Have a question about participating in 2020 One-Day Individual Tournament of Champions (Sr. If so, please email event manager: Kevin Haracz. State Individual Wrestling Tournament Preview. THE TOURNAMENT: The 77th Annual Wisconsin Interscholastic Athletic Association Individual Wrestling Tournament will be held Thursday-Saturday, Feb. 27-29, at the Kohl Center on the campus of the University of Wisconsin in Madison, Wis. TICKET INFORMATION: Ticket prices per tournament session are $10. Tickets are available on-line on the WIAA website with a $2 service fee until two hours prior to any respective session with the printing of the. In addition, the best player in each tournament from teams not qualified also advance to the next round as individual competitors. It is a stroke play team competition; starting in 2009, the competition changed to a stroke play/ match play competition with the top eight teams after 54 holes of stroke play being seeded and concluding with an.

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A tournament is a competition involving 4 or more teams, or a large number of competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses:

1. One or more competitions held at a single venue and concentrated into a relatively short time interval.
2. A competition involving a number of matches, each involving a subset of the competitors, with the overall tournament winner determined based on the combined results of these individual matches. These are common in those sports and games where each match must involve a small number of competitors: often precisely two, as in most team sports, racket sports and combat sports, many card games and board games, and many forms of competitive debating. Such tournaments allow large numbers to compete against each other in spite of the restriction on numbers in a single match.

These two senses are distinct. All golf tournaments meet the first definition, but while match play tournaments meet the second, stroke play tournaments do not, since there are no distinct matches within the tournament. In contrast, association football leagues like the Premier League are tournaments in the second sense, but not the first, having matches spread across many states in their past over a period of up to a season. Many tournaments meet both definitions; for example, the Wimbledon tennis championship. Tournaments 'are temporally demarcated events, participation in which confers levels of status and prestige amongst all participating members'.^[1]

A tournament-match (or tie or fixture or heat) may involve multiple game-matches (or rubbers or legs) between the competitors. For example, in the Davis Cuptennis tournament, a tie between two nations involves five rubbers between the nations' players. The team that wins the most rubbers wins the tie. In the later rounds of UEFA Champions League , each fixture is played over two legs. The scores of each leg are added, and the team with the higher aggregate score wins the fixture, with away goals used as a tiebreaker and a penalty shoot out if away goals cannot determine a winner of the game.

Knockout tournaments[edit]

A knockout tournament or elimination tournament is divided into successive rounds; each competitor plays in at least one fixture per round. The top-ranked competitors in each fixture progress to the next round. As rounds progress, the number of competitors and fixtures decreases. The final round, usually known as the final or cup final, consists of just one fixture; the winner of which is the overall champion.

In a single-elimination tournament, only the top-ranked competitors in a fixture progress; in 2-competitor games, only the winner progresses. All other competitors are eliminated. This ensures a winner is decided with the minimum number of fixtures. However, most competitors will be eliminated after relatively few matches; a single bad or unlucky performance can nullify many preceding excellent ones.

A double-elimination tournament may be used in 2-competitor games to allow each competitor a single loss without being eliminated from the tournament. All losers from the main bracket enter a losers' bracket, the winner of which plays off against the main bracket's winner.

A triple elimination tournament allows a competitor to lose two games and creates a third bracket or fourth bracket which are usually followed by a playoff. It is usually used in curling tournaments.

Some elimination tournaments are in a best-of-n series, requiring a competitor to lose a majority of n games (in a series against the same opponent) before being eliminated (e.g. in a best-of-7 games series, the winner must win 4 games).

Some formats use a repechage, allowing losers to play extra rounds before re-entering the main competition in a later round. Rowingregattas often have repechage rounds for the 'fastest loser' from the heats. The winners of these progress, but are at a disadvantage in later rounds owing to the extra effort expended during the repechage.

A family of tournament systems that grew from a system devised for the Victorian Football League, the historic predecessor to the Australian Football League (AFL), allow the teams with the best record before the playoffs to lose a game without being eliminated, whereas lesser qualifiers are not. Several of the most prominent leagues in Australia use such a system, such as the AFL and the National Rugby League in rugby league. The A-League of association football also used such a system through its 2011–12 season, but now uses a pure knockout playoff. Similar systems are used in cricket's Indian Premier League and most curling tournaments, and were also used by the Super League of European rugby league before being scrapped after the 2014 season.

