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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? Alternatively, perhaps, in a pairs game, sixteen players wish to partner each other once (and therefore oppose each other twice) over fifteen rounds at four venues? Various such tournament designs are published here, for the use of tournament organisers and others.
Contents: All-play-all; Carry-over; and Individual Pairs.
Also see the more detailed explanations: all-play-all, carry-over and individual-pairs.
Publication history: Earlier versions of many 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 first publication on the web. Usual disclaimer and copyright terms apply.
All-play-all is a classic tournament design, in which each of the players plays each of the others exactly once. If there are an odd number of players, each will have one bye; if an even number, then (for most tournament sizes) there will not be any byes.
See the much more detailed all-play-all explanation, and the page of all the all-play-all links. All-play-all tournament designs are published for various numbers of players from twenty-six down to the (utterly 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 also ‘excess-venue’ designs (which use more venues than the minimum necessary) 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).
Sometimes it happens that there are too many participants for an all-play-all to be attempted within the allotted time. An alternative is for the competition to be split into two all-play-all leagues, with the top so-many from each league going through to the second round. The second round is scored as an all-play-all, except that those pairs who have already played each other use the scores from the first round. The only additional games that take place are between those who have not yet played each other.
See the much more detailed carry-over explanation, and the page of all the carry-over links. Carry-over tournament designs are published for various numbers of players from 16+16=32 down to trivial sizes: 16+16=32, 15+15=30, 14+14=28, 13+13=26, 12+12=24, 11+11=22, 10+10=20, 9+9=18, 8+8=16, 7+7=14, 6+6=12, 5+5=10, 4+4=8, 3+3=6, 2+2=4 and 1+1=2.
In an individual pairs tournament, each player partners everybody once and opposes everybody twice. This is a highly sociable format for an informal tournament, though tends not to be used in ‘serious’ competition.
It is not possible to produce an individual pairs tournaments for all numbers of players. To see this start by calculating, for any given number of players, the number of possible partnerships. Each game in a tournament uses two of these partnerships, so if the number of partnerships is odd, an individual pairs cannot be produced for that number of players.
The calculation can be altered by allowing self-partnering. But self-partnering only makes sense in certain games. It works in the likes of croquet, snooker, billiards and tiddlywinks, in which a partnership consists of players taking alternate turns. But in games such as bridge, and aerobic sports such as double tennis, two-versus-one doesn’t work. An individual pairs tournament in which every game is two-versus-two is said to be ‘pure’, and one which includes self-partnering is said to be ‘mixed’.
In a pure individual pairs tournament with n players, there are n(n–1)/2 partnerships. This is even iff n is 0 or 1 modulo 4.
In a mixed individual pairs tournament with n players, in which each player self-partners once, there are n(n–1)/2+n partnerships. This is even iff n is 0 or 3 modulo 4. This result continues to hold if each player self-partners any odd number of times. If players self-partner an even number of times, then n must be either 0 or 1 modulo 4.
So pure individual pairs are possible if n is 0 or 1 modulo 4; mixed individual pairs are possible if n is 0 or 1 or 3 modulo 4; and if n is 2 modulo 4 then the design must be (at last slightly) asymmetric.
See the much more detailed individual-pairs explanation, and the page of all the individual-pairs links. Also see the pages of pure individual-pairs tournament designs: 4 5 8 9 12 13 16 16* 17 20 21 24 25; those of mixed designs: 3 7 9 11 15 19 23; and the asymmetric designs for numbers of players that are 2 modulo 4: 6 10 14 18 22 26.
Questions are occasionally received. From November 2011 answers also appear at www.jdawiseman.com/papers/tournaments/tournaments_questions.html.
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