D:I J:F Q:R 20 S T:O C:A M:Q R:N B:D F:E P:U G:L I:H K:J 21 T A:B Q:K R:S J:E M:I H:D N:F O:U L:P C:G |
Each player plays each of the others exactly once, and each player has one bye.
No player plays two consecutive games at the same venue.
Each player plays exactly once on each side of each venue.
Hence each player plays twice at each venue, and each player plays ten times on each side (left and right).
It is seeded, so that important games come late in the tournament if player A is the best, B the second best, etc.
It has a permutation score of 160708.6114834, which is hopefully not too far from the maximum.
It has a left-right asymmetry measure of 21.572786, which is minimal given the assignment of games to rounds and venues.
It is constructed as follows. Players are split into three groups of seven. In rounds 1, 4, 7, …, 19, (later to be permuted) all the players in the first group play against all those in the second at venues 4 to 10, while the players in the third play among themselves at venues 1 to 3. The third-group all-play-all is done so that each plays twice at each venue (with consecutivity not being an issue), and the bipartite tournament between the first two groups is done so that each player plays once at each venue. Both of these are achieved using the cyclic group of order 7. The groups are cycled to do likewise in rounds 2, 5,…, 20, and in 3,6, …, 21. In the bipartite tournaments no player should play consecutively at any of venues 4 to 10, which is easy because there are lots of small integers coprime with 7. This structure is then hidden by the permuting of rounds and players to maximise the permutation score, and by the rearrangement of venues so that the more important games tend to happen at lower-number venues.
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