In athletics meetings, fastest losers may progress in a running event held over several rounds; e.g. the qualifiers for a later round might be the first 4 from each of 6 heats, plus the 8 fastest losers from among the remaining runners.

An extreme form of the knockout tournament is the stepladder format where the strongest team (or individual, depending on the sport) is assured of a berth at the final round while the next strongest teams are given byes according to their strength/seeds; for example, in a four team tournament, the fourth and third seed figure in the first round, then the winner goes to the semifinals against the second seed, while the survivor faces the first seed at the final. Four American sports organizations either currently use this format, or have in the past:

• Since the mid-1960s, most ten-pin bowling events use a stepladder final, usually involving five bowlers.
• Two U.S. college conferences operate a tournament format in basketball that combines two stepladder tournaments into one—that is, both halves of the bracket are organized as stepladder tournaments. When eight teams are involved in the tournament, the bottom four teams play in the first round; the survivors will face the #3 and #4 seeds, and the winners of those matches take on the top two seeds in the semifinals. This format was used by the West Coast Conference (WCC) for its men's and women's tournaments from 2003 through 2013, and has been used by the Ohio Valley Conference (OVC) for men since 2011 and women from 2011 through 2014. From 2019 forward, the WCC tournaments will return to the aforementioned format, but add an extra round so that all 10 current conference members will participate (note that the OVC tournament does not involve all of the league's members, currently 12).
• In the Philippines, the UAAP Basketball Championship and the NCAA Basketball Championship both use the stepladder format if a team wins all elimination round (group stage) games. This format has been adopted to other sports in both leagues, and to other leagues. If no team wins all elimination round games, the playoffs remain in the usual two-round playoff format.
• The now-defunct Women's Professional Soccer used this format in all of its three seasons of existence. For an example of its playoff system, see 2009 Women's Professional Soccer Playoffs.

Group tournaments[edit]

A group tournament, league, division or conference involves all competitors playing a number of fixtures (again, a fixture is one name for a tournament-match that determines who, out of two or three or more, will advance; a fixture may consist of one or more game-matches between competitors). Points are awarded for each fixture, with competitors ranked based either on total number of points or average points per fixture. Usually each competitor plays an equal number of fixtures, in which case rankings by total points and by average points are equivalent. The English County Championship in cricket did not require an equal number of matches prior to 1963.^[2]

In a round-robin tournament, each competitor plays all the others an equal number of times, once in a single round-robin tournament and twice in a double round-robin tournament. This is often seen as producing the most reliable rankings. However, for large numbers of competitors it may require an unfeasibly large number of rounds. A Swiss system tournament attempts to determine a winner reliably, based on a smaller number of fixtures. Fixtures are scheduled one round at a time; a competitor will play another who has a similar record in previous rounds of the tournament. This allows the top (and bottom) competitors to be determined with fewer rounds than a round-robin, though the middle rankings are unreliable.

There may be other considerations besides reliability of rankings. In some professional team sports, weaker teams are given an easier slate of fixtures as a form of handicapping. Sometimes schedules are weighted in favour of local derbies or other traditional rivalries. For example, NFL teams play two games against each of the other three teams in their division, one game against half of the other twelve teams in their conference, and one game against a quarter of the sixteen teams in the other conference.

American sports are also unusual in providing fixtures between competitors who are, for ranking purposes, in different groups. Another, systematic, example of this was the 2006 Women's Rugby World Cup: each of the teams in Group A played each of the teams in Group B, with the groups ranked separately based on the results. (Groups C and D intertwined similarly.) An elaboration of this system is the Mitchell movement in duplicate bridge, discussed below, where North-South pairs play East-West pairs.

