Tournaments and Competitions

Written and reviewed by Rohit Singh, Computer Science & Commerce faculty at UniDrill (MCA, Osmania University · 6 years teaching). Based on NCERT Health and Physical Education Class XI, Chapter 10.

How does a school organise a cricket cup for 13 teams?

Imagine you are the sports captain and 13 classes have entered an inter-class cricket cup. The principal asks two questions: how many matches will be played, and can it finish before the exams? A different tournament format answers each differently. A knock out needs only 12 matches, a league needs 78, and the choice depends on time, money, grounds and officials. This chapter gives you the formats, the arithmetic and the major games, and most CUET questions here are one line formula problems.

A tournament is a competition among different teams in a particular sport, held to a fixed schedule, where a winner is decided. Before choosing a type, an organiser must consider the season, the time available, the play fields and equipment, the type of activity, the officials and the budget.

The four families in one line: Knock out (lose and leave) → League (everyone plays everyone) → Combination (groups, then a final stage) → Challenge (players climb a ladder)

Single knock out: lose once and you are out

In a single knock out tournament, a defeated team is eliminated and gets no second chance. Every match produces exactly one loser, and every team except the champion loses exactly once. So the number of matches equals the number of eliminations.

$$\text{Matches} = N - 1$$

Here $N$ is the number of teams. For 13 teams the matches are $13 - 1 = 12$.

Byes and the power of two. A fixture works smoothly only when the number of teams is a power of two: 2, 4, 8, 16, 32, 64 and so on. If it is not, some teams get a bye, which means they move to the next round without playing. Byes are given in the first round only.

$$\text{Byes} = (\text{next higher power of 2}) - N$$

The number of rounds depends on the power of two the teams fall under: 5 to 8 teams give 3 rounds, 9 to 16 give 4, 17 to 32 give 5, 33 to 64 give 6 and 65 to 128 give 7.

Worked example: a 13 team fixture.

The wrong step students make is to think byes equal the number of teams that skip the whole tournament. A bye only skips round one. Check: round one has 5 matches, then 8 teams remain (5 winners and 3 bye teams), and $4 + 2 + 1$ more matches follow, giving $5 + 7 = 12$ in total.

Halves and quarters. The fixture is split into an upper half and a lower half. For $N$ teams, each half holds $N/2$ teams, so with 8 teams each half has 4 and the total is $8 - 1 = 7$ matches. When the draw is split into four quarters, the table tells you how many teams go to each quarter. Let $Q$ be the quotient of $N \div 4$.

For 13 teams, $13 \div 4$ gives $Q = 3$ and remainder 1, so the quarters hold 4, 3, 3, 3 teams.

Where the byes go. Byes are placed in a fixed order, and if there are more than four, the same order continues.

Pause and predict: a school enters 25 teams in a single knock out. How many byes, how many matches, and where does the first bye go?

Check your answer: byes are $32 - 25 = 7$, matches are $25 - 1 = 24$, and the first bye goes to the bottom of the lower half.

Seeding. If the draw is made purely by lots, two strong teams may meet in an early round and one is knocked out, while weaker teams reach the semi finals. That makes the competition unfair and dull. Seeding sorts the strongest teams and places them in the fixture so that they do not meet in the early rounds. If the seeded teams are of similar standard, they are distributed in the fixture by lots among them.

Special seeding. When top ranked teams or players enter, they are fixed to join directly at the quarter final or semi final stage. In a 24 team draw with 4 teams specially seeded straight into the quarter finals, the byes are found as follows.

Beyond the textbook. For the exam: seeding stops strong teams meeting early, and the number of byes comes from the next power of two. In reality: big tennis tournaments use the same ideas. A 128 player draw is a power of two ($2^7$), has 127 matches and seeds the top 32 players, so the best players cannot meet until later rounds. The formula $N - 1$ works there too, which is a good way to check your understanding of it.

Consolation tournaments: a second chance

A consolation tournament lets defeated teams play again for subsidiary honours. More matches are played, so it is considered superior to the single knock out.

$$\text{Consolation Type II matches} = 2N - 3$$

For 16 teams: $2 \times 16 - 3 = 29$ matches. Of the two methods of drawing a Type II fixture, the first is recommended.

Double knock out: you must lose twice

A team is eliminated only after two defeats, and the tournament continues until every team but one has been defeated twice. It is superior to the single knock out and the consolation type because it decides the true winner. It extends consolation Type II. The regular round winner meets the consolation round winner to decide the champion.

$$\text{Matches} = 2N - 2 \ \text{ or } \ 2N - 1$$

The two answers arise because the final may or may not need a second match. If the regular winner beats the consolation winner, the champion is clear and the total is $2N - 2$. If the consolation winner wins, both teams have lost once, so they play again, and the total is $2N - 1$. For 10 teams the answer is 18 or 19.

