**Reasoning Under Constraints – Multi-variable Puzzle Sets**

Reasoning Under Constraints – Multi-variable Puzzle Sets

A Premium CAT DILR Study Guide

What This Is

Multi-variable puzzle sets (MVPS) are constraint-based reasoning questions where 4–7 variables (people, objects, days, etc.) must be arranged under 5–10 rules. These appear in ~20% of CAT DILR sets (2–3 sets per paper) and are high-scoring if solved systematically. A single mistake can collapse the entire solution, but a structured approach guarantees 90%+ accuracy in 8–10 minutes per set.

Example (CAT 2021 Slot 2):
Seven books (A–G) are arranged on a shelf from left to right. Constraints: 1. A is to the left of B. 2. C is immediately to the left of D. 3. E is not adjacent to F. 4. G is at one of the ends. Question: Which book must be in the middle?

Key Concepts & Techniques

1. Variable Listing
2. What: List all variables (e.g., people, days, positions) and their possible values (e.g., "Monday–Sunday").

3. When: First step in every MVPS. Prevents missing variables mid-solution.

4. Constraint Classification

5. Hard Constraints: Absolute rules (e.g., "A is left of B"). Always enforce first.

6. Soft Constraints: Relative rules (e.g., "A is adjacent to B"). Use after hard constraints narrow options.

7. Fixed-Position Anchors

8. What: Identify variables locked into specific positions (e.g., "G is at an end").

9. When: Immediately after listing variables. Reduces complexity by eliminating possibilities.

10. Adjacency Chains

11. What: Group variables that must be next to each other (e.g., "C is immediately left of D" → "CD").

12. When: For rules involving "immediately left/right" or "adjacent." Treat the chain as a single unit.

13. Elimination Grids

14. What: A table where rows = variables, columns = positions. Cross out impossible placements.

15. When: For 3+ variables with overlapping constraints (e.g., scheduling problems).

16. If-Then Deductions

17. What: Assume a variable’s position and check for contradictions (e.g., "If A is 3rd, then B must be 4th–7th").

18. When: When stuck after applying all hard constraints. Use sparingly (time-consuming).

19. Answer Choice Exploitation (MCQs Only)

20. What: Plug in answer choices to verify constraints (e.g., "Option C says A is 2nd. Does this violate Rule 1?").
21. When: Only for MCQs when the question asks for a specific position (e.g., "Which must be 4th?").

Step-by-Step Strategy

Step 1: Read the Problem Twice

• First pass: Identify variables, positions, and all constraints.
• Second pass: Note hard vs. soft constraints and fixed anchors.

Step 2: Draw a Skeleton

• Sketch positions (e.g., 7 slots for 7 books: _ _ _ _ _ _ _).
• Label fixed anchors (e.g., "G is at an end" → G _ _ _ _ _ _ or _ _ _ _ _ _ G).

Step 3: Apply Hard Constraints

• Enforce absolute rules first (e.g., "A is left of B" → A must be in a lower-numbered slot than B).
• Cross out impossible positions in your skeleton.

Step 4: Build Adjacency Chains

• Group variables that must be together (e.g., "CD" for "C is immediately left of D").
• Treat chains as single units (e.g., "CD" occupies 2 slots).

Step 5: Fill in Deductions

• Use process of elimination to place remaining variables.
• For soft constraints, test possible positions (e.g., "E is not adjacent to F" → if E is 3rd, F cannot be 2nd or 4th).

Step 6: Verify All Constraints

• Double-check every rule before finalizing the arrangement.
• For TITA questions, ensure no ambiguity (e.g., "Which must be in the middle?" → only one possible answer).

Step 7: Answer the Question

• Refer to your final arrangement to answer the question.
• For MCQs, eliminate options that violate constraints.

Fully Worked Example (CAT-Style)

Problem:
Six friends (P, Q, R, S, T, U) are seated in a row from left to right. Constraints: 1. P is to the left of Q.
2. R is immediately to the left of S.
3. T is not at either end.
4. U is to the right of Q.
5. P is not adjacent to R.

Question: Which of the following must be true? A) Q is 3rd.
B) S is 5th.
C) T is 4th.
D) U is 6th.

Step 1: Read Twice

• Variables: P, Q, R, S, T, U.
• Positions: 6 slots (_ _ _ _ _ _).
• Hard constraints:
• P left of Q.
• R immediately left of S (→ "RS").
• T not at ends.
• U right of Q.
• Soft constraint: P not adjacent to R.

Step 2: Skeleton

No fixed anchors, but note: - "RS" is a chain (occupies 2 slots).
- T cannot be 1st or 6th.

Step 3: Apply Hard Constraints

1. P left of Q: P must be in a lower-numbered slot than Q.
2. U right of Q: U must be in a higher-numbered slot than Q.
3. T not at ends: T can be 2nd–5th.

Step 4: Build Adjacency Chains

• "RS" is a single unit. Possible positions:
• Slots 1–2, 2–3, 3–4, 4–5, or 5–6.

Step 5: Fill Deductions

Case 1: "RS" in 1–2.
- Slots: R S _ _ _ _ - P must be left of Q, and U right of Q.
- Possible for P: 3rd or 4th.
- If P is 3rd, Q must be 4th–6th.
- U must be right of Q → U is 5th or 6th.
- T cannot be 1st or 6th → T is 4th or 5th.
- But U is 5th or 6th → T cannot be 5th (conflict).
- So T is 4th, U is 6th, Q is 5th.
- Arrangement: R S P T Q U
- Check soft constraint: P (3rd) not adjacent to R (1st) → Valid.
- If P is 4th, Q must be 5th–6th.
- U must be right of Q → U is 6th, Q is 5th.
- T cannot be 1st or 6th → T is 3rd.
- Arrangement: R S T P Q U
- Check soft constraint: P (4th) not adjacent to R (1st) → Valid.

