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

Games, Tournaments and Pairing Logic short notes for CAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: The Tournament Master Sheet

The Tournament Master Sheet

Knockout Formulas

Total Matches = $N - 1$
Byes = $P - N$ (where $P$ is next power of 2)
Total Rounds = $lo g_{2} (P)$

Round Robin Formulas

Single RR Matches = $\frac{N ( N - 1 )}{2}$
Double RR Matches = $N (N - 1)$

Points & Qualification

Total Points (3-1-0) = $3 \times Matches - Draws$
Guarantee Qualification: Assume top teams draw against each other to minimize the cutoff score.
Maximize Elimination: Assume top teams win all matches against bottom teams to maximize the cutoff score.

Summary: Pairing and Earnings Master Sheet

Pairing and Earnings Master Sheet

Single Round Pairings

For $2 n$ players, total ways = $(2 n - 1)!!$

Tracking

Always use a pairing matrix to mark who played with whom in each round.

Earnings Strategy

Maximize: Greedily pick the highest-value valid pair each round.
Minimize: Greedily pick the lowest-value valid pair each round.

Golden Rule: A valid pair is one that did not play together in the immediately preceding round.

The 3-Step Matrix Solver Checklist

1. Deconstruct & Build

Calculate total strategies: (Options 1) $\times$ (Options 2).
Draw the grid: Rows = Player A, Cols = Player B.
Verify the payoff unit (Vote share? Margin? Zero-sum?).

2. Reduce (If matrix is large)

Check for row dominance (A wants higher). Eliminate dominated rows.
Check for column dominance (B wants lower). Eliminate dominated columns.

3. Solve for Equilibrium

Simultaneous: Find Maximin and Minimax. If equal $\to$ Saddle Point.
Sequential (A first): Find min of each row, then take the maximum of those minimums.

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 short notes in this unit

chapter

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

Games, Tournaments and Pairing Logic short notes for CAT: 3 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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