Question

Anita prepared a fixture based on the league competition. She distributed this fixture to the coaches of all the teams. When Khushi received this fixture, it was misprinted and showed only the matches of rounds three and four. Study the fixture given below:
III Round
5 Bye
4 6
3 7
2 1
IV Round
4 Bye
3 5
2 6
1 7
Based on the above fixture and your previous knowledge answer the following questions:
(i) In which round the match between number 4 and 3 teams will be held?
(ii) How many rounds will be played in the tournament?
(A) 6 (B) 7 (C) 5 (D) 4
(iii) In which round team number 2 will get a bye?
(iv) How many matches will be played in the above fixture?
(A) 21 (B) 23 (C) 19 (D) 18
OR
(iv) The other name for a league tournament is _____.
(For Visually Impaired Candidates only - Alternative case on Keshav organising State level Kabaddi tournament with four committees)
(i) Which of the following committee is responsible for publishing the news?
(A) Technical committee (B) Finance committee (C) Logistics committee (D) Marketing committee
(ii) Which committee is responsible for maintaining the play fields while organising the tournament?
(A) Marketing committee (B) Logistics committee (C) Finance committee (D) Technical committee
(iii) Write any 2 post sports event responsibilities of finance committee?
OR
(iii) Purchase of sports equipment is a function of the \underline{\hspace{2cm}} committee.
(A) Logistics (B) Finance (C) Marketing (D) Technical
(iv) \underline{\hspace{2cm}} committee arranges refreshments, boarding and lodging. }

Hint:

In a cyclic league fixture with an odd number of teams, the number of rounds is always equal to the number of teams ($n$). If the number of teams is even, the number of rounds is $n-1$.

Answer 1

Step 1: Understanding the Concept:
By observing the fixture, we see that the numbers are rotating in a Cyclic Method. In this method, one team (or 'Bye') is fixed at the top right, and others rotate clockwise. Since team numbers up to 7 are mentioned, there are $n = 7$ teams. Because 7 is an odd number, the number of rounds is $n = 7$, and one team gets a 'Bye' in each round.
Step 2: Key Formula or Approach:

1. Total Rounds (for odd $n$) $= n$.

2. Total Matches $= \frac{n(n-1)}{2}$.

3. In Cyclic rotation, if $n$ is odd, 'Bye' is fixed and numbers rotate.
Step 3: Detailed Explanation:

1. (i) Round for Match 4 vs 3: Following the cyclic pattern (where numbers shift one position clockwise each round), if Round IV is [4-Bye, 3-5, 2-6, 1-7], in Round V the Bye goes to
3. In Round VI the Bye goes to
2. In Round VII, the pairing would be [Bye-1, 7-2, 6-3, 5-4]. Thus, the match between 4 and 3 is in Round VII.
2. (ii) Total Rounds: Since $n = 7$ (odd), Rounds $= 7$. Correct Option: (B) 7.
3. (iii) Round for Team 2 Bye: In this pattern, the 'Bye' moves sequentially through the numbers. Round III (5-Bye), Round IV (4-Bye), Round V (3-Bye). Therefore, in Round VI, Team 2 will get a 'Bye'.
4. (iv) Total Matches: $\frac{7(7-1)}{2} = \frac{7 \times 6}{2} = 21$. Correct Option: (A) 2
1.
5. OR (iv) Other name: The other name for a league tournament is the Berger System (or Round Robin).
Step 4: Final Answer:
(i) Round VII (ii) (B) 7 (iii) Round VI (iv) (A) 21 OR (iv) Berger System. % Visually Impaired Candidates Section For Visually Impaired Candidates Only:
Step 5: Explanation for Committee Questions:

1. (i) Publishing News: This is a promotional task handled by the (D) Marketing committee (specifically the publicity sub-committee).
2. (ii) Maintaining Playfields: This involves the technical setup and rules of the game, handled by the (D) Technical committee.
3. (iii) Post-event Finance Duties: (a) Preparation of the final balance sheet and financial report. (b) Making final payments to vendors and settling all outstanding bills.
4. OR (iii) Purchase of Equipment: Handled by the (B) Finance committee (as it involves fund allocation and procurement).
5. (iv) Refreshments/Lodging: This is the responsibility of the Logistics committee (specifically the boarding and lodging sub-committee).

Answer 2

This question combines a fixture-reading problem with a set of short applied questions on tournament organisation and committee responsibilities, with an alternative version for visually impaired candidates. Reconstructing the entire round-robin schedule directly, rather than only extending it forward, makes every part easy to read off.

Setting up the schedule: The numbers used go up to 7, so there are \( n = 7 \) teams. Since \( n \) is odd, the tournament runs for \( n = 7 \) rounds, with exactly one team getting a bye in each round, so every team sits out exactly once across the whole schedule. Comparing the given Round III, Bye 5; pairs 4-6, 3-7, 2-1, with Round IV, Bye 4; pairs 3-5, 2-6, 1-7, shows that every number, including the bye, decreases by 1, wrapping from 1 to 7, as the round number goes up by one. Applying this same shift consistently in both directions reconstructs the complete schedule:

Round I: Bye 7; 1-6, 2-5, 3-4
Round II: Bye 6; 1-4, 2-3, 5-7
Round III: Bye 5; 1-2, 4-6, 3-7
Round IV: Bye 4; 1-7, 2-6, 3-5
Round V: Bye 3; 1-5, 2-4, 6-7
Round VI: Bye 2; 1-3, 4-7, 5-6
Round VII: Bye 1; 2-7, 3-6, 4-5

(i) Round for the match between team 4 and team 3: Scanning the reconstructed table, the pairing 3-4 appears in Round I. This table also accounts for all 21 possible pairings exactly once, confirming the reconstruction is complete and consistent.

(ii) Total rounds: As established, an odd number of teams, 7, always produces exactly 7 rounds, each with one bye, so option (B) 7 is correct.

(iii) Round in which team 2 gets a bye: Reading down the Bye column of the reconstructed table, team 2 sits out in Round VI.

(iv) Total matches in the fixture: With \( n = 7 \) teams playing every other team exactly once, the total number of matches is \( \dfrac{n(n-1)}{2} = \dfrac{7 \times 6}{2} = 21 \), so option (A) 21 is correct.
OR (iv) Other name for a league tournament: A league format, where every team plays every other team once, is also known as the Round Robin or Berger System.

For Visually Impaired Candidates, Keshav's organising committees:

(i) Publishing the news: Getting event coverage and results into the press and media is a communication and promotional task, which falls under the Marketing committee (D), since this committee handles publicity for the tournament.
(ii) Maintaining the play fields: Ground marking, layout and the technical playing conditions of the venue are handled by the Technical committee (D), since this committee is responsible for everything related to how the sport is actually conducted on the field.
(iii) Two post-event responsibilities of the finance committee: After the tournament ends, the finance committee prepares and submits an audited financial statement to the school authorities, and it clears all outstanding dues owed to vendors, officials and equipment suppliers.
OR (iii) Purchase of sports equipment: Because buying equipment involves allocating and releasing funds, this task is handled by the Finance committee (B).
(iv) Arranging refreshments, boarding and lodging: Since this involves the physical arrangement and supply of facilities for participants, it is handled by the Logistics committee.