Problem Solving: Seating, Ordering & Grouping Puzzles
Overview
Problem Solving is one of the most scoring yet challenging sections in the Maharashtra Police Bharti reasoning paper. These puzzles test your ability to organise information systematically and draw logical conclusions. Unlike direct formula-based questions, puzzles require you to build a mental (or written) model from scattered clues.
The three main types—seating arrangement, ordering (ranking/sequencing), and grouping—appear regularly. Mastering the technique of tabulation and elimination is essential; guessing rarely works here.
The key skill tested is methodical thinking under time pressure. Police work demands exactly this: processing multiple pieces of information, identifying relationships, and reaching accurate conclusions. Approach these puzzles with patience and a clear system, and they become your advantage.
Key Concepts
- Seating Arrangement: People sit around a circular table or in a linear row. You must determine each person's exact position and often their facing direction (inward/outward for circular; left-right orientation for linear).
- Linear vs Circular: In linear arrangement, there are two ends; in circular, everyone has exactly two neighbours with no endpoints.
- Ordering/Ranking: Arranging items or people based on a parameter—height, age, score, floor number, or sequence of events. Focus on relative positions (A is taller than B) rather than absolute values.
- Grouping/Distribution: Assigning people to teams, departments, or categories based on given conditions. Often combined with additional attributes like gender, profession, or city.
- Definite vs Comparative Clues: Definite clues fix a position ("Ramesh sits at the left end"). Comparative clues show relation ("Suresh sits to the right of Ramesh"). Always start with definite clues.
- Immediate vs Non-immediate Neighbours: "A sits next to B" means adjacent. "A sits second to the left of B" means one person between them.
- Negative Clues: "A does not sit adjacent to B" eliminates possibilities. Use these after placing definite positions.
- Multiple Possibilities: Some puzzles have 2–3 valid arrangements until a later clue fixes the final one. Track all possibilities or you may miss the correct answer.
Key Facts / Rules to Remember
- Circular table facing centre: "Left of X" means clockwise from X; "Right of X" means anticlockwise from X.
- Circular table facing outward: Directions reverse—"Left of X" is anticlockwise.
- Linear row facing you: Leftmost position is Position 1 when people face you (north). From their perspective, it flips.
- Total positions = Total people in standard puzzles. If 8 people sit in a row, there are exactly 8 seats.
- "Between" means excluding endpoints: "B sits between A and C" means A–B–C or C–B–A, with B in the middle.
- Ranking formula: If A's rank from top is R1 and from bottom is R2, then Total = R1 + R2 − 1.
- Grouping constraints: When distributing n people into k groups, ensure each person appears in exactly one group unless stated otherwise.
- Floor-based puzzles: Ground floor is typically Floor 1; higher floor number = higher position. Clarify in the question whether numbering starts from 0 or 1.
Worked Examples
Example 1: Linear Seating (6 people in a row)
Puzzle: Six friends P, Q, R, S, T, U sit in a row facing north.
- R sits at one of the ends.
- Q sits second to the right of R.
- S sits third to the left of T.
- P does not sit adjacent to Q.
Step-by-step:
- Draw 6 positions: _ _ _ _ _ _
- R sits at an end → two cases: R at Position 1 or Position 6.
- Q sits second to the right of R.
- If R at Position 1: Q at Position 3. ✓
- If R at Position 6: Position 8 does not exist. ✗
- So R at Position 1, Q at Position 3.
- Arrangement so far: R _ Q _ _ _
- S sits third to the left of T.
- Possible pairs: (S at 1, T at 4), (S at 2, T at 5), (S at 3, T at 6).
- Position 1 = R, Position 3 = Q. So valid: S at 2 and T at 5, OR T at 6 with S at 3 (but 3 = Q). Only S at 2, T at 5 works.
- Arrangement: R S Q _ T _
- Remaining: P, U for positions 4 and 6.
- P not adjacent to Q (Position 3). Adjacent to Q is Position 4.
- So P cannot be at 4 → P at 6, U at 4.
- Final: R S Q U T P (Positions 1–6)
Answers to typical questions:
- Who sits at Position 6? → P
- Who is second to the right of S? → U
- Who sits between Q and T? → U
Example 2: Circular Arrangement (5 people facing centre)
Puzzle: A, B, C, D, E sit around a circular table facing the centre.
- A sits second to the left of B.
- C sits to the immediate right of A.
- D is not adjacent to B.
Step-by-step:
- Fix B at 12 o'clock (arbitrary starting point for circular).
- A is second to the left of B → Move anticlockwise 2 positions from B → A at 8 o'clock position.
- C is immediate right of A → Right = anticlockwise when facing centre → C at 6 o'clock.
- Filled: B, A, C. Remaining: D, E for positions 10 o'clock and 4 o'clock (neighbours of B are 10 and 2, but 2 o'clock is between B and A—wait, let's recount).
Actually in 5-seat circle:
- B at position 1
- Positions clockwise: 1(B), 2, 3, 4, 5 back to 1.
- Second to left of B (anticlockwise): Position 4.
- So A at 4. C immediate right of A (anticlockwise) = Position 3.
- Remaining positions: 2 and 5. D and E to fill.
- Adjacent to B: positions 2 and 5.
- D not adjacent to B → D cannot be at 2 or 5 → Contradiction?
- Re-check: Only 2 and 5 remain, both adjacent to B. This means puzzle has additional interpretation or error.
Exam tip: If stuck, reread clues. Sometimes "not adjacent" has exceptions or you misplaced someone. Here, ensure your circle numbering is correct.
Example 3: Ordering (Ranking)
Puzzle: In a class of 35 students, Amit ranks 11th from the top. Vijay ranks 26th from the bottom. How many students are between them?
Solution:
- Amit from top = 11
- Vijay from bottom = 26 → Vijay from top = 35 − 26 + 1 = 10
- Amit at rank 11, Vijay at rank 10 (from top).
- Students between them = 11 − 10 − 1 = 0.
Answer: No student between them; they are adjacent in ranking.
Common Mistakes
- Confusing left-right in circular arrangements: Students forget that "left" changes direction based on whether people face inward or outward. → Always note facing direction first and fix your convention.
- Ignoring negative clues until the end: Waiting too long to apply "X does not sit next to Y" causes rework. → Apply negative clues immediately after each placement to eliminate invalid options.
- Miscounting "second to the right": "Second to the right" means skip one person, not two. → Count: immediate = 1 step, second = 2 steps.
- Assuming only one arrangement exists: Many puzzles have two valid configurations until a final clue resolves them. → Maintain parallel tracks (Case 1, Case 2) until one is eliminated.
- Rank miscalculation formula error: Forgetting the "−1" in Total = Top rank + Bottom rank − 1 leads to off-by-one errors. → Always verify with a small example (e.g., 5 people, rank 2 from top = rank 4 from bottom: 2+4−1=5 ✓).
Quick Reference
- Circular facing centre: Left = Clockwise, Right = Anticlockwise.
- Linear row: Always clarify who is facing which direction before solving.
- Start with definite clues (fixed positions), then comparative, then negative.
- Ranking formula: Total = Rank from top + Rank from bottom − 1.
- Between A and C: Excludes both A and C from the count.
- Draw the diagram—never solve seating puzzles purely in your head.