Bagnall Wild: deciding the first three places

This tournament can truly decide the first three places, whereas a plain knock out decides only the champion.

League or round robin: everyone plays everyone

In a single league each team plays every other team once. In a double league each team plays every other team twice.

$$\text{Single league: } \dfrac{N(N-1)}{2} \qquad \text{Double league: } N(N-1)$$

Where the formula comes from. Each team plays $N - 1$ opponents. Counting every team gives $N(N-1)$, but each match has been counted twice (once for each side), so a single league divides by 2. A double league plays every pairing twice, which cancels the division.

Worked example: league matches.

The common wrong step is to use $N - 1$ for a league. That formula counts eliminations in a knock out, not pairings.

Pause and predict: 13 teams in a single knock out need 12 matches. How many would a single league need?

Check your answer: $\dfrac{13 \times 12}{2} = 78$ matches. This is why a league needs far more time and facilities.

Drawing a league fixture. Three methods are given.

For 5 teams in the cyclic method, N is odd, so there are 5 rounds, with one team getting a bye in each round, and $\dfrac{5 \times 4}{2} = 10$ matches. For 6 teams, N is even, so there are 5 rounds and $\dfrac{6 \times 5}{2} = 15$ matches.

Deciding the winner. In league tournaments the winner is decided by points: win 2 points, draw 1 point to each team, loss 0. The team with the most points wins. If teams tie, the rules of the game's federation or the tournament authority, fixed before the tournament starts, break the tie.

Beyond the textbook. For the exam: a win carries 2 points, a draw 1 and a loss 0. In reality: the point system is decided by the sport and the competition, and many football leagues give 3 points for a win. So never carry one system over to another game. The 2, 1, 0 system is the one to use for this chapter.

Combination tournaments: zones first, then a final stage

Combination tournaments are used when matches are played on a group or zonal basis. They are recommended for inter school tournaments at district, state or zone level because they save time and money. A stage played within each zone is followed by an inter zonal stage.

Worked example: four zones of four teams each. Count the matches for each combination.

Challenge tournaments: ladders and pyramids

Challenge tournaments suit games like badminton, table tennis and squash. They run over a specified time without a fixed schedule and help select the best individual or dual players.

Other ladder rules: the period of the tournament is fixed, challenges are played in the order made, the time to play an accepted challenge is specified, a player who has played a challenge cannot challenge again, and no excuse for failure is accepted.

Major competitions

Participation in international competitions such as the Olympic, Asian and Commonwealth Games helps people to know each other and promotes friendship across continents. Sports participation builds unity of mind and body for optimum health, promotes social peace, justice and democracy, and helps break barriers between social classes and nations.

Beyond the textbook. For the exam: the first Paralympic Games were held in Rome in 1960. In reality: the Paralympic movement grew out of a sports competition for wounded war veterans at Stoke Mandeville in England in 1948, and the 1960 Rome Games were the first to be recognised as the Paralympics. Both facts fit together, but only the Rome date is the exam answer.

India at the Olympic Games. India first took part in 1900, when Norman Pitchard won two silver medals in the 200 m and 200 m hurdles. The Indian men's hockey team has won 8 gold medals (six in a row from 1928 to 1956, then 1964 and 1980), 1 silver (Rome 1960) and 2 bronze (Mexico 1968 and Munich 1972).

India at the Paralympics and other Games. At the 2016 Paralympics in Rio de Janeiro, India won 2 gold (Mariyappan Thangavelu in men's high jump and Devendra Jhajharia in men's javelin), 1 silver (Deepa Malik, women's shot put) and 1 bronze (Varun Singh Bhati, men's high jump). At the Commonwealth Games, India's best medal tally listed in the chapter is at Delhi 2010 (38 gold, 27 silver, 36 bronze, 101 in all, ranked 3rd). At the Asian Games, India took 17 gold, 17 silver and 18 bronze (52 in all) in 1951 at New Delhi, ranked 2nd.

Exam traps

60-second revision

Where this chapter connects

• Previous: Measurement and Evaluation covers the tests that tell you who is ready to enter a competition.
• Next: Adventure Sports moves from organised contests to outdoor, risk based activities.
• Related: Team Games and Individual Games give the sports for which these tournaments are organised.