Games & Tournaments | Logical Reasoning (LR) and Data Interpretation (DI) - CAT PDF Download

Introduction

Games & Tournament problems often involve calculating matches, distributing points, and determining rankings across different formats. The three main types—Round-Robin, Knockout, and Hybrid—require a clear understanding of rules, point systems, and progression stages. Mastering these concepts helps in solving questions quickly and accurately by applying the right formulas and logical steps.

Round-Robin Tournaments

A round-robin tournament is a format where each participant plays every other participant exactly once (single round-robin) or twice (double round-robin). 

Total Matches

For n teams:

• Single Round-Robin: Total matches are given by the combination formula:
  Example: For n = 8, total matches 
• Double Round-Robin: Each team plays every other team twice: n(n-1)
  Example: For n = 8, 
  total matches = 8 x 7=56.

Points and Rankings

Points System: Typically, a win awards 2 points, a draw/tie awards 1 point, and a loss awards 0 points.

Total Points: Each match distributes exactly 2 points (either 2 to the winner or 1 to each team in a draw). Thus: Total Points = 2 x Total Matches

Wins, Losses, and Draws: Let:

• M: Total number of matches.
• W: Number of win-loss matches.
• D: Number of draw matches.

Then:  M = W + D

Across all teams:

• Total wins (W_total) = W, as each win-loss match contributes one win. 
• Total draws (D_total) = 2D, as each draw match contributes one draw to each of two teams. 
• Total losses (L_total) = W, as each win-loss match contributes one loss.

The relationship is:

​Solved Examples

Example: Tournament with 16 Teams
Consider a tournament with 16 teams divided into two groups of 8 each. Each group plays a single round-robin within itself. The top 4 teams from each group advance to Round 2, where each team from one group plays every team from the other group. After Round 2, the top 4 teams overall advance to the semi-finals, followed by the final. Points: Win = 2, Tie/Draw =1, Loss =0. Tie-breaker: Better run-rate.
Key Points:

• Total teams: 16
• Groups: A and B, each with 8 teams
• Points: Win = 2, Tie/Draw =1, Loss=0
• Advancement: Top 4 from each group to Round 2
• Round 2: Each team from Group A plays each team from Group B
• Final Stage: Top 4 overall to semi-finals, then final
• Tie-breaker: Run-rate

(i) Total number of matches played in the tournament?

Ans: 75

Group A Matches:

Group B Matches:

Round 2 Matches: Top 4 from A vs. Top 4 from B: 4 x 4 = 16

Semi-Finals: 2 matches

Final: 1 match

Total: 28 + 28 + 16 + 2 + 1 = 75

(ii) Least number of points to advance to Round 2? 

Ans: To minimize points for a team to be in the top 4:
Maximize points for top 3 teams:

• Team A (1st): Wins all 7 matches: 7 x 2 = 14 points
• Team B (2nd): Wins all except against A: 6 x 2 = 12 points
• Team C (3rd): Wins all except against A and B: 5 x 2 = 10 points

Total points in group:

Points used by top 3: 14 + 12 +10 = 36
Remaining points for bottom 5: 56- 36 = 20
Distribute among bottom 5:

• Bottom 5 play  = 10 mutual matches.
• If all are draws, each match gives 1 point to each team.
• Total points: 10x 2=20
• Each team plays 4 mutual matches, all draws: 4x 1 = 4 points
• Ranking: A: 14, B: 12, C: 10, D-H: 4 each
• A team with 4 points can be 4th with a better run-rate.

(iii) Maximum possible number of points for a team eliminated at the first round? 

Ans: 10
To maximize points for the 5th-placed team:
Minimize points for bottom 3 teams:

• They play = 3 mutual matches.
• Each wins one match: Total points = 3 x 2 = 6

Remaining points for top 5: 56 - 6 = 50
Distribute equally: 50 ÷ 5 = 10 points each
The 5th team has 10 points but is eliminated due to run-rate.

(iv) Minimum number of matches won to reach the finals?

Ans:1
If all matches before semi-finals are draws, a team advances via run-rate.
Winning only the semi-final (1 match) reaches the final.

(v) Maximum number of matches won to reach the finals?

Ans: 12
Win all group stage matches: 7
Win all Round 2 matches: 
Win semi-final: 1
Total: 7+4+1=12

Knockout Tournaments

A knockout tournament eliminates one participant per match until a champion emerges. 
Key properties:

• Total Matches: N - 1, where N is the number of participants.
• Rounds: If N = 2^k, there are k rounds. Otherwise, byes are needed.
• Seeding: Higher seeds play lower seeds (e.g., #1 vs. #N).
• Upsets: Lower-seeded participant defeats a higher-seeded one.