In 2-competitor games where ties are rare or impossible, competitors are typically ranked by number of wins, with ties counting half; each competitors' listings are usually ordered Wins–Losses(–Ties). Where ties are more common, this may be 2 points for a win and 1 for a tie, which is mathematically equivalent but avoids having too many half-points in the listings. These are usually ordered Wins–Ties–Losses. If there are more than two competitors per fixture, points may be ordinal (for example, 3 for first, 2 for second, 1 for third).

Multi-stage tournaments[edit]

Many tournaments are held in multiple stages, with the top teams in one stage progressing to the next. American professional team sports have a 'regular season' (group tournament) acting as qualification for the 'post season' or 'playoffs' (single-elimination tournament). A group stage (also known as pool play or the pool stage) is a round-robin stage in a multi-stage tournament. The competitors are divided into multiple groups, which play separate round-robins in parallel. Measured by a points-based ranking system, the top competitors in each group qualify for the next stage. In most editions of the FIFA World Cup finals tournament, the first round has been a group stage with groups of four teams, the top two qualifying for the 'knockout stage' played as a single-elimination tournament. This format is common in many international team events, such as World Cups or Olympic tournaments. Some tournaments have two group stages, for example the 1982 FIFA World Cup or the 1999–2000 UEFA Champions League. As well as a fixed number of qualifiers from each group, some may be determined by comparing between different groups: at the 1986 FIFA World Cup and UEFA Euro 2016, the best four of six third-place sides qualified; at the 1999 Rugby World Cup the best one of five third-place sides did so.

Sometimes, results from an earlier phase are carried over into a later phase. In the Cricket World Cup, the second stage, known as the Super Eight since 2007 and before that the Super Six, features two teams from each of four preliminary groups (previously three teams from two preliminary groups), who do not replay the teams they have already played, but instead reuse the original results in the new league table. Formerly in the Swiss Football League, teams played a double round-robin, at which point they were split into a top 'championship' group and a bottom 'relegation' group; each played a separate double round-robin, with results of all 32 matches counting for ranking each group. A similar system is also used by the Scottish Premiership and its historic predecessor, the Scottish Premier League, since 2000. After 33 games, when every club has played every other club three times, the division is split into two halves. Clubs play a further 5 matches, against the teams in their half of the division. This can (and often does) result in the team placed 7th having a higher points total than the team placed 6th (because their final 5 games are considerably easier), nevertheless, a team in the bottom half never receives a higher final ranking than a team which qualified for the top half.

A multi-stage pool system was implemented by Curling Canada for the Canadian championship curling tournaments (the Scotties Tournament of Hearts for women and the Tim Hortons Brier for men) starting in 2018. The change was intended to allow the expansion of the main stage of the tournament from twelve to sixteen teams while keeping the round robin at eleven games. The teams are seeded using a ranking system in which points are calculated based on the teams' results in all competitive bonspiels using a complicated formula. Seeds 1, 4, 5, 8, 9, 12, 13 and 16 and placed in Pool A while seeds 2, 3, 6, 7, 10, 11, 14 and 15 are placed in Pool B. After each team has played seven games, the top four teams from each pool advance to the 'Championship Pool.' Carrying over their entire round robin records with them, Championship Pool teams play one game against each of the four teams in the opposite pool, with the top four teams qualifying for the page playoffs. In contrast, teams that fail to qualify for the Championship Pool play only one additional 'Placement Round' game against the team that finished in the same position in the opposite pool for the purposes of determining final tournament ranking. For these teams, there is little else to play for since there is no form of relegation (and, with the expansion of the field to sixteen teams, no 'Pre-Qualifying Tournament') and seeding is based solely on the performances of the participating teams and not the past results of the provinces and territories they represent.

The top Slovenian basketball league has a unique system. In its first phase, 12 of the league's 13 clubs compete in a full home-and-away season, with the country's representative in the Euroleague (an elite pan-European club competition) exempt. The league then splits. The top seven teams are joined by the Euroleague representative for a second home-and-away season, with no results carrying over from the first phase. These eight teams compete for four spots in a final playoff. The bottom five teams play their own home-and-away league, but their previous results do carry over. These teams are competing to avoid relegation, with the bottom team automatically relegated and the second-from-bottom team forced to play a mini-league with the second- and third-place teams from the second level for a place in the top league.

