DILR
Linear Arrangements
A one-page CAT 2026 reference for linear arrangement puzzles, covering setup and counting, facing directions, fixed and gap clues, immediate-neighbour blocks, ordering, parallel rows, elimination…
View
Circular Arrangements
CAT’26 DILR CHEATSHEET
Every setup, count and facing trick you need for CAT 2026 circular sets.
Circular arrangement looks like linear arrangement with the two ends taped together, and that one change is where most of the marks are won or lost. People sit around a round table, so there are no corners to anchor to, rotations of the same seating count as one, and ‘to the left’ suddenly depends on whether everyone faces the centre or faces out. Get those three ideas straight and the puzzles turn mechanical. This sheet walks through the setups and counts you actually need, from the (n minus 1) factorial base and fixing a person, through opposites, blocks and necklaces, to the facing and direction traps that quietly flip every clue. Every box carries one worked mini puzzle, so you practise the move rather than just recognising it.
Every circular arrangement rule you need
1Circular Setup and Total Ways
n people sit around a round table with no fixed seats.
Distinct circular arrangements = (n − 1)!
Example. 5 people around a round table can sit in (5 − 1)! = 4! = 24 ways, since rotations count as the same seating.
CAT Hack. Fix any one person first, then the other (n − 1) fill the rest in (n − 1)! ways.
2Facing Centre vs Outward
Facing the centre keeps left and right, facing out flips them.
Face centre, left is anticlockwise. Face outward, left is clockwise
Example. If all face the centre, A's left neighbour is the next seat anticlockwise. Turn everyone to face outward and A's left becomes the clockwise seat.
Common Mistake. Most circular errors come from ignoring the facing line, so read whether they face in or out first.
3Left and Right Around the Circle
‘To the left’ means one turn direction, set by the facing.
Facing centre, moving left goes anticlockwise around the ring
Example. With everyone facing centre, ‘2nd to the right of P’ means count two seats clockwise from P, landing on a definite seat.
CAT Insight. Draw the arrow for left and right once on your diagram, then every position clue reads off it.
4Everyone Has Two Neighbours
In a circle there are no ends, so each person has two neighbours.
Every seat has a left and a right neighbour, no exceptions
Example. In a circle of 6, if C sits between A and B then A and B are C's two neighbours, unlike a row where an end person has only one.
CAT Favourite. The ‘no ends’ rule is the biggest difference from linear sets, so never hunt for a corner seat.
5Fix One Person to Remove Rotation
Pinning one person turns the circle into a countable line.
Fix 1 seat, arrange the other (n − 1) in (n − 1)! ways
Example. Seat H at the top of a table of 5, then the remaining 4 fill the other seats in 4! = 24 ways, matching (n − 1)!.
CAT Hack. Fixing the most constrained person, not a random one, collapses the cases fastest.
6K Between Gives Two Arcs
‘K people between X and Y’ can be counted either way round.
K between X and Y → one arc, and (n − 2 − K) on the other
Example. In a circle of 8 with 2 people between X and Y on one side, the other arc holds 8 − 2 − 2 = 4 people, so both splits are possible.
CAT Insight. In a circle ‘between’ has two directions, so a gap clue usually leaves two cases, not one.
7Exactly Opposite (Even n)
Directly opposite exists only when the number of seats is even.
Opposite seat is n/2 places away, defined only for even n
Example. At a round table of 10, the person opposite seat 1 is 10/2 = 5 seats away, at seat 6. For odd n no seat is exactly opposite.
Common Mistake. Do not look for an ‘opposite’ person when n is odd, because none exists.
8Group Together as a Block
k people who must sit together act as one seat in the circle.
k together in a circle = (n − k)! × k!
Example. 3 of 6 people must sit together, so (6 − 3)! × 3! = 3! × 6 = 6 × 6 = 36 ways.
CAT Favourite. The circular block count is (n − k)! × k!, one factorial lower than the linear version.
9Necklace and Reflection
If clockwise and anticlockwise look identical, halve the count.
Necklace arrangements = (n − 1)! / 2
Example. For 5 distinct beads on a necklace, (5 − 1)! / 2 = 24 / 2 = 12, because a flip gives the same necklace.
CAT Insight. Halve only when a mirror image counts as the same, like beads. Seats at a table do not flip.
10Mixed Facing In and Out
Mixed facings mean left and right differ from person to person.
Set left and right seat by seat, using each person's own facing
Example. If A faces centre and B faces out, A's right is anticlockwise while B's right is clockwise, so a shared clue splits into two readings.
Common Mistake. Never apply one left-right rule to the whole table when the facings are mixed.
11Distinct or Numbered Seats
If the seats are labelled, rotations are no longer the same.
Numbered seats around a circle = n!, not (n − 1)!
Example. 8 people in 8 numbered chairs around a table can sit in 8! = 40320 ways, since seat 1 differs from seat 2.
CAT Insight. Read whether the seats are labelled, since labels switch the count from (n − 1)! back to n!.
12Counting Kth to the Right
Count K seats in the stated direction, wrapping around the circle.
Kth to the right of X = move K seats clockwise, wrap seat n to 1
Example. In a circle of 7, the 4th person to the right of seat 6 is 6, then wrap to 7, 1, 2, 3, landing on seat 3.
CAT Hack. When the count runs past the last seat, keep going from the first, because the ring has no end.
13Case-Splitting on Facing
When facing or direction is unclear, build each case and test it.
2 possible readings → draw both circles, keep the one that fits
Example. If a clue works clockwise or anticlockwise, draw both rings, apply the rest of the clues, and drop the ring that breaks a rule, often leaving one.
CAT Hack. Split on facing or direction early, because guessing it wrong wastes the whole diagram.
Exam shortcuts, common traps and revision
14
CAT Exam Shortcuts
- Circular arrangements of n people = (n − 1)!
- Fix one person first to remove rotation
- Numbered seats change the count to n!
- k people together = (n − k)! × k!
- Necklace with reflection = (n − 1)! / 2
- Opposite seat is n/2 away, only for even n
- Read the facing line before any left or right clue
15
Most Common CAT Traps
- Using n! for a plain round table instead of (n − 1)!
- Forgetting that facing outward flips left and right
- Looking for an ‘opposite’ person when n is odd
- Reading ‘between’ one way when a circle allows both arcs
- Applying one left-right rule when the facings are mixed
- Not wrapping the count past the last seat back to the first
16
30-Second Revision Box
- Free seats = (n − 1)!, numbered seats = n!
- Fix one person, then arrange the rest
- Face centre keeps left and right, face out flips them
- k together = (n − k)! × k!, necklace = (n − 1)! / 2
- Opposite exists only for even n, at n/2 away
- ‘Between’ has two arcs, so expect two cases
Circular sets reward a clean diagram more than raw speed, so draw the ring, mark the facing arrow once, and let every clue read off it. Fix the most constrained person, watch for the two arcs a gap clue opens up, and the rest usually falls into place. When you want to see where you actually stand, run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the DILR breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor and shape one to your target.