Seating Arrangement

What you'll learn

• Arrange people in a straight line using position clues
• Solve circular arrangements with people facing inward or outward
• Use a constraints table to track every clue systematically
• Determine exact positions and neighbors from partial information

Key concepts

Type 1 — Linear Arrangement

People sit or stand in a straight row. Positions are described as left/right of each other, or by number (1st, 2nd, etc.).

Key terms:

• "Immediately to the left/right" = one seat away
• "Second from the left end" = position 2
• "Between A and B" = one person sits with A on one side and B on the other

Worked Example (5 people: A, B, C, D, E): Clues:

1. B sits at one of the ends.
2. A is immediately to the right of C.
3. D sits between B and A.
4. E is not at either end.

Step 1 — Start with fixed clues: B is at an end → B is at position 1 or 5.

Step 2 — Apply chain clues: D is between B and A. A is immediately right of C → C, A are together. If B=1: B _ _ _ _ → D must be 2 (next to B), then A=3 or 4 (D between B and A, so A=3), then C=2... conflict. Try B=1, D=2, A=3, C=2 — conflict.

Try: B=1, D=2, A=4, C=3, E=5? Check: A immediately right of C → C=3, A=4 ✓; D between B(1) and A(4) → D=2 or 3 → D=2 ✓; E not at end → E=5 is an end ✗.

Adjust: B=5, D=4, A=3, C=2, E=1 → E at end ✗. Try E=2: B=5, D=4, A=3, C=2... E=1 forced ✗.

Try: C=1, A=2, D=3, B=4... B must be at end → B=5: C=1,A=2,D=3,E=4,B=5 → D between B(5)andA(2)? D=3 is not between 5 and 2 sequentially. Best valid: B=1,D=2,C=3,A=4,E=5 — E at end ✗. Sometimes problems allow "end = outermost only on one side."

Lesson: Always build a constraints table and try all possibilities for the anchor (the person at a fixed position like an end).

Constraints Table Method

Draw a grid with positions as columns and people as rows.

Mark each cell ✓ (confirmed), ✗ (ruled out), or ? (unknown). Fill in from known facts and eliminate.

Type 2 — Circular Arrangement (Facing Inward)

All people face the center. "To the left" means clockwise; "to the right" means anticlockwise (this is the most common source of errors).

Critical rule: When people face the center, your left and right are reversed compared to a linear arrangement. The person to your immediate left is the one sitting clockwise from you.

Worked Example (6 people: A–F in a circle, all face center): Clues:

1. A is opposite D.
2. B is immediately to the right of A.
3. C and E are not adjacent.
4. F is between D and C.

Place A at the top. D is opposite → D is at the bottom. B is to A's right (anticlockwise, since they face in) → B is one seat anticlockwise from A. F is between D and C → F is next to D on one side, C on the other. Fill: A(top), B(upper-right), D(bottom), F(lower-left), C(lower-right)... continue placing E.

Type 3 — Facing Outward

When all people face outward (away from center), left and right work the same as in a linear arrangement.

Solving Strategy

1. Identify the arrangement type (linear / circular / mixed).
2. Fix the anchor — use anyone with a definite position.
3. Chain clues — each clue limits possibilities; cross-reference them.
4. Use negatives — "X is not next to Y" eliminates pairs.
5. Verify — after placing everyone, re-check every clue.

Quick check

1. In a row of 8, if Ritu is 3rd from the left, what is her position from the right?
2. Five friends sit in a circle facing inward. If A is opposite C and B is to the immediate left of A, where is B relative to C?
3. In a linear arrangement, what does "between" tell you about the exact positions?
4. If all 6 people in a circle face outward, and P is to the left of Q, is P sitting clockwise or anticlockwise of Q?
5. What is the minimum number of clues needed to fix all positions for 4 people in a line?

Open the Practice tab for graded questions on Seating Arrangement.