Visual Representation of Knockout Tournament (8 Teams)

In this image, 
The numbers 1 through 8 represent the seedings of the players or teams:

• 1 = Top seed (strongest competitor based on rankings/performance).

• 8 = Lowest seed (weakest competitor).

• Seeds are arranged to avoid early clashes between top contenders (e.g., 1 and 2 can only meet in the final).

Bracket Logic

The bracket follows a mirrored pairing system to ensure fairness:

Top Half:

• Match 1: 1 (1st seed) vs. 8 (8th seed)

• Match 2: 4 (4th seed) vs. 5 (5th seed)

Bottom Half:

• Match 3: 6 (6th seed) vs. 3 (3rd seed)

• Match 4: 7 (7th seed) vs. 2 (2nd seed)

This ensures:

• The highest seed (1) faces the lowest seed (8) in the first round.

• The second-highest seed (2) faces the second-lowest seed (7).

• Mid-tier seeds (3-6) face each other to balance competition.

Round 1 (Quarterfinals)

• Four matches are played:

  1. 1 vs. 8 → Winner advances, loser eliminated.

  2. 4 vs. 5 → Typically the most balanced matchup.

  3. 6 vs. 3 → Potential for an upset (lower seed 6 could defeat higher seed 3).

  4. 7 vs. 2 → Higher seed (2) is favored, but upsets can happen.

Round 2 (Semifinals)

• Winners from the top half (1/8 vs. 4/5) face each other.

• Winners from the bottom half (6/3 vs. 7/2) face each other.

Final Round (Championship)

• The last two remaining competitors play for the title of "Winner".

Solved Examples

Example: For a tournament with 75 players, each match is a knockout. If there are an odd number of players in any round, the top-seeded player gets a bye.
(i) If no byes from Round 2 onwards, how many matches in Round 1?

Ans: 11
Rounds: 2⁶ < 75 < 27, so 7 rounds. 
To have no byes from Round 2, Round 2 must have 2k players. 
Closest 2^k ≤ 75: 2⁶ = 64. 
Players eliminated in Round 1: 75 - 64 = 11. 
Matches in Round 1: 11

(ii) If top-seeded players in quarterfinals and semi-finals are defeated, and one bye per round, what is the sum of ranks of finalists?

Ans: 6
Round 1: 75 players, 11 matches, 53 byes, 64 advance.
Assume top seed (#1) gets bye in rounds with odd players.
Quarterfinals (8 players): Top seed (#1) defeated.
Semi-finals: Next top seed (#2) defeated.
Finalists: #1 (via bye) and #5 (defeated #4 in semi-finals).
Sum of ranks: 1+5= 6.

Hybrid Tournaments

Hybrid tournaments combine round-robin and knockout formats:

• Group Stage: Round-robin within groups.
• Knockout Stage: Top teams advance to playoffs.
• Visual Representation of Hybrid Tournaments
  
  

In this image, there are following stages 

Stage 1: Group Phase (Round-Robin)

• Teams are split into two groups (e.g., Group 1: A, B, C, D; Group 2: E, F, G, H).

• Each team plays every other team in their group.

• The results (e.g., A1-B2, C1-D2) represent matches where:

  • The first letter (A, C, E, G) is the "home" or higher-seeded team.

  • The second letter (B, D, F, H) is the "away" or lower-seeded team.

  • The numbers (1, 2) likely indicate game scores or rankings (e.g., A1 = Team A’s first win, B2 = Team B’s second loss).

Stage 2: Crossover Knockout Matches
​After the group stage, the top teams from each group face off in inter-group matches:

• B1-A2 → 2nd-place from Group 1 vs. 1st-place from Group 2.

• D1-C2 → 4th-place from Group 1 vs. 3rd-place from Group 2.

• And so on.

This ensures:

• Strong teams advance further.

• Every match remains competitive (no dead rubbers).

Solved Example

Example: Win-Loss Table For a 6-team round-robin tournament (15 matches):

• Points: Win = 2, Tie/Draw =1, Loss=0

Conditions:

• Two teams have the same points.
• Two pairs have the same number of wins.
• One pair has the same number of losses.

Solution:

Tips and Strategies

Use diagrams for knockout tournaments to track opponents.

• Use tables for a round-robin to track points and wins.
• Memorise key formulas:
  
• For minimum points/wins, distribute from top to bottom.
• For maximum points/wins, distribute from bottom to top.
• Practice regularly to recognise patterns and solve efficiently.