CAT (Common Admission Test) · Data Interpretation & Logical Puzzles
Seating and Linear-Circular Arrangement
Placing people or objects in rows, circles or grids from clues.
Seating puzzles are frame-first: pick the layout, fix facing, then place absolute clues before relative ones. Left and right are not page left and right until facing is named.
- CAT (Common Admission Test)
- Medium level
- 7 concepts
- 17 practice questions
1Frame before clues
Before reading any left/right clue, decide the structure: single row, two facing rows, or a circle facing in or out. Draw the empty seats and mark facing on every person who has one. The frame is what makes left, right and opposite mean something; applying those words to the wrong frame is the usual early collapse.
Figure. Draw the empty seats and facing arrows before any name. Left/right only means something after the frame is fixed.
Order of attack
- Name the frameSingle row, facing rows, or circle — and whether facing is inward, outward or mixed.
- Draw empty seatsMark ends, opposites and wrap-around before anyone is placed.
- Only then place peopleAbsolute seats first; relative left/right only after facing is written on the sketch.
A set says 'B is immediate left of A' and never states facing. What must you fix before that clue can be used?
- Whether A and B are male or female
- The frame and each person's facing direction
- How many seats are empty
Left of A reverses between centre-facing and outward-facing (and between the two rows in a facing-row pair). Gender and empty-seat count do not define left.
2Between and immediate
Convert every gap clue into a seat count before placing anyone. 'Exactly two persons between A and B' means two people sit strictly between them, so the seat-number difference is 3. 'Immediately left/right' means adjacency: zero people between. Loose talk that drops 'exactly' or 'immediate' is how a later clue silently changes meaning.
Figure. Exactly two persons between A and B means two seats strictly between, seat-number difference 3. Immediate would mean zero seats between.
Convert then place
- Quote the wordingKeep 'exactly' and 'immediate' attached to the clue; do not paraphrase them away.
- Write the gapPeople strictly between = (seat difference) − 1 once seats are numbered.
- Place the pairOnly after the gap is a number — not a vibe — mark both legal orientations if the clue does not fix order.
| Clue wording | People strictly between | Seat-number gap |
|---|---|---|
| Immediately left/right of | 0 | 1 (adjacent) |
| Exactly one between | 1 | 2 |
| Exactly two between | 2 | 3 |
| Persons between = (seat difference) − 1 | formula | use after numbering |
In a numbered row, A is at seat 2 and B at seat 5. How many people sit strictly between them?
- 3
- 2
- 4
Persons between = (5 − 2) − 1 = 2. Counting three seats from 2 to 5 inclusive is the inclusive span, not the between-count.
3Anchor absolute clues
Place absolute positions before relative ones: an end seat, a named seat number, or a fixed opposite pair. Each anchor cuts the case count. Relative clues ('B is somewhere left of A') then collapse onto the fixed skeleton instead of generating a tree of free placements that you prune too late.
Figure. Equal seats in left-to-right order R–S–Q–P–T. Q was the first absolute lock (seat 3); P rode on adjacency; R and S were forced by the end / not-end pair.
Fix then branch
- Harvest absolutesEnds, numbered seats, 'opposite', 'second from the left' — place these first.
- Lock adjacenciesImmediate pairs that touch an anchor go next; they inherit a side from the fixed person.
- Leave soft clues last'Somewhere left of' and 'not at an end' wait until the board is almost full.
Five in a row, north-facing
Five friends P, Q, R, S, T sit in a straight row facing north. Q is third from the left. P is immediately right of Q. S is not at an end. R is immediately left of S. Who sits at the extreme right?
- Q is third from the leftQ → seat 3
- P immediately right of Q (facing north)P → seat 4
- S not at an end; open seats {1,2,5}S → seat 2 (only non-end left)
- R immediately left of SR → seat 1; T → seat 5
- Extreme rightT
Pro tip. After Q and P lock seats 3–4, 'S not at an end' leaves only seat 2 for S — then the immediate-left clue finishes the row without a fork.
In a five-seat row facing north, Q is third from the left and P is immediately right of Q. Which seats are already forced before any end-clue is read?
