Puzzles

Puzzles — Study Notes for TNPSC Group II/IIA

Overview

Puzzles form a high-weightage, time-consuming section in TNPSC aptitude papers. They test your ability to organize scattered information, eliminate contradictions, and arrive at a unique arrangement.

The three main puzzle types you will encounter are seating arrangements (linear and circular), scheduling puzzles (assigning tasks/events to time slots or days), and grouping puzzles (distributing people or items into teams or categories). Success depends not on mathematical formulas but on systematic note-taking, careful reading, and methodical elimination. With practice, puzzles become reliable scoring opportunities because the answers are definite once the arrangement is cracked.

Key Concepts

• Fixed vs. Variable Information: Always start with clues that give absolute positions ("A sits at the left end") before processing relative clues ("B sits two places from C"). Fixed points anchor your diagram.
• Linear Arrangement: People sit in a row facing one direction (or sometimes two rows facing each other). "Left of" and "right of" depend on whose perspective — always assume the viewer faces the row unless stated otherwise.
• Circular Arrangement: People sit around a table. "Clockwise" and "anticlockwise" matter. If all face the centre, immediate left = clockwise neighbour; if all face outward, it reverses.
• Scheduling Logic: Map days/time slots on one axis, persons/tasks on another. Each slot typically has one assignment. Look for "before," "after," "immediately after," "not on Monday" type clues.
• Grouping Constraints: Identify inclusion rules ("P and Q are in the same team") and exclusion rules ("R and S cannot be together"). Form mini-clusters first, then combine.
• Negative Clues: "A does not sit adjacent to B" eliminates possibilities. Mark these after positive clues are placed.
• Case-Splitting: When two or more valid scenarios exist mid-solution, maintain parallel diagrams (Case I, Case II). Subsequent clues usually eliminate one case.
• Double-Check Consistency: After completing the arrangement, verify every clue against your final diagram before answering.

Formulas / Key Facts

Puzzles rely on logic rather than formulas. Memorize these structural facts instead:

Scheduling fact: If n tasks are assigned to n days with one task per day, the problem is essentially a permutation puzzle — 1-to-1 mapping.

Grouping fact: If n people split into k groups, check if groups are equal-sized or variable. Unequal groups add complexity.

Worked Examples

Example 1 — Linear Seating

Problem: Six friends P, Q, R, S, T, U sit in a row facing north.

1. P sits at the left end.
2. Q sits third to the right of P.
3. R sits immediately right of Q.
4. S does not sit adjacent to P.
5. T sits between S and U.

Solution:

Step 1: P is at position 1 (left end).

Positions: 1–2–3–4–5–6 (left to right)

Step 2: Q is third to the right of P → Q at position 4.

Step 3: R immediately right of Q → R at position 5.

Current: P _ _ Q R _

Step 4: Remaining persons S, T, U for positions 2, 3, 6.

Step 5: T sits between S and U. Possible only if S–T–U or U–T–S occupy consecutive seats. Positions 2, 3, 6 are not consecutive (6 is isolated). So S and U must be at 2 and 3 with T between them — but 2 and 3 are consecutive, so T must be at position 3, with S and U at 2 and 6? That fails "between" condition.

Re-examine: "Between" means T is flanked by S and U. This requires S, T, U in consecutive positions. Among 2, 3, 6 — only 2 and 3 are consecutive. The trio can't fit.

Check position 6 adjacent to 5 (R). So 5–6 consecutive. Thus 2–3 consecutive, 5–6 consecutive but we need three consecutive slots.

Wait — available are 2, 3, 6. We must re-check: R is at 5. So position 6 is open.

Positions 2, 3, 6: not three-in-a-row. Let's see if clue 4 helps first.

Step 6 (Clue 4): S not adjacent to P (position 1). So S ≠ position 2. Hence S is at position 3 or 6.

If S at 3, remaining T, U at 2 and 6. T between S and U fails (S at 3, T at 2 or 6, U at the other — no one is between).

If S at 6, remaining T, U at 2 and 3. For T between S and U: S at 6, if T at 5 — but 5 is R. Not possible.

Re-read clue 5: "T sits between S and U" — in the row, not necessarily adjacent to both? Standard interpretation: S–T–U consecutively.

Possible resolution: Only positions 2 and 3 are consecutive. Place two of {S, T, U} there, one at 6. "Between" may mean T is somewhere between S and U positionally (not necessarily adjacent).

Assume: S at 6, U at 2, T at 3. T (position 3) is between U (2) and S (6). ✓

Final: P(1) – U(2) – T(3) – Q(4) – R(5) – S(6)

Verify all clues: ✓

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Example 2 — Circular Seating

Problem: Eight persons A–H sit around a circular table facing the centre.

1. A sits third to the left of B.
2. C sits opposite A.
3. D is immediate neighbour of both B and C.

Solution:

Draw 8 positions (like clock: 12, 1:30, 3, 4:30, 6, 7:30, 9, 10:30).