Case 2: "RS" in 2–3.
- Slots: _ R S _ _ _ - P must be left of Q, and U right of Q.
- P can be 1st or 4th.
- If P is 1st, Q must be 4th–6th.
- U must be right of Q → U is 5th or 6th.
- T cannot be 1st or 6th → T is 4th or 5th.
- If Q is 4th, U is 5th or 6th, T is 5th → Conflict (U and T cannot both be 5th).
- If Q is 5th, U is 6th, T is 4th.
- Arrangement: P R S T Q U
- Check soft constraint: P (1st) not adjacent to R (2nd) → Violates "P not adjacent to R".
- If P is 4th, Q must be 5th–6th.
- U must be right of Q → U is 6th, Q is 5th.
- T cannot be 1st or 6th → T is 2nd or 3rd.
- But "RS" is in 2–3 → T cannot be 2nd or 3rd → Invalid.

Case 3: "RS" in 3–4.
- Slots: _ _ R S _ _ - P must be left of Q, and U right of Q.
- P can be 1st or 2nd.
- If P is 1st, Q must be 5th–6th.
- U must be right of Q → U is 6th, Q is 5th.
- T cannot be 1st or 6th → T is 2nd.
- Arrangement: P T R S Q U
- Check soft constraint: P (1st) not adjacent to R (3rd) → Valid.
- If P is 2nd, Q must be 5th–6th.
- U must be right of Q → U is 6th, Q is 5th.
- T cannot be 1st or 6th → T is 1st or 3rd.
- If T is 1st: T P R S Q U → P (2nd) adjacent to R (3rd) → Violates soft constraint.
- If T is 3rd: P T R S Q U → P (2nd) adjacent to R (3rd) → Violates soft constraint.

Case 4: "RS" in 4–5 or 5–6.
- Similar logic shows no valid arrangements without violating constraints.

Step 6: Verify

Valid arrangements from Cases 1 and 3: 1. R S P T Q U 2. P T R S Q U

Step 7: Answer the Question

Check options against both valid arrangements: - A) Q is 3rd: False in both.
- B) S is 5th: False in both.
- C) T is 4th: True in R S P T Q U; False in P T R S Q U → Not always true.
- D) U is 6th: True in both arrangements → Must be true.

Answer: D) U is 6th.

Common Mistakes

1. Mistake: Ignoring adjacency chains.
2. Why it happens: Students treat "R is immediately left of S" as two separate rules.

3. Correct approach: Always group adjacency pairs (e.g., "RS") and treat as a single unit.

4. Mistake: Not checking all constraints after filling positions.

5. Why it happens: Overconfidence after placing most variables.

6. Correct approach: Always verify every rule before finalizing the answer.

7. Mistake: Assuming a variable’s position without testing alternatives.

8. Why it happens: Students pick the first possible arrangement without checking others.

9. Correct approach: For must-be-true questions, ensure the answer holds in all valid arrangements.

10. Mistake: Misapplying "left of" vs. "right of."

11. Why it happens: Confusing "P is left of Q" (P < Q) with "P is right of Q" (P > Q).
12. Correct approach: Write inequalities (e.g., P < Q) to avoid confusion.

CAT Traps & Time Management

Traps:

1. Hidden Fixed Anchors:
2. Trap: A rule like "G is at an end" seems obvious, but students forget to lock it first.

3. Avoid: Always mark fixed anchors immediately (e.g., G _ _ _ _ _ or _ _ _ _ _ G).

4. Overlapping Constraints:

5. Trap: Rules like "A is left of B" and "B is left of C" imply "A < B < C," but students miss the transitive deduction.

6. Avoid: Chain inequalities (e.g., A < B < C → A must be left of C).

7. Adjacency Confusion:

8. Trap: "A is not adjacent to B" is misread as "A is adjacent to B."
9. Avoid: Underline "not" in constraints to avoid misreading.

Time Management:

• Easy set (4–5 variables, 5–6 rules): 6–8 minutes.
• Medium set (6 variables, 7–8 rules): 8–10 minutes.
• Hard set (7+ variables, 9+ rules): 10–12 minutes.
• If stuck after 5 minutes: Move on and return later. Never spend >12 minutes on a single set.

Quick Practice

Question:
Five people (A, B, C, D, E) are seated in a row. Constraints: 1. A is to the left of B.
2. C is immediately to the right of D.
3. E is not at either end.
4. B is not adjacent to C.

Which of the following must be true? A) A is 1st.
B) D is 2nd.
C) E is 3rd.
D) C is 5th.

Answer: C) E is 3rd.
Explanation: The only valid arrangement is A D C E B or D C A E B. In both, E is 3rd.

Last-Minute Cram Sheet

1. List variables + positions first. Never skip this.
2. Hard constraints > soft constraints. Enforce absolute rules first.
3. Adjacency pairs = single unit. Treat "CD" as one block.
4. Fixed anchors (ends, middle) reduce options. Lock them early.
5. For "must be true" questions, test all valid arrangements.
6. "Left of" = lower number; "right of" = higher number.
7. If stuck, assume a position and check for contradictions.
8. Always verify all constraints before finalizing.
9. MCQs: Plug in answer choices to eliminate options.
10. Time trap: Spend max 12 minutes per set. Guess if needed.

Final Tip: MVPS are pattern-recognition games. The more you practice, the faster you’ll spot deductions. Do 2–3 sets daily to build speed and accuracy.