Seating Arrangement Problems: Linear, Circular, Square

Seating arrangement is one of the most scoring topics in the reasoning/mental ability section of Kerala PSC exams. Once you master the rules and approach, these 3-5 mark questions become easy pickings. This note covers all three types: linear, circular, and square arrangements.

1. Types of Seating Arrangements

Type	Description	Key Rule
Linear	People sit in a straight row/line	Facing direction matters (north/south)
Circular	People sit around a round table	No ends; everyone has exactly 2 neighbours
Square/Rectangular	People sit around a table with corners	Corner and side positions differ

2. Important Terms and Conventions

Term	Meaning
Immediate left/right	Directly next to (adjacent)
Second to the left	Two positions away to the left
Opposite	Directly across (in circular/square)
Between A and B	Seated in the space separating A and B
Faces north	Person looks towards north
Faces the centre	In circular, person looks inward
Faces away from centre	In circular, person looks outward

Direction Conventions

If facing North	Left = West, Right = East
If facing South	Left = East, Right = West
If facing East	Left = North, Right = South
If facing West	Left = South, Right = North

3. Linear Arrangement

Rules for Linear (Row) Arrangement

Rule	Explanation
Single row facing north	Left of a person = their west side; Right = east side
Single row facing south	Left of a person = their east side; Right = west side
Two rows facing each other	Row 1 faces south, Row 2 faces north (typically)
Position from end	Count from left end or right end as specified

Formula for Position

If a person is Mth from the left and Nth from the right:

Total number of people = M + N - 1

If two people are Mth from left and Nth from right, and their positions are not the same:

Minimum people = M + N - 1 (if no overlap)

Solved Example 1: Linear Arrangement

Problem: Six people A, B, C, D, E, F sit in a row facing north.

• B sits third from the left end
• D sits immediately to the right of B
• A sits at the right end
• C is not adjacent to A
• E is not adjacent to D

Solution:

Step 1: Place B at position 3 (from left) Position: _ _ B _ _ _

Step 2: D is immediately right of B Position: _ _ B D _ _

Step 3: A sits at right end (position 6) Position: _ _ B D _ A

Step 4: C is not adjacent to A, so C cannot be at position 5 C must be at position 1 or 2

Step 5: E is not adjacent to D, so E cannot be at position 5 E must be at position 1 or 2

Step 6: Both C and E go to positions 1 and 2; F goes to position 5 Position: C/E E/C B D F A

Step 7: Since E is not adjacent to D (position 4), and position 5 is adjacent to D… wait, position 3 and 5 are adjacent to D (position 4). E cannot be at position 5 (already F there) or position 3 (B there). So E at position 1 or 2 is fine.

Final arrangement (from left): E C B D F A or C E B D F A

(Both satisfy all conditions — further clues would narrow to one answer.)

Solved Example 2: Position Calculation

Problem: In a row of 40 students, Ravi is 15th from the left end. What is his position from the right end?

Solution: Position from right = Total - Position from left + 1 = 40 - 15 + 1 = 26th from the right

4. Circular Arrangement

Rules for Circular Arrangement

Rule	Explanation
All face centre	Left = clockwise direction for the viewer; Right = anticlockwise
All face outward	Left = anticlockwise; Right = clockwise (reversed from facing centre)
Mixed facing	Each person’s left/right depends on their individual facing direction
No fixed ends	Unlike linear, there is no “left end” or “right end”
Opposite in circle of N	The person N/2 positions away (in either direction)

Key Principle

When all people face the centre:

• Immediate right = next person in anticlockwise direction
• Immediate left = next person in clockwise direction

When all people face outward (away from centre):

• Immediate right = next person in clockwise direction
• Immediate left = next person in anticlockwise direction

Opposite Positions in Circular

Total People	Opposite = positions apart
6 people	3 positions apart
8 people	4 positions apart
10 people	5 positions apart
N people	N/2 positions apart

Solved Example 3: Circular Arrangement

Problem: Eight people P, Q, R, S, T, U, V, W sit around a circular table facing the centre.

• S sits third to the left of P
• T sits opposite to S
• Q sits immediately to the right of T
• V sits second to the right of Q
• U sits opposite to Q
• R is not adjacent to S

Solution:

Step 1: Fix P at any position (say position 1 — top of circle). Since it is circular, we can fix one person.

Step 2: S sits third to the left of P. Counting left (clockwise): P(1) → clockwise → position 2 → 3 → 4 S is at position 4.

Step 3: T sits opposite to S. In 8 people, opposite = 4 positions away. T is at position 8.

Step 4: Q sits immediately to the right of T. Right = anticlockwise from T. Q is at position 7.

Step 5: V sits second to the right of Q. Right = anticlockwise. Q(7) → 6 → 5. V is at position 5 (wait, let me recount with proper positioning).

Let me use clock positions 1-8 going clockwise:

• P = 1
• S = 4 (third to left of P; left = clockwise: 2, 3, 4)
• T opposite S: 4 + 4 = 8. T = 8.
• Q immediately right of T: right = anticlockwise = position 7. Q = 7.
• V second to right of Q: right of Q anticlockwise: 6, 5. V = 5. Wait — let me be more careful.

