Seating Arrangement

high importance~3-5 Q in Tier 14 formulas⚡ 8 shortcuts4 subtopics

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How seating puzzles are set

⏱ 5 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

A seating puzzle gives a group of people and clues about where they sit. You rebuild the one seating that fits every clue, then answer a question such as who sits opposite D. Rows and circles are the two families. In both, the direction each person faces decides which side is left.

What a seating puzzle is

You get a group of people and a list of clues. You draw the seats first. Then you place each person the clues fix. At the end, one seating is left, and every question is a lookup on it.

There is no calculation here. The skill is to read each clue the same way every time.

Left and right in a row

The row ends belong to you, the viewer. The left end is on your left when you draw the row.

Facing north: a person's left is your left. Their right is your right.
Facing south: it flips. A person's left is your right.

The ends never flip. Only the person's own left and right flip.

Watch: If the puzzle says "facing south", draw a small down arrow on every seat. Read left and right from that arrow, not from your own hands.

Left and right in a circle

People around a table face the centre unless the puzzle says otherwise.

Facing the centre: left neighbour = clockwise. Right neighbour = anticlockwise.
Facing outward: it flips. Left = anticlockwise. Right = clockwise.

A circle has no first seat. Put the first named person at the top. Every other seat is fixed by that person.

"Opposite" exists only when the number of seats is even. The person opposite you is $\dfrac{n}{2}$ places away.

Counting words

Small words change the count by one, so read them slowly.

"Second to the right of X" means one person sits between X and that seat.
" $k$ -th to the left" means $k$ steps. So $k - 1$ people are in between.
"Immediately between X and Y" means X and Y are the two neighbours.
"Between X and Y" alone means somewhere inside, not always next to them.
People between seats $a$ and $b$ = $|a - b| - 1$ .
The $p$ -th person from the left in a row of $n$ is $n - p + 1$ from the right.

Rule: "Neighbour" allows two sides. "Immediate right" allows one side only.

The three-step method

Fix the frame. Draw $n$ boxes for a row or $n$ marks for a circle. Write the facing direction on it.
Place the sure clues. Ends, the middle, and opposite pairs need no guessing.
Hang the chains. Attach each new person to someone already placed. Use "not" clues last.

A short worked row

Five people sit in a row facing north. T is in the middle. U is second to the right of T. V is immediate left of T.

T is seat 3. U is seat 5. V is seat 2.
Seats 1 and 4 are left for W and X.
If W is not at an end, W is seat 4 and X is seat 1.

Now face the same row south. U is second to the right of T, so U is seat 1. V is immediate left of T, so V is seat 4. One word changed the whole answer.

Example: Same clues, different facing, different row. Always find the facing word first.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Facing direction and left or right

How to spot it:

The row says 'facing north' or 'facing south'. A left or right neighbour or an end is asked.

\text{south-facing: right = west, left = east}

Method

Underline the facing word first.
Draw a small arrow on each seat.
Read left and right from the arrow.
Number the seats west to east and place each clue.

Why it works:

The ends stay with the viewer, but each person's hands turn with the facing.

Try this

Five people P, Q, R, S, T sit in a row facing south. R sits in the middle. S sits immediate right of R. P sits at the west end of the row. T does not sit at an end. Who sits immediate left of R?

Show solution

Seats 1 to 5 from the west. R = seat 3.
South-facing right is west. So S = seat 2. P = seat 1.
T is not at an end, so T = seat 4 and Q = seat 5.
South-facing left is east. The immediate left of R is seat 4.

Answer

Type 2very common3 practice Q

Counting people between two seats

How to spot it:

The question asks how many people sit between two people, or how many sit to the right of someone.

|a - b| - 1

Method

Write each person's seat number.
Subtract the smaller from the larger.
Subtract 1.

Why it works:

The two people themselves are not between, so one seat is dropped.

Try this

Eight people sit in a row facing north. A is third from the left and H is second from the right end. How many people sit between A and H?

Show solution

H is $8 - 2 + 1 = 7$ th from the left.
Between: $7 - 3 - 1 = 3$ .

Answer

Type 3very common3 practice Q

Position from either end

How to spot it:

A person is 'third from the left' or 'second from the right end'. The other end is asked.

n - p + 1

Method

Write the number of seats.
Subtract the rank from the number of seats.
Add 1.