Promotion and relegation[edit]

Where the number of competitors is larger than a tournament format permits, there may be multiple tournaments held in parallel, with competitors assigned to a particular tournament based on their ranking. In Chess, Scrabble, and many other individual games, many tournaments over one or more years contribute to a player's ranking. However, many team sports involve teams in only one major tournament per year. In European sport, including football, this constitutes the sole ranking for the following season; the top teams from each division of the league are promoted to a higher division, while the bottom teams from a higher division are relegated to a lower one.

This promotion and relegation occurs mainly in league tournaments, but also features in Davis Cup and Fed Cup tennis:

• In the Davis Cup:
  • The first-round losers in the top-level World Group compete in playoff ties against the winners of the second-round ties in Group I of the competition's three regional zones, with the winners of each playoff tie remaining in or promoted to the World Group.
  • In the three regional zones, Group II is conducted in a knockout format. The winner of the knockout tournament is promoted to Group I of its zone. The first-round losers then play relegation ties, with the losers relegated to Group III.
  • Groups III and IV in each zone are contested in a round-robin format. The top two teams in each group are promoted, while the bottom two teams are relegated (assuming there is a lower group in their zone).
• In the Fed Cup:
  • The four first-round losers in World Group I compete in playoff ties against the four winners in World Group II, with the winners remaining in or promoted to World Group I.
  • The losers in World Group II play ties against the four zonal Group I winners (two from Europe/Africa and one each from Asia/Oceania and Americas), with the winners playing in World Group II the following season.
  • Groups I and II in all zones, plus Group III in the Europe/Africa Zone only, are conducted in a round-robin format. The bottom two teams in each group are relegated to the next group down, assuming one exists, while the top two teams in Groups II and III are promoted to the next-higher group.

The hierarchy of divisions may be linear, or tree-like, as with the English football league pyramid.

Bridge tournaments[edit]

In contract bridge a 'tournament' is a tournament in the first sense above, composed of multiple 'events', which are tournaments in the second sense. Some events may be single-elimination, double-elimination, or Swiss style. However, 'Pair events' are the most widespread. In these events, a number of deals (or boards) are each played several times by different players. For each such board the score achieved by each North-South (NS) pair is then measured against all the other NS pairs playing the same board. Thus pairs are rewarded for playing the same cards better than others have played them. There is a predetermined schedule of fixtures depending on the number of pairs and boards to be played, to ensure a good mix of opponents, and that no pair plays the same board or the same opponents twice (see duplicate bridge movements).

Poker tournaments[edit]

In poker tournaments, as players are eliminated, the number of tables is gradually reduced, with the remaining players redistributed among the remaining tables. Play continues until one player has won all of the chips in play. Finishing order is determined by the order in which players are eliminated: last player remaining gets 1st place, last player eliminated gets 2nd, previous player eliminated gets 3rd, etc.

In a 'shootout' tournament, players do not change tables until every table has been reduced to one player.

Alternatives to tournament systems[edit]

While tournament structures attempt to provide an objective format for determining the best competitor in a game or sport, other methods exist.

Challenge

In this format, champions retain their title until they are defeated by an opponent, known as the challenger. This system is used in professional boxing (see lineal championship), and the World Chess Championship. The right to become a contender may be awarded through a tournament, as in chess, or through a ranking system: the ranking systems used by boxing's governing bodies are controversial and opaque. If the champion retires or dies, then the current top challenger may be declared champion or the title may be vacant until a match between two challengers is held. Prior to 1920, the reigning Wimbledon champion received a bye to the final; the official name of the FA Challenge Cup reflects a similar arrangement which applied only in that tournament's very early years. The America's Cup is decided between the winners of separate champion and challenger tournaments, respectively for yachts from the country of the reigning champion, and of all other countries. The Ranfurly Shield in New Zealand rugby union is a challenge trophy between provincial teams, in which the holders of the Shield retain it until they are beaten by a challenging province.

