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Linear Arrangement (Single Row)

Introduction

The Linear Arrangement (Single Row) puzzle is one of the most common and fundamental reasoning patterns. It tests your ability to arrange people or objects in a straight line based on given clues about their relative positions.

This pattern is important because it builds the foundation for more advanced puzzles like double-row, circular, and mixed arrangements.

Pattern: Linear Arrangement (Single Row)

Pattern

The key concept is to position people or items in a straight line using relational clues such as "to the left of", "to the right of", "between", or "next to".

The arrangement may involve everyone facing the same direction (usually North) or opposite directions. Understanding the facing direction is crucial to correctly interpret "left" and "right".

Step-by-Step Example

Question

Five friends - A, B, C, D, and E - are sitting in a row facing North. A is to the immediate left of B, C is not next to D, and D is not at any end. Who are sitting at the extreme ends?

  • A. A and B
  • B. C and E
  • C. B and D
  • D. C and A

Solution

  1. Step 1: Identify the total positions

    There are 5 people (A, B, C, D, E), so positions are 1 to 5 from left to right (facing North).

  2. Step 2: Place A and B

    A is to the immediate left of B → A and B must be together like [A B].

  3. Step 3: Analyze D’s position

    D is not at any end → D must be in one of the middle positions (2, 3, or 4).

  4. Step 4: Use C’s clue

    C is not next to D → wherever D sits, C must not be adjacent.

  5. Step 5: Try logical placement

    Possible middle position for D is 3. If D is at 3, A and B can be at (1,2) or (4,5). To ensure C is not next to D, C must be placed at one extreme.

  6. Step 6: Final possible order

    The correct sequence satisfying all clues is: C - A - B - D - E

  7. Final Answer:

    C and E → Option B

  8. Quick Check:

    A is left of B ✅ D is not at any end ✅ C not next to D ✅ All conditions satisfied ✅

Quick Variations

1. All facing the same direction (simplest).

2. Some facing North, others South (adds complexity).

3. People can be arranged based on both position and condition (e.g., profession or color).

4. Multi-row linear puzzles (Two Rows) extend this pattern.

Trick to Always Use

  • Draw placeholders (like boxes 1-5) before starting.
  • Note all direct clues (e.g., "A is left of B") and mark them immediately.
  • Use elimination for indirect clues (e.g., "C is not next to D").
  • Mark facing directions first - it changes left/right interpretation.

Summary

Summary

  • Identify total positions and note who faces which direction.
  • Start with fixed relations (like immediate left/right clues).
  • Apply “not” clues to eliminate wrong placements.
  • Always cross-verify with all conditions before finalizing order.

Example to remember: 5 people facing North - use placeholders and link pairs like A-B first, then fill remaining.

Practice

(1/5)
1. Six persons - A, B, C, D, E, and F - are sitting in a row facing North. B is to the immediate right of A. C is two places to the right of B. E is not at any end. Who is sitting exactly between D and C?
easy
A. E
B. B
C. F
D. A

Solution

  1. Step 1: Place A and B

    B is immediately right of A ⇒ a pair [A B].

  2. Step 2: Place C

    C is two places to the right of B ⇒ positions include [A B _ C].

  3. Step 3: Use E’s condition

    E is not at any end ⇒ E occupies one of the middle positions (position 3 or 4 in a 6-person row).

  4. Step 4: Construct a valid arrangement

    One arrangement that satisfies all clues is: D - A - B - E - C - F.

  5. Step 5: Identify the exact middle between D and C

    Between D and C are A, B, and E; the middle of these three is B.

  6. Final Answer:

    B → Option B

  7. Quick Check:

    B is immediately right of A ✅ C is two places right of B ✅ E not at an end ✅
Hint: Start from immediate pairs, place the two-step relation, then fill the remaining middles respecting 'not at end' clues.
Common Mistakes: Counting 'between' people incorrectly - when there are three people in between, pick the middle one (not the first or last).
2. Seven people - P, Q, R, S, T, U, and V - are seated in a line facing North. P is to the left of R. Only two people sit between R and T. S sits to the immediate right of Q. How many persons sit between Q and T?
easy
A. 2
B. 3
C. 4
D. 5

Solution

  1. Step 1: Use R-T relation

    Two persons between R and T → pattern [R _ _ T].