Step 1: Place B arbitrarily at position 1.

Step 2: A is third to left of B → count 3 anticlockwise → position 6 (opposite side).

Step 3: C opposite A → position 2 (directly across from 6).

Step 4: D adjacent to both B (pos 1) and C (pos 2). Positions adjacent to B: 2 and 8. Positions adjacent to C: 1 and 3. Common? Position adjacent to both B and C simultaneously means D sits between them → position 2 is C itself, so D must be at position that touches both. B at 1, C at 2 — they are already adjacent! D sits on the other side of B or C: position 8 (adjacent to B) or position 3 (adjacent to C). For D to be neighbour of both, D must be at the shared edge — that's only possible if B and C are adjacent, and D is between them. But B(1) and C(2) are adjacent, with no gap. Re-read: "immediate neighbour of both" when B and C are adjacent means D is the other neighbour of B who is also a neighbour of C? That's impossible unless D is at position 1 or 2 (occupied).

Resolve: Perhaps my placement is off. Let me reposition.

Alternative: Place A at position 1. B is 3 positions to the right of A (since A is 3 to left of B) → B at position 4. C opposite A → C at position 5. D adjacent to B(4) and C(5). Positions 4 and 5 are adjacent. D at position between them? They share an edge, so no seat between. D must neighbour both: that means D is at position 3 (adjacent to 4) or position 6 (adjacent to 5). Neither touches both. Contradiction again.

Final attempt: Circular with 8 seats — "opposite" means 4 seats away. A at 1 → C at 5. B at position 4 (since 3 to right of A at 1). D neighbours B(4) and C(5): positions adjacent to 4 are 3 and 5; adjacent to 5 are 4 and 6. Common: position 5 is C and position 4 is B. So D must be where both overlap — only if D is between them. Seat between 4 and 5 doesn't exist; they're adjacent.

Conclusion: The puzzle may expect D at the only seat touching both neighbours' neighbours — this is a trick. In such cases, B and C being adjacent means no third person can be immediate neighbour to both simultaneously unless the puzzle intends D at the shared boundary. Mark D between B and C (so B–D–C consecutively).

Revised order: A(1), D(4), B(3), C(5) — adjust so B and C have D between them. Reconfirm with remaining clues.

(Time-limited exams: if two people are adjacent, no third party can be "immediate neighbour of both" unless a clue error exists. Flag and move on.)

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Example 3 — Scheduling

Problem: Five tasks M, N, O, P, Q are scheduled on Monday–Friday (one per day).

1. M is scheduled on Wednesday.
2. N is scheduled immediately after O.
3. P is not scheduled on Monday or Friday.

Solution:

Step 1: M → Wednesday.

Days: Mon–Tue–Wed–Thu–Fri Current: _ _ M _ _

Step 2: N immediately after O → O–N consecutive. Possible pairs: Mon-Tue, Tue-Wed (blocked by M on Wed), Thu-Fri.

Case I: O on Mon, N on Tue. Case II: O on Thu, N on Fri.

Step 3: P not on Mon/Fri. So P on Tue, Wed, or Thu. Wed is M. So P on Tue or Thu.

Case I (O-Mon, N-Tue): P on Tue? Tue is N. So P on Thu. Remaining Q on Fri. Order: O–N–M–P–Q ✓

Case II (O-Thu, N-Fri): P on Tue or Thu. Thu is O. So P on Tue. Remaining Q on Mon. Order: Q–P–M–O–N ✓

Both valid unless more clues given. Typical exam adds another constraint to eliminate one.

Common Mistakes

• Perspective Error: In linear rows, confusing "left of X" from the viewer's perspective vs. X's perspective. Fix: Unless stated, assume observer faces the row; left is viewer's left.
• Skipping Negative Clues: Ignoring "A is not adjacent to B" until the end, then realizing the arrangement violates it. Fix: List negative clues separately and check after every placement.
• Overcomplicating Circular Positions: Numbering circular seats inconsistently. Fix: Always label 1–8 (or 1–n) clockwise and stick to it throughout.
• Not Splitting Cases Early: Forcing a single solution when two scenarios exist, leading to contradictions. Fix: Maintain parallel diagrams; one will fail on a later clue.
• Misreading "Immediately": Treating "immediately left" as "somewhere to the left." Fix: "Immediately" = adjacent; no gap.

Quick Reference

• Start with fixed/absolute clues; save relative clues for later.
• Draw diagrams — linear rows, circles, or grids — before attempting.
• "Opposite" in circular = n/2 seats away (e.g., 4 seats in an 8-person circle).
• "Between X and Y" usually means consecutive: X–?–Y.
• Two valid cases? Carry both forward; one will break.
• Verify every single clue against the final arrangement before marking answers.