Ladder tournament

The ladder is an extension of the challenge system. All competitors are ranked on a 'ladder'. New contestants join the bottom of the ladder. Any contestant can challenge a competitor ranked slightly higher; if the challenger wins the match (or the challenge is refused) they swap places on the ladder. Ladders are common in internal club competitions in individual sports, like squash and pool. Another ladder system is to give competitors a certain number of ranking points at the start. If two competitors play each other, then the winner will gain a percentage of the loser's ranking points. In this way competitors that join later will generally start in the middle, since top competitors already have won ranking points and bottom competitors have lost them.

Selection

A champion may be selected by an authorised or self-appointed group, often after a vote. While common in non-competitive activities, ranging from science fairs to cinema's Oscars, this is rarely significant in sports and games. Though unofficial, the polls run by the Associated Press and others were prestigious titles in American college football prior to the creation in 1998 of the Bowl Championship Series, a quasi-official national championship (to this day, the NCAA does not officially award a championship in the top division of college football). From 2005 until the final season of the BCS in 2013, the AP Poll operated independently from the BCS, and two other polls were part of the BCS formula. The BCS was replaced by the College Football Playoff, a four-team tournament whose participants are chosen by a selection committee, in 2014; since then, all polls have operated independently from the CFP.

Tournaments of value[edit]

Tournaments of value have come to legitimise what are often seen as marginalised practices that sit outside of popular culture.^[3] For example, the Grammy Award ceremony helped to shape country music as a viable commercial field,^[4] and Booker Prize ceremony helped to create new fields of literary fiction.^[5] Tournaments of value go beyond game show and simple contests as the journey itself emerges as being more significant,^[6] bestowing status and prestige on the winner and, in the process, shaping industry practices and acting as institutional mechanisms for shaping social fields.^[7]

See also[edit]

References[edit]

1. ^Thompson, Alex; Stringfellow, Lindsay; Maclean, Mairi; MacLaren, Andrew; O’Gorman, Kevin (2015-03-24). 'Puppets of necessity? Celebritisation in structured reality television'(PDF). Journal of Marketing Management. 31 (5–6): 478–501. doi:10.1080/0267257X.2014.988282. ISSN0267-257X.
2. ^
3. ^Garud, Raghu (2008-09-01). 'Conferences as Venues for the Configuration of Emerging Organizational Fields: The Case of Cochlear Implants'. Journal of Management Studies. 45 (6): 1061–1088. doi:10.1111/j.1467-6486.2008.00783.x. ISSN1467-6486.
4. ^Anand, N.; Watson, Mary R. (2004-02-01). 'Tournament Rituals in the Evolution of Fields: the Case of the Grammy Awards'. Academy of Management Journal. 47 (1): 59–80. doi:10.2307/20159560. ISSN0001-4273. JSTOR20159560.
5. ^Anand, N.; Jones, Brittany C. (2008-09-01). 'Tournament Rituals, Category Dynamics, and Field Configuration: The Case of the Booker Prize'. Journal of Management Studies. 45 (6): 1036–1060. doi:10.1111/j.1467-6486.2008.00782.x. ISSN1467-6486.
6. ^Taheri, Babak; Gori, Keith; O’Gorman, Kevin; Hogg, Gillian; Farrington, Thomas (2016-01-02). 'Experiential liminoid consumption: the case of nightclubbing'. Journal of Marketing Management. 32 (1–2): 19–43. doi:10.1080/0267257X.2015.1089309. ISSN0267-257X.
7. ^Thompson, Alex; Stringfellow, Lindsay; Maclean, Mairi; MacLaren, Andrew; O’Gorman, Kevin (2015-03-24). 'Puppets of necessity? Celebritisation in structured reality television'(PDF). Journal of Marketing Management. 31 (5–6): 478–501. doi:10.1080/0267257X.2014.988282. ISSN0267-257X.

Julian D. A. Wiseman, June 2003 and April 2004

Abstract: Some number of players wish to play an individual-pairs tournament, in which each partners each other once and hence opposes each other twice. How can this be arranged?

Contents: Introduction;Presentations and file formats;Technical notes (unfairness measure, permuting players);Also see.

Publication history: Earlier versions of some 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.

Quick individual-pairs links:complete list; pure 458912131616*1720212425;mixed 37911151923;asymmetric 61014182226.