Actually in facing-centre arrangement: right = anticlockwise. So:

• Right of Q(7): anticlockwise → 6, then 5. V = 6? No — “second to the right” means two positions to the right.
• First to right of Q = position 6. Second to right of Q = position 5. But wait — if going anticlockwise from 7: next is 6, then 5? No.

Let me re-establish: If positions are numbered 1-8 clockwise, and everyone faces centre, then “right” of any person = the next position in anticlockwise direction = previous number.

So: right of Q(7) = 6. Second to right of Q = 5. V = 5.

Step 6: U sits opposite Q. Q = 7, so U = 7 - 4 = 3. U = 3.

Step 7: R is not adjacent to S(4). Adjacent to S = positions 3 and 5. Position 3 = U, Position 5 = V. Already filled. So this condition is automatically satisfied.

Remaining people: R and W for positions 2 and 6. R is not adjacent to S(4). Position 5 = V (adjacent to S), Position 3 = U (adjacent to S). Positions 2 and 6 are both NOT adjacent to S. So R can be at either 2 or 6.

Final: P(1), ?(2), U(3), S(4), V(5), ?(6), Q(7), T(8) where R and W fill positions 2 and 6 (both valid since neither is adjacent to S).

5. Square/Rectangular Arrangement

Rules

Position Type	Characteristics
Corner seats	Face the centre diagonally; have one person on each side
Middle seats	Face the centre straight; have one person on each side
Opposite	Directly across the table

For 8 People around a Square Table

• 4 corner positions + 4 middle positions
• Corner people face centre diagonally
• Middle people face centre straight
• Each person has 2 immediate neighbours

Solved Example 4: Square Arrangement

Problem: Eight people sit around a square table (2 on each side). A, B, C, D sit at corners. E, F, G, H sit at middles. All face the centre.

• A and C sit at opposite corners
• E sits between A and B
• G sits opposite to E
• F is immediately to the right of C

Step 1: Place A at top-left corner. C is opposite = bottom-right corner.

Step 2: E sits between A and B. Since A is at top-left corner, E is on the top side (middle). B must be at top-right corner.

Step 3: D takes the remaining corner = bottom-left.

Step 4: G is opposite to E. E is at top-middle. Opposite = bottom-middle. G = bottom middle.

Step 5: F is immediately to the right of C. C is at bottom-right corner, facing centre (facing towards top-left). Right of C (from C’s perspective facing centre) = the person on C’s right side.

For someone at bottom-right corner facing centre: their right side is towards the right side of the table = the right-side middle position.

F = right-side middle position.

Step 6: H takes the remaining middle position = left-side middle.

6. Shortcuts and Tips

General Approach (All Types)

Step	Action
1	Read ALL clues first before drawing
2	Identify definite/fixed positions (e.g., “sits at the end”, “sits opposite”)
3	Place definite clues first
4	Use elimination for remaining positions
5	Verify ALL conditions after completing arrangement

Common Tricks in PSC Questions

Trick	How to Handle
”Left” without specifying whose left	Usually means left of the subject mentioned
”Between A and B”	Could be A_X_B or B_X_A — check other clues
Negative information (“not adjacent to”)	Use elimination after placing definite clues
”Second to the right” vs “second from the right"	"Second TO the right” = 2 seats away in right direction; “Second FROM the right” = position 2 counting from right end
Mixed facing directions	Draw arrows showing each person’s facing direction

Quick Formulas

Scenario	Formula
Total people in a row	Left position + Right position - 1
Position from other end	Total - Given position + 1
People between two in a row	Difference of positions - 1
Opposite in circle of N	N/2 positions apart
People between two in circle	Count in both directions; question specifies which

7. Practice Strategy

Level	What to Practice
Basic	4-5 people in a row; single direction
Intermediate	6-8 people; circular with all facing centre
Advanced	Mixed facing; two rows; square with corners

Common Mistakes to Avoid

Mistake	Correction
Confusing left/right when facing south	Draw the arrangement; mark directions clearly
Forgetting facing direction affects left/right in circular	Always note: facing centre OR facing outward
Not checking all conditions	After solving, verify every single given condition
Assuming only one valid arrangement	Some problems have multiple valid arrangements — answer choices will tell you which aspect is being asked

8. Sample PSC-Style Questions

Q1: In a row of children facing north, Arun is 12th from the left and Binu is 18th from the right. If they interchange positions, Arun becomes 25th from the left. How many children are in the row?

Solution:

• After interchange, Arun is at Binu’s original position = 25th from left
• Binu’s original position from left = 25
• Binu was 18th from right
• Total = 25 + 18 - 1 = 42 children

Q2: Six people sit in a circle facing the centre. A is second to the left of B. C is opposite A. D is between B and C. Who is to the immediate right of A?

Solution: (Work through systematically using the rules above.)

9. Key Points for PSC

• Always draw a diagram — never try to solve mentally
• In circular arrangement facing centre: right = anticlockwise, left = clockwise (as seen by an observer looking down)
• “Opposite” in a circle of 8 = 4 positions apart (either direction)
• Linear arrangement: always confirm the facing direction before identifying left/right
• “Between” in linear = must be physically positioned between the two named people
• “Adjacent” = immediately next to (no gap)
• Kerala PSC typically gives 6-8 person arrangements
• Time management: spend maximum 2-3 minutes per seating arrangement question
• If stuck, try fixing one uncertain element and check if contradictions arise