Games, Tournaments and Pairing Logic Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Games, Tournaments and Pairing Logic notes for CAT: 38 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Games, Tournaments and Pairing Logic

Chapter Roadmap

1. Tournament Formats & Outcomes

Knockout and Round Robin structures
Byes, seeding, and mixed formats
Points systems and qualification math

2. Pairing Rounds & Earnings

Dynamic pairing constraints
Maximizing and minimizing earnings

3. Election Campaign Matrices

Vote distribution and intensity levels
Strategic allocation of resources

The Architecture of Tournaments

The Goal

Translate the rules of the game into mathematical constraints.

The Core Skill

Map the structure, count the matches, and track the points.

The Formats

Whether it is a sudden death knockout or a grueling round robin, the underlying logic remains the same.

The Knockout Format: One Loss and You Are Out

The Knockout Format

In a single-elimination tournament, every match produces exactly one loser who is immediately eliminated.

Total Matches = N - 1

To find a single winner from

N

teams

Intuition: Every match eliminates exactly one team. Therefore, the number of matches equals the number of eliminations required.

Handling Non-Powers of Two: The Concept of Byes

The Concept of Byes

Find the next highest power of 2

Let this be

P

(e.g., 8, 16, 32, 64...)

Calculate the number of byes

Byes = P - N

Calculate first round matches

First Round Matches = \frac{N - Byes}{2}

Example: For 13 teams,

P = 16

. Byes =

16 - 13 = 3

. First round matches =

(13 - 3) /2 = 5

Games, Tournaments and Pairing Logic: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Data Interpretation and Logical Reasoning MCQ

A single-elimination knockout tournament starts with 64 teams. How many rounds will be played in total to determine the winner?

A.
4
B.
5
C.
6
D.
7

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct formula application for the number of rounds in a knockout tournament, recognizable because the number of teams is a power of 2.

Step 1: In a knockout tournament where the number of teams $N$ is a power of 2, the number of rounds is the exponent of that power.

Step 2: We need to find $k$ such that $2^{k} = N$ .

Step 3: Here, $N = 64$ . We know that $2^{6} = 64$ .

Step 4: Therefore, the total number of rounds is 6.

Answer: 6

Question 2 · Data Interpretation and Logical Reasoning MCQ

In a single round-robin tournament with 8 teams, a win gives 2 points, a draw gives 1 point, and a loss gives 0 points. The top 4 teams qualify for the semi-finals. What is the minimum number of points a team must score to guarantee qualification to the semi-finals, regardless of the results of other matches?

A.
9
B.
10
C.
11
D.
12

Correct Answer:

Step-by-Step Solution

Key idea: This is a "Guaranteeing Qualification" question, recognisable because it asks for the minimum points to ensure advancement regardless of other results. The method involves constructing the worst-case scenario where the top teams maximize their points, while accounting for the hidden constraint that the bottom teams must play each other.

Step 1: Calculate the total points in the tournament.

Number of teams $N = 8$ .

Total matches = $(2 8) = 28$ .

In a 2-1-0 system, every match awards exactly 2 points in total.

Total points in the tournament = $28 \times 2 = 56$ .

Step 2: Construct the worst-case scenario for the top 5 teams.

To find the maximum possible score for the 5th place team (the highest score that fails to qualify), we must minimize the points of the bottom 3 teams.

The bottom 3 teams play $(2 3) = 3$ matches among themselves.

Regardless of the outcomes, these 3 matches will generate exactly $3 \times 2 = 6$ points for the bottom 3 teams.

They can get 0 points from matches against the top 5 teams (by losing all of them).

Thus, the minimum possible total points for the bottom 3 teams is exactly 6.

Step 3: Calculate the maximum points available for the top 5 teams.

Maximum points for top 5 = Total points - Minimum points for bottom 3

Maximum points for top 5 = $56 - 6 = 50$ .

Step 4: Find the maximum score for the 5th place team.

If the top 5 teams share the 50 points as equally as possible, their average score is $50/5 = 10$ .

It is possible for all 5 teams to score exactly 10 points. In this case, the 5th place team has 10 points and is eliminated (due to tie-breakers).

Therefore, 10 points does NOT guarantee qualification.

Step 5: Determine the guaranteeing score.

If a team scores 11 points, can 4 other teams also score 11 or more?

The sum of the top 5 teams would need to be at least $5 \times 11 = 55$ .

However, the maximum available points for the top 5 teams is 50.

It is impossible for 5 teams to score 55 points. Thus, at most 4 teams can score 11 or more points.

A team with 11 points is guaranteed to be in the top 4.

Answer: C

More notes in this unit

chapter

Games, Tournaments and Pairing Logic Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Games, Tournaments and Pairing Logic notes for CAT: 38 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

A single-elimination knockout tournament starts with 64 teams. How many rounds will be played in total to determine the winner?

Question 2

In a single round-robin tournament with 8 teams, a win gives 2 points, a draw gives 1 point, and a loss gives 0 points. The top 4 teams qualify for the semi-finals. What is the minimum number of points a team must score to guarantee qualification to the semi-finals, regardless of the results of other matches?

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