Circular Arrangement
Overview
Unlike linear arrangements where positions have fixed endpoints, circular arrangements involve persons seated around a round table with no beginning or end—this creates a continuous loop where the "first" and "last" positions are adjacent.
At Clerk level, expect straightforward single-variable puzzles: 5–8 persons seated in a circle, all facing the center (or all facing outward). The complexity stays manageable—no double-row circles or mixed facing directions that appear in PO exams. Mastering this topic requires understanding relative positioning (who sits to whose left/right) and building the arrangement step-by-step from definite clues.
Success here depends on two skills: correctly interpreting "left" and "right" based on facing direction, and anchoring your diagram with the most concrete clue first.
Key Concepts
- Circular has no fixed reference point: Unlike linear rows, you can place any person anywhere initially. Fix one person's position, then build others relative to them.
- Facing Center vs Facing Outward: When facing center, the person's left is clockwise from your viewing perspective. When facing outward, left becomes anticlockwise. This single rule causes maximum confusion.
- "Immediate left/right" means adjacent: No gap between the two persons. "Second to the left" means one person sits between them.
- Counting direction matters: Count positions in the direction specified (left or right of a person), not your viewing direction.
- Opposite positions: In an 8-person circle, opposite means 4 positions away. In a 6-person circle, opposite means 3 positions away. Formula: n/2 positions away for even n.
- Definite vs Conditional clues: Start with clues that give exact positions ("A sits second to the left of B"). Save clues with "or" conditions for later verification.
- Neighbour means immediate neighbour: "A and B are neighbours" means they sit adjacent to each other.
Key Facts
- In a circle of n persons, each person has exactly 2 neighbours (immediate left and immediate right).
- For facing-center arrangement: moving clockwise = moving to the left of each person.
- For facing-outward arrangement: moving clockwise = moving to the right of each person.
- "A sits third to the right of B" and "A sits (n−3)th to the left of B" describe the same position in an n-person circle.
- Two persons sitting opposite each other have n/2 persons between them on either side (for even n).
- If only relative positions matter, fix any one person at the "top" of your circle diagram—this doesn't change relationships.
- Total arrangements of n persons in a circle = (n−1)! but for exams, you only need to find one valid arrangement.
Worked Examples
Example 1: Basic Facing-Center Problem
Problem: Eight persons A, B, C, D, E, F, G, H sit around a circular table facing the center.
- D sits second to the left of A.
- F sits opposite to D.
- B sits third to the right of F.
- C is an immediate neighbour of both D and B.
- E sits opposite to B.
- G is not a neighbour of E.
Find: Who sits to the immediate right of A?
Solution:
Step 1: Draw a circle with 8 positions. Fix A at top (arbitrary reference).
Step 2: D sits second to the left of A.
- Facing center, left = clockwise from person's view
- Count 2 positions clockwise from A → place D there
Step 3: F sits opposite to D.
- Opposite = 4 positions away
- Place F directly across from D
Step 4: B sits third to the right of F.
- Right = anticlockwise from F's view
- Count 3 positions anticlockwise from F → place B
Step 5: C is immediate neighbour of both D and B.
- Check: D and B should have a common adjacent position
- Place C in that shared neighbour spot
Step 6: E sits opposite to B.
- Place E directly across from B
Step 7: G is not a neighbour of E.
- Two positions remain: one adjacent to E, one not
- Place G in the non-adjacent position, H takes the remaining spot
Final arrangement (clockwise from A): A, _, D, C, B, _, F, _ Fill remaining: A, H, D, C, B, G, F, E
Answer: H sits to the immediate right of A.
Example 2: Facing Outward Problem
Problem: Six persons P, Q, R, S, T, U sit around a circular table facing outward.
- P sits second to the right of Q.
- R sits opposite to P.
- T is an immediate neighbour of R.
- S sits third to the left of T.
Find: Who sits opposite to Q?
Solution:
Step 1: Fix Q at top position.
Step 2: Facing outward, right = clockwise direction.
- P sits second to the right of Q → 2 positions clockwise from Q
Step 3: R sits opposite to P.
- In 6-person circle, opposite = 3 positions away
- Place R
Step 4: T is immediate neighbour of R → two possibilities, hold for now.
Step 5: S sits third to the left of T.
- Left (facing outward) = anticlockwise
- Try each T position, verify S placement
Step 6: U fills the remaining spot.
Final arrangement (clockwise from Q): Q, _, P, _, R, T Complete: Q, S, P, U, R, T
Answer: U sits opposite to Q.
Common Mistakes
- Reversing left/right for facing outward: Students apply facing-center logic universally. Fix: Draw a small person figure at one position, mark their actual left hand—that's their left regardless of your viewpoint.
- Counting the starting person: "Second to the left" means count 2 persons, not including the starting person. Counting A as position 1 instead of 0 gives wrong placement.
- Ignoring "immediate": Treating "to the left" same as "immediate left." Without "immediate," it means anywhere on the left side, requiring other clues to pin down exact position.
- Not verifying all clues: Building arrangement with first few clues, then not checking if remaining clues hold. Always verify every statement against final diagram.
- Drawing flat diagrams for circles: Sketching a line instead of an actual circle. This causes errors when visualizing opposite positions. Always draw a proper circular figure.
Quick Reference
- Facing center: Clockwise = to each person's left
- Facing outward: Clockwise = to each person's right
- Opposite in n-person circle = n/2 positions away
- Fix one person first, build others relatively
- "Immediate" = adjacent; "neighbour" = adjacent
- Always verify every clue after completing the arrangement