- Only seat 3
- Seats 3 and 4
- Seats 1 and 5
Third from the left is seat 3; immediate right of that seat (facing north) is seat 4. Ends are still free.
4Centre-facing left and right
Around a circle facing the centre, each person's left is clockwise on the page and each person's right is anticlockwise. That is the reverse of the outward-facing convention. Getting left/right backwards here flips the whole arrangement even when every other clue was read correctly.
Figure. Circle unfolded left-to-right in clockwise order: P, empty, empty, S, empty, Q. Facing centre, so Q at the right end wraps to sit at P's immediate right (anticlockwise neighbour).
Facing before left/right
- Write facingCentre-facing (inward) for every seat in this concept — do not mix yet.
- Map the sidesInward: clockwise = left; anticlockwise = right. Say it once on the sketch.
- Place relative cluesOnly after the map is written, place 'immediate right/left' neighbours.
Opposite then immediate right
Six people sit around a circular table facing the centre. P is opposite S. Q is to the immediate right of P. Is Q the clockwise or anticlockwise neighbour of P?
- Facing = centre; left = clockwise, right = anticlockwiseconvention fixed
- Place P; S opposite PS three seats away on either numbering
- Q immediate right of Pright ⇒ anticlockwise neighbour
- Clockwise vs anticlockwiseQ is anticlockwise from P
Pro tip. Fix opposite pairs first, then resolve immediate right/left with the facing sentence already written — never from page-left.
Six people face the centre. P is at the top of your sketch. Who sits to P's immediate left?
- The anticlockwise neighbour of P
- The clockwise neighbour of P
- Whoever sits opposite P
Centre-facing: left = clockwise. Opposite is three seats away, not a neighbour.
5Outward and mixed facing
If everyone faces outward, clockwise becomes each person's right — the centre-facing map flipped. Harder still is mixed facing: some face in, some face out, so left/right reverses per person. Applying one global left/right rule to a mixed circle is the trap that survives a careful first read.
Figure. Unfolded ring: inward sitters take clockwise as left; outward sitters reverse it. Tag every seat before any left/right clue.
Per-seat facing
- Tag every seatWrite In or Out on each chair before any left/right clue.
- Resolve one clueFor the person named in the clue, use only that person's facing to pick clockwise vs anticlockwise.
- Never globaliseA neighbour who faces the other way keeps the opposite map for clues about them.
| Facing | Clockwise neighbour is | Anticlockwise neighbour is |
|---|---|---|
| Centre (inward) | that person's left | that person's right |
| Outward | that person's right | that person's left |
| Mixed | depends on that seat's facing | depends on that seat's facing |
A and B are neighbours. A faces the centre; B faces out. A clue says 'C is immediate left of A'. Which way from A is C?
- Clockwise from A (A's inward left)
- Anticlockwise from A, because B faces out
- Opposite A, because facing is mixed
The clue is about A's left, so only A's facing matters. Inward ⇒ left = clockwise. B's outward facing waits for clues about B.
6Branch and prune
When a clue still permits two placements after the anchors are down, split into cases and carry both. Do not guess. The next constraint should kill one branch; if both survive, you have under-used a clue or the puzzle is not unique. A valid final layout satisfies every clue at once — re-check the discarded branch's killer clue against the kept arrangement.
Figure. Fork on the undecided seats, then let the next clue strike one branch. Never guess — prune.
Split then kill
- Name the forkWrite Case I / Case II with the single undecided seat or side.
- Push one constraintApply the next unused clue to both; strike the case that breaks.
- Re-check the keeperRead every clue against the surviving layout before answering.
After placing anchors, clue X fits seat 1 or seat 5. Clue Y is unused. What next?
- Pick seat 1 and move on — one case is enough
- Open Case I (seat 1) and Case II (seat 5), then use clue Y to prune
- Erase the anchors and restart from clue Y
Two legal placements after anchors means a fork. Carry both until a later clue kills one; guessing a single case is how contradictory finals appear.
7Two facing rows
In two parallel rows that face each other, each person in row 1 has exactly one opposite in row 2. Anchor one facing pair; every other opposite pair is then fixed by alignment. Left and right still mean from each person's facing — so the two rows reverse page-left when both are drawn on a north-up sketch.