Introduction

Twelve players (or teams) are all to play a pairs-style tournament, in which each game consists of one partnership against another. This can be done by having each player partner each of the others exactly once, and hence oppose each of the others exactly twice. Each player has to play eleven games, so this tournament requires at least eleven rounds. With three games happening simultaneously, there must be at least three venues. On the right can be found an example 12-player individual-pairs at three venues venues over eleven rounds.

Individual pairs are not possible 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 makes sense only in non-aerobic games in which turns are taken by players who may confer. So it works in the likes of tiddlywinks, croquet, snooker, and billiards. But self-partnering doesn’t work in games such as bridge and doubles tennis. 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’.

Individual Tournament Tennis

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 neither type is possible.

Pure individual-pairs tournament designs are published for various numbers of players from four to twenty-five: 4, 5, 8, 9, 12, 13, 16, (and in another version, 16*), 17, 20, 21, 24, and25. Mixed individual-pairs tournament designs are published for various numbers of players from three to twenty-three: 3, 7, 9, 11, 15, 19, and23. Asymmetric individual-pairs tournament designs are published for various numbers of players from six to twenty-six:6, 10, 14, 18, 22 and 26.Also see the complete list of individual-pairs links.

Meaning of terms

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

• Players are labelled with upper-case roman letters: A, B, C, D, etc.

• The specific meaning of the word ‘venue’ will depend on context. In snooker or billiards, a venue refers to a particular table; in tiddlywinks (for which these tournament designs were originally devised), to a particular mat; in tennis to a court; in croquet to a lawn or to a set of balls; and in other games to a location or set of equipment as appropriate. Venues are labelled with lower-case Roman numbers: i, ii, iii, iv, etc.

• Several games happen simultaneously during a round, each of which is labelled with an Arabic numeral: 1, 2, 3, 4, etc.

• A particular game might be described as ‘A+B:C+D’, meaning that player A is partnering B against C and D (in a particular round and at a particular venue). This is sometimes abbreviated to ‘AB:CD’. However, ‘A+B:C+D’ is not necessarily the same game as ‘C+D:A+B’. In tiddlywinks, the left team should take the dominant corners.In games in which one team starts, left starts or deals first. In general, the left team should have the advantage.

• It is intended that other symmetries should also be meaningful. So, in games in which this is meaningful, ‘A+B:C+D’ could have a different order of play to ‘A+B:D+C’, the order being first+third:second+fourth. In games such as doubles tennis this might apply to the serve order. In tiddlywinks, it is suggested that the tournament organiser chooses which of the two dominant corners is to be taken by the first player, so differentiating ‘A+B:C+D’ from ‘B+A:D+C’. In other games it is for the tournament organiser to determine the meaning of left-versus-right, of the play order, and of the residual symmetry. To encourage consistency, tournament organisers are invited to inform the author of any conventions which may be agreed.

Presentations and file formats

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

• Embedded in the master HTML file for each size of individual-pairs is a PRE-formatted text grid, giving the description of which game takes place in each round at each venue. This is also available in a separate text file.

• A similar description of the game schedule is available in PDF, with versions optimised for A4, A3 and US Letter (USL).

• A game schedule is also available as a table, with one player per column and one round per row, showing the venue to which that player should be going. The players for that game are shown in small characters in the corners of the cell, and hence this format is particularly appropriate for games played on a rectangular surface.

• A score-sheet is available in PDF, again in A4, A3 and USL. Players go both horizontally and vertically; enter into each cell the score achieved by the player of that row when partnering the player of that column.Each cell shows the game description, as well as the round and venue.

• A more detailed score-sheet is available in PDF (in A4, A3 and USL), with extra columns into which a player’s running total of scores may be entered. With a large number of players the extra columns can crowd the page. The running-total cells show the description of the game just played.

• Both the less- and more-detailed score sheets are also available in ‘blank variants’, which don’t have the game descriptions in the cells, and so can be used with other tournament cells.

• Finally, each tournament is available in a machine-readable text format, with one row per game. Each row is in the form RR V A+B:C+D, where RR is the round, V is the venue (for ease of machine-readability exceptionally shown in ‘Arabic’ numerals rather than lower-case Roman numbers), and A+B:C+D is the game description. It is from this file that the others are generated: readers sending improved tournament designs are asked to do so in this format.