  2. Step 2: Place Q and S

    S immediately right of Q → pair [Q S].

  3. Step 3: Place P

    P is left of R → [P ... R].

  4. Step 4: Combine logically

    Possible order: P - Q - S - R - U - V - T.

  5. Step 5: Count between Q and T

    Between Q and T lie S, R, U, and V → 4 persons.

  6. Final Answer:

    4 → Option C

  7. Quick Check:

    S right of Q ✅ R-T gap two ✅ P left of R ✅
Hint: Count gaps after full arrangement; do not assume one person 'exactly between' if even count.
Common Mistakes: Interpreting 'exactly between' when the count is even - always verify numerically.
3. Five students - J, K, L, M, and N - are sitting in a row facing North. M is to the immediate right of L. N is not at any end. K is to the left of J but not adjacent. Who sits exactly in the center?
easy
A. L
B. M
C. N
D. K

Solution

  1. Step 1: Place L and M

    M immediately right of L → [L M].

  2. Step 2: N’s condition

    N not at any end → must occupy a middle position.

  3. Step 3: K-J relation

    K left of J but not adjacent → [K _ J].

  4. Step 4: Build order

    Final order: K - L - M - N - J.

  5. Step 5: Identify center

    In a 5-person row, the 3rd position is center → M.

  6. Final Answer:

    M → Option B

  7. Quick Check:

    M right of L ✅ N not at end ✅ K left of J non-adjacent ✅
Hint: Always test central positions explicitly for odd counts to verify the exact middle person.
Common Mistakes: Selecting the wrong middle when counting from both ends.
4. Seven persons - A, B, C, D, E, F, and G - are sitting in a straight line facing North. A sits second to the right of D. C sits to the immediate left of E. F is not at any end and is to the left of B. Who sits exactly between D and E?
medium
A. A
B. F
C. C
D. B

Solution

  1. Step 1: Decode A-D relation

    A second to right of D → [D _ A].

  2. Step 2: Decode C-E relation

    C immediately left of E → pair [C E].

  3. Step 3: F’s position

    F not at any end and sits to the left of B → [F ... B].

  4. Step 4: Combine logically

    Valid sequence: D - F - A - C - E - B - G.

  5. Step 5: Identify person between D and E

    Between D and E are F, A, and C → the exact middle is A.

  6. Final Answer:

    A → Option A

  7. Quick Check:

    A 2nd right of D ✅ C left of E ✅ F not at end, left of B ✅
Hint: When multiple people lie between two persons, the 'exactly between' person is the middle of those.
Common Mistakes: Choosing the first person between instead of finding the middle one.
5. Eight persons - P, Q, R, S, T, U, V, and W - are sitting in a row facing North. Q is second to the left of S. R sits to the immediate right of T. V is not at any end. U sits third to the right of Q. Who sits at the extreme right end?
medium
A. P
B. U
C. W
D. S

Solution

  1. Step 1: Translate Q-S and U position

    Q is second to the left of S ⇒ if S is at position p, Q is at p-2. U is third to the right of Q ⇒ U is at (Q+3) = p+1.

  2. Step 2: Determine feasible S positions

    For an 8-seat row, p can be 3..7 so that Q and U fall within 1..8.

  3. Step 3: Place T-R pair

    R is immediate right of T ⇒ place an adjacent pair [T R] in any available adjacent slots.

  4. Step 4: Respect V’s condition

    V is not at an end, so V must occupy one of positions 2-7.

  5. Step 5: Build one valid arrangement

    Choosing S = 3 gives Q = 1 and U = 4. Placing [T R] at positions 5-6 and placing V at position 7 yields a valid seating: Q - P - S - U - T - R - V - W (here P fills the remaining slot).

  6. Step 6: Identify extreme right

    The extreme right end (position 8) is W.

  7. Final Answer:

    W → Option C

  8. Quick Check:

    Q is 2 left of S (pos1 & 3) ✅ U is 3 right of Q (pos4) ✅ R right of T (pos6 right of pos5) ✅ V not at end (pos7) ✅
Hint: Place the fixed-distance pair (Q & S) first, compute dependent positions (U), then fit adjacency pairs and non-end constraints.
Common Mistakes: Assuming the first valid fill is unique - multiple valid orders may exist but the extreme-right person can still be determined in the example above.

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