Seating Arrangements

Seating Arrangements

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Seating Arrangement problems ask you to place people (sometimes with vacant seats) on a line, circle, or rectangle using direct and indirect clues. The three geometry types you’ll meet are linear, circular, and rectangular/polygonal, with persons either facing the centre or facing outside. The most-tested tricks are the “second to the left” phrasing (count two jumps, not one), the left/right reversal when people face inward, and the circular shortcut that moving k places left equals (n − k) places right. Always anchor one person using the strongest definite clue (e.g., “A sits at the extreme end” or “B sits in the middle”), then build outward. HAT-UG almost always pairs seating with a blood-relation sub-clue, so expect chain statements like “the mother of A sits to the left of the father of B.”

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Geometry Types

• Linear row — n persons in a line, all facing the same direction (usually north). “Immediate left” = the seat directly beside; “second to the left” = two seats away.
• Circular table — n persons around a round table. There is no fixed starting point, so answers are typically given relative to a named anchor. The shortcut k left = (n − k) right lets you convert direction without redrawing.
• Rectangular / polygonal — persons on the four (or more) sides; vacant seats between people are counted like a circle, and the side of the table determines who faces whom.

Facing the Centre vs Facing Outside

This single rule causes more errors than any other: when a person faces the centre, their left is your right as you read the diagram. Draw a small arrow on top of each person during practice, or rotate the page 180° mentally to convert to the “facing outside” orientation. The same person at the same seat has opposite left/right depending on facing direction.

Worked Pattern: Six-Person Circular Set

A, B, C, D, E, F sit around a circle facing the centre. B is second to the left of A. D sits immediately to the right of B. C is opposite A.

1. Fix A anywhere; B goes two jumps counter-clockwise (left) from A.
2. D sits one jump clockwise (right) of B.
3. With six persons, “opposite A” = three jumps either way, so C lands directly across from A.
4. The two remaining seats are filled by E and F; any negative clue (e.g., “E is not adjacent to D”) then breaks the tie.

Negative and Coded Clues

A negative clue (“A is not next to C”) eliminates positions just as decisively as a positive one — never skip it. Coded clues swap names for variables or roles (e.g., “the doctor sits between the engineer and the lawyer”) and require a two-pass solve: first fix the people, then map the professions.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Transitive Closure and Sub-Case Branching

Transitive closure is the engine of multi-statement problems: if A is to the left of B and B is to the left of C, then A is to the left of C — even when no single clue mentions A and C together. Build a directed graph on scratch paper and check it for chains before committing to a seating chart. When a clue permits two placements (say, an end seat in a linear row), branch into sub-cases and propagate each against the remaining clues; one branch will usually violate a negative statement, collapsing the case.

Vacant Seats and Counting

Problems that say “A, B, C, D, E sit in a row of seven seats” — or “six persons around a circular table of eight chairs” — require you to count gaps, not persons. Treat each empty chair as a position; the phrase “three seats to the left of B” means three chair-shifts, not three person-shifts. A classic HAT-UG trap asks for the seat number of a person; the answer is the position index, which depends entirely on whether numbering starts at 1 or 0 and from which end.

Edge Cases Worth Memorising

• Odd-numbered circles have no exact opposite. With 5 or 7 persons, “opposite A” is impossible — the clue will instead say “second to the left” or specify a relative position.
• Rectangular tables have two distinct “opposites”: the person directly across (same number on the long side) and the diagonal. Read whether the clue says “facing” or “sitting opposite.”
• Blood-relation fusion appears in roughly one of every three HAT-UG seating sets; sketch a family tree first, assign roles to names, then place them.

Common Traps

1. Reversing left/right because facing direction was misread.
2. Using k jumps when the clue says immediate.
3. Forgetting the (n − k) circular equivalence.
4. Concluding after the first arrangement without checking that a second valid arrangement exists.

Practice Prompts

1. Eight persons sit around a circular table facing the centre. P is third to the right of R. Q is second to the left of P. S is opposite R. T sits between Q and U. V is not adjacent to R. W and X occupy the remaining seats. Who sits exactly opposite T?
2. Linear row of seven seats, A at seat 1, G at seat 7. B sits two seats to the right of C. D is immediately to the left of E. F sits in seat 4. In how many valid arrangements can the row be completed, and which seat is empty?

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