Question

When drafting league tournament fixtures containing an odd number of participating teams, how many total operational rounds must be scheduled if the coordinator chooses the standard 'Staircase Method'?

• A. \( N - 1 \text{ rounds} \)
• B. \( N \text{ rounds} \)
• C. \( N + 1 \text{ rounds} \)
• D. \( \frac{N(N-1)}{2} \text{ rounds} \)

Hint:

The Staircase Method is the most straightforward way to schedule a league tournament because it completely avoids the need to distribute "Byes," and the number of rounds always simplifies to \( N - 1 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A league (or round-robin) tournament ensures that every team plays against every other team.
There are three main methods to draw these fixtures:
1. Cyclic Method: Teams rotate around a fixed point.
2. Tabular Method: Pairs are drawn using a grid/table.
3. Staircase Method: Matches are arranged in a descending step-like visual sequence.
Key Formula or Approach:
In the Cyclic and Tabular methods, the number of rounds depends on whether \( N \) is even or odd:
- If \( N \) is even, rounds = \( N - 1 \).
- If \( N \) is odd, rounds = \( N \).
However, the Staircase method is an exception to this general rule.
Step 2: Detailed Explanation:
In the Staircase Method, the fixtures are arranged such that you start with one team and pair it with all others in a vertical column, then move to the next team and do the same, omitting previously scheduled pairs.
For example, if \( N = 5 \) (Teams A, B, C, D, E):
Step 1: A-B, A-C, A-D, A-E (4 matches)
B-C, B-D, B-E (3 matches)
Step 2: C-D, C-E (2 matches)
Step 3: D-E (1 match)
The number of vertical columns (steps) used to organize these matches is \( 5 - 1 = 4 \).
Each "Step" in the staircase represents an operational round of scheduling.
The Staircase method is popular because it is the easiest to draw and it does not require the calculation of "Byes" for odd numbers of teams.
The structural logic remains that to connect \( N \) points to every other point exactly once without repetition, you need \( N - 1 \) logical layers of pairings.
Option (D) \( \frac{N(N-1)}{2} \) is the formula for the total number of matches, not the number of rounds.
Step 3: Final Answer:
For the Staircase method, regardless of whether the number of teams is even or odd, the number of rounds is always \( N - 1 \).