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. According to the above statement, study the fixture given below.
Answer the following Questions based on the above fixture and your prior knowledge:
Question: 1
In which round the match between number 4 and 3 teams will be held?
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In a standard cyclic scheduling system with $N$ teams (where $N$ is odd), team $A$ plays team $B$ in the round $R$ that satisfies the modular condition: $(A + B - 1) \pmod N$. Here, $(4 + 3 - 1) = 6 \pmod 7$, which matches the first round configuration.
Step 1: Analyzing the Provided Misprinted Fixture Fragment:
We are given two columns of a cyclic or staircase league fixture: Round III and Round IV.
• Round III matches: • Bye vs 5
• 4 vs 6
• 3 vs 7
• 2 vs 1
• Round IV matches: • Bye vs 4
• 5 vs 3
• 6 vs 2
• 7 vs 1
Step 2: Identifying the Cyclic Structural Rotation:
In a cyclic league tournament with an odd number of teams ($N = 7$), the total number of stations per round is $(N+1)/2 = 4$. The position “Bye” remains fixed at the top-left slot, while all teams rotate sequentially in a clockwise pattern. Let us trace the positions across rounds to find where team 4 pairs with team 3.
• In Round III, team 4 plays team 6, and team 3 plays team 7.
• In Round IV, team 4 gets a Bye, while team 3 plays team 5.
Step 3: Calculating backwards/forwards to locate the Match (4 vs 3):
Let's reconstruct the systematic clockwise shifting:
• Round V: Moving clockwise from Round IV, the positions shift down by one slot. Team 7 moves up to play team 2, team 6 plays team 3, team 5 plays team 4. Thus, in Round V, the matches are: Bye vs 3, 4 vs 2, 5 vs 1, 6 vs 7.
• Round II: Let's look before Round III. Shifting counter-clockwise from Round III:
• Round II configuration: Top position has Bye vs 6. The remaining pairs shift upwards: 5 vs 7, 4 vs 1, 3 vs 2.
• Round I: Moving counter-clockwise from Round II:
• Round I configuration: Top position has Bye vs 7. The remaining pairs: 6 vs 1, 5 vs 2, 4 vs 3.
Hence, the match between team number 4 and team number 3 occurs during Round I.
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Question: 2
How many rounds will be played in the tournament?
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When the number of teams ($N$) is odd in a league format, each team needs one round to sit out on a “Bye”. Because every team must take a turn resting, the total number of rounds always equals $N$.
Step 1: Identifying the Tournament Format and Parameters:
The text states that Anita prepared a fixture based on a league competition format. Looking at the pairings, the highest team number shown is 7, indicating there are $N = 7$ competing teams.
Step 2: Applying the Formula for Total Rounds:
In a Single League (Round-Robin) tournament, the calculation for the total number of rounds depends on whether the number of teams ($N$) is even or odd:
• If $N$ is even, Total Rounds = $N - 1$.
• If $N$ is odd, Total Rounds = $N$.
Step 3: Calculating the Value:
Since $N = 7$ is an odd number, the total number of rounds to be played is exactly equal to 7. This matches option (B).
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Question: 3
In which round team number 2 will get a bye?
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To verify your work quickly: notice that the team receiving a bye decreases by 1 each round (Round 3 $\rightarrow$ Team 5, Round 4 $\rightarrow$ Team 4, Round 5 $\rightarrow$ Team 3, Round 6 $\rightarrow$ Team 2).
Step 1: Understanding the Mechanism of Byes in a Cyclic League Fixture:
In a cyclic league fixture involving an odd number of teams, the word “Bye” is locked at the first top-left position. The teams then rotate clockwise around this fixed slot. A team receives a bye when it rotates into the position directly opposite or paired with the “Bye” slot.
Step 2: Tracking the Bye Position across Rounds:
Let's trace which team is paired with the “Bye” position across the rounds using the clockwise rotation pattern:
• Round III: The given fixture explicitly shows Bye vs 5. (Team 5 gets a bye).
• Round IV: The given fixture explicitly shows Bye vs 4. (Team 4 gets a bye).
• Round V: Continuing the clockwise sequence, the numbers shift down by one. The pairing becomes Bye vs 3. (Team 3 gets a bye).
• Round VI: Shifting down by one more number, the pairing becomes Bye vs 2.
Therefore, team number 2 is paired with the Bye position during Round VI.
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Question: 4
How many matches will be played in the above fixture?
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Alternatively, you can multiply the number of matches per round by the total number of rounds. With 7 rounds and 3 active matches per round (since 1 of the 4 slots is a Bye), $7 \times 3 = 21$ matches.
Step 1: Extracting the Structural Parameters:
The tournament uses a single league format with a total number of teams $N = 7$.
Step 2: Applying the Formula for Total League Matches:
In a single round-robin/league tournament, the mathematical formula used to determine the total number of matches ($M$) is:
$$M = \frac{N \cdot (N - 1)}{2}$$
Step 3: Calculating the Value:
Substitute $N = 7$ into the equation:
$$M = \frac{7 \cdot (7 - 1)}{2} = \frac{7 \cdot 6}{2} = \frac{42}{2} = 21$$
Therefore, exactly 21 matches will be played in the tournament. This matches option (A).
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Question: 5
The other name for a league tournament is ______________.
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Remember: Knock-out tournaments are called Elimination tournaments, while League tournaments are called Round-Robin tournaments.
Step 1: Identifying Tournament Nomenclature:
A league tournament is a competitive framework where every participating team plays against every other team an equal number of times, regardless of whether they win or lose individual matches.
Step 2: Tracing the Historical and Technical Term:
This system is formally known across sports sports science frameworks as a Round-Robin Tournament. Additionally, in specific historical contexts or regional formats, it is also referred to as the Berger System (named after Johann Berger, who designed the pairing matrices).
Step 3: Conclusion:
Therefore, the most widely accepted alternative name for a league tournament is a Round-Robin tournament.