Why it works:

The person is counted once from each end, so add the shared seat back.

Try this

Twelve people sit in a row. Kabir is fifth from the left end. What is his position from the right end?

Show solution

$12 - 5 + 1$ .
$= 8$ .

Answer

8th from the right

Type 4very common3 practice Q

Circle facing rules

How to spot it:

People sit around a table and face the centre or face outward. A left or right neighbour is asked.

\text{centre: left = clockwise}

Method

Put the first person at the top and number seats clockwise.
Facing the centre: left is +1, right is -1.
Facing outward: swap the two.

Why it works:

One fixed rule turns every clue into a seat number.

Try this

Six people A to F sit around a table facing the centre. B is immediate left of A. C is second to the right of A. D sits opposite A. E is immediate right of D. Who is immediate right of A?

Show solution

A = seat 0, numbered clockwise.
B = seat 1. C = seat $0 - 2 = 4$ . D = seat 3.
E = $3 - 1 = 2$ . F takes seat 5.
Immediate right of A is seat $0 - 1 = 5$ .

Answer

Type 5common

Counting words: between, second to the right

How to spot it:

Clues use 'second to the right', 'third to the left' or 'between'. You must place someone exactly.

k\text{-th to the right} \Rightarrow k - 1 \text{ between}

Method

Turn 'k-th to the right' into seat + k for a north-facing row.
Count the people between as $k - 1$ .
Use 'immediately' only when the clue says it.

Why it works:

Each small word shifts the count by exactly one seat.

Try this

Seven people sit in a row facing north. M is second from the left. N is third to the right of M. How many people sit between M and N?

Show solution

M = seat 2. N = seat $2 + 3 = 5$ .
Between: $5 - 2 - 1 = 2$ , which is $3 - 1$ .

Answer

Formula sheet

Circular neighbour rule

L(i) = (i+1) \bmod n,\quad R(i) = (i-1) \bmod n

Seats numbered clockwise, everyone facing the centre. Both swap if facing outward.

Opposite seat

i \leftrightarrow (i + n/2) \bmod n

Only when n is even. An odd table has no opposite seat.

People between two seats

\text{between} = |a - b| - 1

a and b are seat numbers in the same row.

Viewer versus person

\text{person faces south} \Rightarrow \text{their left} = \text{your right}

Applies to the person's own left and right, never to the row ends.

Shortcuts that save time

⚡ Pin the absolutes first

Ends, the middle and opposite seats fix a person outright. Fill them before you touch any chain. The remaining people usually fall into one or two slots.

Example

Six people sit in a row facing north. B is at the left end and F is at the right end. D is immediate right of B. C is immediate left of F. A is not next to D. Find the row.

Show solution

B = seat 1, D = seat 2, F = seat 6, C = seat 5.
A and E take seats 3 and 4.
Seat 3 is next to D, so A = seat 4 and E = seat 3.

Answer

B D E A C F

⚡ Test one clue both ways

If you are unsure of the facing, place one clue for north and for south. The two seats differ, and this shows you which one you must use.

Example

T sits in seat 3 of a row of five. U is second to the right of T. Where is U if the row faces north? Where is U if it faces south?

Show solution

North: right is east, so $3 + 2 = 5$ .
South: right is west, so $3 - 2 = 1$ .

Answer

Seat 5 facing north, seat 1 facing south

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Reading a south-facing person's left as your own left.

Their left is your right. Draw an arrow on each seat.

Mistake 02

Flipping the row ends when the row faces south.

The ends stay with the viewer. Only the person's own left and right flip.

Mistake 03

Taking 'second to the right' as two people in between.

It means one person in between. $k$ -th means $k - 1$ between.

Mistake 04

Looking for an opposite seat at an odd table.

Opposite exists only for an even number of seats.

Mistake 05

Assuming 'between' means 'immediately between'.

Only the word 'immediately' forces neighbours.

Quick revision

Read this the night before the exam.

North-facing left = your left. South-facing left = your right.
Circle facing the centre: left = clockwise. Facing outward: left = anticlockwise.
$k$ -th to the right means $k - 1$ people in between.
People between = $|a - b| - 1$ . Other end = $n - p + 1$ .
Opposite is $\dfrac{n}{2}$ places away, and only for even $n$ .
Draw the frame, place the sure clues, hang the chains, use 'not' clues last.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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