Here’s a structured English solution for a hypothetical CAT 2022 DILR Slot 3 question involving a logical game scenario. Since the exact question isn’t provided, I’ll create a generic example based on common DILR patterns (e.g., seating arrangements, matrix problems, or logical grouping). Adjustments can be made based on specific details from the actual paper.
CAT 2022 DILR Slot 3: Logical Game Problem
Question:
A team of 8 players (A, B, C, D, E, F, G, H) participates in a tournament. They are divided into two groups of 4: Group 1 (Sports) and Group 2 (Arts). The following conditions apply:
A and B cannot be in the same group.
C and D must be in the same group.
E and F cannot be in the same group.
G must be in Group 1 if H is in Group 2.
If E is in Group 1, then F must be in Group 2.
Task:
Determine the possible composition of both groups.
Solution:
Step 1: List All Players
Players: A, B, C, D, E, F, G, H
Step 2: Analyze Constraints
A and B are in different groups.
C and D are in the same group.
E and F are in different groups.
G is in Group 1 only if H is in Group 2.
Logical Form: ( G \in \text{Group 1} \rightarrow H \in \text{Group 2} )
Contrapositive: If ( H \in \text{Group 1} ), then ( G \in \text{Group 2} ).
If E is in Group 1, then F must be in Group 2.
Logical Form: ( E \in \text{Group 1} \rightarrow F \in \text{Group 2} ).
Step 3: Assign Variables
Let’s denote:
( G1 ): Group 1 (Sports)
( G2 ): Group 2 (Arts)
Step 4: Test Scenarios
Case 1: Assume C and D are in Group 1
Then, Group 1 has C, D, and needs 2 more players.
A and B must be split. Suppose A is in G1 and B in G2.
Now, Group 1: A, C, D, ?
Remaining players: E, F, G, H.
E and F cannot be together.
If E is in G1, F must be in G2 (Rule 5).
Rule 4: If H is in G2, then G must be in G1.
Subcase 1a: Assign E to G1
Then F must be in G2 (Rule 5).
Remaining: G, H.
If H is in G2, G must be in G1 (Rule 4).
Group 1: A, C, D, E, G → Invalid (exceeds 4 members).
If H is in G1, then G can be in G2 (Rule 4 contrapositive).
Group 1: A, C, D, E, H → Invalid (exceeds 4).
Subcase 1b: Assign E to G2
Then F must be in G1 (since E and F are split).
Group 1: A, C, D, F
Remaining: E, G, H.
Assign G and H to G2.
Check Rule 4: If H is in G2, G must be in G1.
Contradiction: G is in G2 here.
Invalid Case.
Conclusion for Case 1: No valid arrangement.
Case 2: Assume C and D are in Group 2
Group 2 has C, D, and needs 2 more players.
A and B must be split. Suppose A is in G1 and B in G2.
Group 1: A, ?, ?, ?
Group 2: B, C, D, ?
Subcase 2a: Assign E to G1
Then F must be in G2 (Rule 5).
Group 1: A, E, ?, ?
Group 2: B, C, D, F
Remaining players: G, H.
Assign G and H.
If H is in G2, G must be in G1 (Rule 4).
Group 1: A, E, G, ? → Requires one more player (H can’t be here).
If H is in G1, G can be in G2.
Group 1: A, E, H, ? → Only one spot left (no players left).
Invalid Subcase.
Subcase 2b: Assign E to G2
Then F must be in G1 (since E and F are split).
Group 1: A, F, ?, ?
Group 2: B, C, D, E
Remaining players: G, H.
Assign G and H.
If H is in G2, G must be in G1.
Group 1: A, F, G, H → Valid (4 members).
Group 2: B, C, D, E → Valid (4 members).
If H is in G1, G can be in G2.
Group 1: A, F, H, G → Valid (Rule 4 satisfied if H is in G1).
Group 2: B, C, D, E → Valid.
Final Arrangements:
Option 1:
G1: A, F, G, H
G2: B, C, D, E
Option 2:
G1: A, F, H, G
G2: B, C, D, E
Answer Summary
The possible compositions are:
Group 1 (Sports): A, F, G, H
Group 2 (Arts): B, C, D, E
OR
Group 1 (Sports): A, F, H, G
Group 2 (Arts): B, C, D, E
Key Takeaways
Use constraints to split players into groups systematically.
Test assumptions (e.g., C and D in Group 1 vs. Group 2).
Apply logical implications (e.g., contrapositive of Rule 4).
Validate all conditions for each arrangement.
Let me know if you need further refinements!
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