Figure. Top row B–A–C facing south; bottom row E–D–F facing north. Vertical links mark opposites. A faces D; B (west of A, A's right when facing south) faces E.
Align then side
- Fix one oppositePlace the named facing pair on matching columns.
- Read left/right in-rowUse that row's facing to decide which side a neighbour sits on.
- Map acrossThe person opposite that neighbour is the aligned seat in the other row.
Who faces B?
Row 1 has A, B, C facing south; Row 2 has D, E, F facing north; each person faces exactly one opposite. Seats run west to east. A faces D. B is immediate right of A. Who faces B?
- A faces D; align columns west→eastA and D share a column
- Row 1 faces south ⇒ person's right = westB is west of A
- Remaining east seat in row 1 is Cwest→east: B, A, C
- Opposites keep columns; D under Awest→east row 2: E, D, F
- Who faces B?E
Pro tip. Anchor one facing pair across the rows, then adjacent people map to adjacent opposites — but only after left/right is taken from that row's facing, not from the page.
Rows face each other. A faces D. In A's row, B is immediate right of A from A's facing. Who faces B?
- Whoever sits opposite A
- The neighbour of D on the same side as B relative to A
- Anyone still unplaced in the other row
Opposites keep columns. B's column is the side of A where B sits, so B faces D's neighbour on that same side.
Notes
- Frame Setup: Decide the structure first - single row, two facing rows, circle facing in/out - and draw the empty slots. The frame determines how left/right and 'opposite' clues apply.
- Start from Definite Clues: Place absolute positions (an end seat, a fixed seat number) before relative clues; this cuts the number of cases dramatically.
- Facing Direction in Circles: For inward-facing people, a person's left is the clockwise neighbour's side reversed - track facing explicitly for every 'left/right' clue.
- Case Branching: When a clue permits two placements, split into cases and carry both until a later clue eliminates one.
- Common trap: In circular puzzles, applying left/right as if all face the same way - mixed facing (some in, some out) reverses left/right per person.
Formulas
- Persons strictly between two seats = (seat difference) - 1.
- Circle of n: neighbours wrap around (seat n is adjacent to seat 1).
- Inward-facing circle: clockwise = left side; outward-facing: clockwise = right side.
- Two facing rows: once one facing pair is aligned, all opposite pairs are fixed.
- For n distinct seats, placing n-1 people forces the last.
Exam traps & shortcuts
- Draw the empty frame and label facing directions before reading relative clues.
- Use absolute clues first; relative clues then collapse onto the fixed anchors.
- Carry two cases when a clue is ambiguous, and prune with the next constraint.
- Re-check all clues against the final layout - a valid arrangement satisfies every clue simultaneously.
Reference tables
Write facing before left/right. Circle wrap: seat n is adjacent to seat 1.
| Setup | Left of a person | Right of a person |
|---|---|---|
| Row facing north | toward west (page left if north is up) | toward east |
| Row facing south | toward east | toward west |
| Circle facing centre | clockwise neighbour | anticlockwise neighbour |
| Circle facing out | anticlockwise neighbour | clockwise neighbour |
Same attack order for linear and circular frames.
| Priority | Clue type | Why first |
|---|---|---|
| 1 | Frame + facing tags | Defines left/right/opposite |
| 2 | Ends, seat numbers, opposites | Absolute anchors cut cases |
| 3 | Immediate / exact-gap pairs | Hard adjacency on the skeleton |
| 4 | Soft relatives and negatives | Prune branches last |
Recap
Night-before pegs for DILR seating.
- Frame
- Empty seats and facing first; left/right are undefined until then.
- Gaps
- Exactly k between ⇒ k people between ⇒ seat difference k + 1. Immediate ⇒ adjacency.
- Anchors
- Ends and opposites before soft relatives; negatives after the board is nearly full.
- Centre circle
- Inward: clockwise = left, anticlockwise = right. Outward flips it. Mixed = per person.
- Facing rows
- Lock one opposite pair; columns carry the rest. In-row left/right uses that row's facing.
- Branches
- Two legal seats ⇒ two cases. Kill one with the next clue; re-check every clue on the survivor.
Practise Seating and Linear-Circular Arrangement
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- 17 exam-style questions on this topic, with explanations
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