Individual Golf Tournaments Near Me

Technical notes: symmetric tournaments

The construction of the symmetric individual pairs relies on a branch of mathematics called design theory. The following paragraph, by Matt Fayers of The Department of Pure Mathematics and Mathematical Statistics at The University of Cambridge, assumes some knowledge of design theory.

An individual pairs tournament is referred to in the mathematical literature as a whist tournament, and the existence of a whist tournament for any number of player congruent to 0 or 1 mod 4 was proved by Baker in 1975[*1] using a variety of construction techniques. For practical purposes, it is often easiest to find lots of possible whist tournaments by seeking cyclic formats. For 4n+1 players, this means indexing the players and the rounds by the integers modulo 4n+1 (so they start at zero), and choosing the games in round 0 and then decreeing that if A+B:C+D is a game in round 0, then (A+i)+(B+i):(C+i)+(D+i) is a game in round i. Of course, the first round must be chosen in such a way that this produces a whist tournament, but the necessary and sufficient condition on the first round is a simple one based on the differences (mod 4n+1) between the partners and opponents in the various games. For 4n players, a similar procedure is adopted, but here we cycle mod 4n–1 and label one of the players ∞ (with the usual convention that ∞+i=∞). In fact, an analogous procedure can be used for any group of order 4n+1 or 4n–1; in the case of 9 players, this is necessary, since there is no cyclic whist tournament for 9 players, although there is one based on the elementary abelian group of order 9.To assign games to venues is a rather different problem. In the case of 4n+1 players, we might dictate that the game (A+i)+(B+i):(C+i)+(D+i) happens at the same venue as A+B:C+D; this guarantees that each player plays exactly four times at each venue, but also guarantees that three players stay at the same venue between consecutive rounds, even if the rounds are re-ordered. For 4n players, adopting this procedure would keep player ∞ at the same venue throughout, and so is only useful when ∞ is a ‘dummy’ player used in a tournament for 4n–1 players (whoever partners the dummy actually self-partners). When there really are 4n players, some shuffling of games between venues is necessary. The number of possibilities is unwieldy, so the approach used here is to label the venues 0,…,n–1, and then assign games to venues for round 0 and dictate that if A+B:C+D happens in round 0 at venue j, then (A+i)+(B+i):(C+i)+(D+i) happens in round j at venue j+i (reduced mod n). In short, games were shifted one place to the right each round. This guarantees that player ∞ plays three times at one venue and four times at each of the others, and it is then a matter of trying various arrangements of the games in round 1 to try to prevent players staying at the same venue between games and to try to ensure that each of the other players has a reasonably even distribution between venues.

This done, it is still possible to permute the players, with the objective of minimising that which is here called ‘unfairness’. In the symmetric tournaments each player partners everybody once, and opposes everybody twice, so a player would consider it a terrible waste to partner the best-ranked player, A, who is the current world champion, against the two worst ranked players. The partnering of A would be, in some sense, wasted on easy opponents. And likewise, it would be highly desirable to partner the worst ranked player against A+B: it gets out of the way a terrible partner and one of the two oppositions against the two top players, for the loss of one game rather than three. So we would like the combinations of opponents and partners to be chosen fairly, and we start by assigning a numerical score to each player: the worst is 1, the second-worst 2, and so on up to A who scores n. For each player, the average of the values of the other players is calculated, and for player Z this is written (Z). The relative advantage in game W+X:Y+Z for player Z is therefore (Z)+Y-X-W, and for each player the total of these is zero. So the total of the cubes of the relative advantages is calculated, knowing that if a player’s games are fairly distributed, then its sum of cubes will be near zero. The variable that is minimised is then the weighted sum of the squares of these sums of cubes; giving a greater weight to better players by weighting the outer summation by the players’ numeric values. For large numbers of players (above 16), it is possible that the absolute minimum has not been found. When there is self-partnering, there is a slight adjustment to the scoring, so that in W+X:Z+Z the relative advantage of Z is 2(Z)-X-W.

Technical notes: asymmetric tournaments

Much of the work of the construction of the symmetric tournaments was done by the mathematics within design theory. The asymmetric tournaments do not have this mathematical base, and for ≥10 players these designs are built by a cascade of optimisations.

• If every pair of players partners, there are an odd number of pairs. So instead one pair of two special players do not partner; all other pairs do. In the 10-player example on the right, C and E never partner. So that there are no byes; there is one venue at which singles games are played; and the special players play singles thrice, the others twice.

• The first optimisation scatters the all-pairs-but-one over the doubles games, and re-arranges to ensure that no player appears more than once in any round. Pairs are then re-arranged within rounds to ensure that most pairs oppose as nearly to twice as possible. For ≥10 players, each pair of players oppose at least once at doubles, and no more than thrice at singles and doubles together. For ≥18 players, further optimisation might result in a slight improvement. (This technique was suggested by Dr Ed Wynn, then of the University of Birmingham.)

• The second optimisation permutes the rounds, and within each round, permutes the games (not the pairs) across venues. The object is to minimise consecutive play at the same venue, and to even the out the numbers of times each player appears at each venue. Ideally, players would never play consecutively at the same venue, but this is achieved only for ≥18 players. Playing consecutive games at the same venue allows a player to become accustomed to that venue’s idiosyncrasies, giving a small advantage. So if there must be consecutive play at the same venue, the optimisation ensures that, in each game, the same number of players in each team were at the same venue in the previous round.Subject to this, it is also desirable that players visit each venue approximately equally, and (again for ≥18 players) each player visits each doubles venue at least three times, and at most five times. For all numbers of players, the sum of the squares of the differences from four has been minimised.

• Within a game, it is still possible to switch the right and left sides. Some pairs of players oppose thrice, and left and right sides have been switched to ensure that for any such pair, each player plays at least once (and hence no more than twice) from each side. For the singles games, each special player plays once from one side and twice from the other, and the non-special players play once from each side. Many pairs of players oppose twice, and ideally these oppositions would be once from each side. This is provably not possible, but the number of exceptions has been minimised.

• Similarly for play order as for left and right: those that oppose thrice at doubles play either twice before and once after, or once before and twice after. For pairs of players who oppose twice at doubles, as many as possible play once before and once after.

• This leaves a remaining symmetry, in that 1^st+3^rd:2^nd+4^th can be exchanged for 3^rd+1^st:4^th+2^nd. In tiddlywinks it is natural to assign a corner to each particular position, so this symmetry has been used to minimise the number of occasions in which a player plays twice consecutively in the same position at the same venue. For 6 and 10 players this happens only once, and for ≥14 not at all (though for ≥18 players it follows from the no-consecutive-play-at-same-venue property). This minimised, it is then natural to ensure that, at each venue considered in isolation, the number of times each player has each position is balanced, in the usual sum-square sense.That done, it is still possible to apply the transformation to all the games at any particular venue, and this is done to balance the total over the venues of the number of times each player has each position.

• It remains to permute the players. Recall that, in the symmetric tournaments each player partners everybody once and opposes everybody twice, so a player’s average opponent and average partner are not changed by permuting the players.But this isn’t true for the asymmetric designs. So the same numerical scores are assigned to each player: the worst is 1, the second-worst 2, and so on up to A who scores n. Again, for each player the average of the values of the other players is calculated, and for player Z this is written (Z), and as above, the relative advantage in game W+X:Y+Z for player Z is therefore (Z)+Y-X-W.These are totalled for each player (uncubed, unlike before), and the weighted sum of the squares of these sums is minimised, where the weights are, again, the players’ numeric values. For large numbers of players (above ≥18), it is possible that the absolute minimum has not been found. It is therefore this minimisation which determines which are the two special players.

Also see

Individual Wrestling Tournaments

Designs for other sizes of tournaments—though requiring much more work from the user—can be found at Durango Bill's Bridge Probabilities and Combinatorics.

Beyblade Burst Individual Tournament

*1 R. Baker, ‘Whist Tournaments’, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Num.14, 89-100.

Mngwl Individual Tournament