Seating Arrangement

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

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Single-row arrangements

⏱ 6 min read🧩 5 question types🎯 16 practice Q

The idea in one minute

Everyone sits in one line, usually facing north. Number the seats from the west end and turn each clue into a small sum on seat numbers. Place the ends first, then the fixed distances, then the pairs. Use the 'not' clues last.

The frame

Number the seats 1 to $n$ from the west end. Write the numbers under the boxes. Now every clue is a small sum, which is safer than moving letters in your head.

For a north-facing row:

"B is $k$ -th to the right of A" gives B = A + $k$ .
"B is $k$ -th to the left of A" gives B = A $-$ $k$ .
"A is at an end" gives A = 1 or A = $n$ .
" $p$ -th from the right end" gives seat $n - p + 1$ .

For a south-facing row the signs flip. " $k$ -th to the right" gives B = A $-$ $k$ .

Order of attack

Absolute clues go in first. Ends, the middle and "third from the left" need no guessing.
Distance clues on a placed person come next. If C is seat 1, "D is fourth to the right of C" gives D = 5.
Floating chains link two people not yet placed. Write the block, such as "P _ Q", on the side. Slide it along the free seats.
Negative clues come last. "Not at an end" or "not next to" often leaves one case.

Tip: If a clue names a placed person, use it now. If it names two unplaced people, park it.

A full worked example

Seven people C, A, B, D, E, F, G sit in a row facing north. C is at the left end. F is fourth to the right of C. A is second to the left of F. D is immediate right of F. G is not at an end. B is next to C.

C = 1. F = $1 + 4 = 5$ . A = $5 - 2 = 3$ . D = 6.
B is next to C, so B = 2.
Seats 4 and 7 remain for E and G. G is not at an end, so G = 4 and E = 7.
Row: C B A G F D E. Third to the right of A is seat 6, which is D.

Case splitting

Sometimes a clue allows two places, such as "A sits at one end". Draw two rows, one for each case. Carry both until a clue kills one.

Watch: Never guess one case and hope. A wrong guess costs a full redraw.

A typical exam puzzle leaves two cases after the first clues. One negative clue then settles it.

Blocks and gaps

"X sits immediately between Y and Z" makes a three-seat block. Y and Z can be on either side. Keep both open.
"Exactly two people sit between A and B" means $|A - B| = 3$ . It does not say who is on the left.
A block of size $k$ in a row of $n$ can start in $n - k + 1$ places. List them and strike out the clashes.

Facing south

You can use the flipped signs, or you can draw the row from east to west. Then a south-facing person's left is your left again. Pick one method and keep it to the end. Mixing the two gives a mirror answer.

Note: A negative clue is weak alone but strong late. With two seats and two people left, one "not at an end" decides both.

Question types you will see

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

Type 1very common2 practice Q

Ends and adjacency

How to spot it:

Clues put people at the ends and name their immediate neighbours.

Method

Place the people at the ends first.
Place each named neighbour next to them.
List the free seats and the free people.
Use the 'not' clue to settle the last two.

Why it works:

Ends and neighbours fix most of the row, so only a small choice is left.

Try this

Six people A to F sit in a row facing north. A is at the left end and B is immediate right of A. F is at the right end and E is immediate left of F. C is not next to B. Who sits third from the left?

Show solution

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

Answer

Type 2very common2 practice Q

Fixed distance chains

How to spot it:

Clues like 'F is fourth to the right of C' and 'A is second to the left of F'.

\text{right} = +k,\ \text{left} = -k \ (\text{north})

Method

Start from the person who is already placed.
Add $k$ for right and subtract $k$ for left.
Place each new person and repeat.

Why it works:

Every relative clue is an arrow of fixed length from a placed person.

Try this

Show solution

C = 1, F = 5, A = 3, D = 6, B = 2.
E and G take seats 4 and 7. G is not at an end, so G = 4 and E = 7.
Third to the right of A is seat $3 + 3 = 6$ .

Answer

Type 3common2 practice Q

South-facing rows

How to spot it:

Everyone in the row faces south.

\text{south: right} = -k \text{ on west-to-east numbers}

Method

Number the seats west to east.
For south-facing people, right is minus and left is plus.
Place the clues and read the question with the same rule.

Why it works:

A south-facing person's hands are turned round, but the seat numbers are not.

Try this

Five people P, Q, R, S, T sit in a row facing south. P is at the east end. Q is second to the right of P. R is immediate left of Q. S is at the west end. Who is immediate left of S?

Show solution

Seats 1 to 5 from the west. P = seat 5.
South-facing right is west, so Q = $5 - 2 = 3$ .
South-facing left is east, so R = 4. S = 1, T = 2.
Immediate left of S is seat $1 + 1 = 2$ .

Answer

Type 4very common2 practice Q

Negative clues settle the case

How to spot it:

Clues say 'not at an end', 'not adjacent to' or 'not a neighbour of'.

Method

Place every positive clue first.
List the free seats for each person.
Cross out the seats that a negative clue forbids.
Fill the person with one seat left, and repeat.

Why it works:

Each 'not' clue removes seats, so the last person has only one place.

Try this

Five people A to E sit in a row facing north. A is in the middle. B is second to the right of A. C is not at an end. D is not next to A. E is not next to B. Who is immediate right of A?

Show solution

A = seat 3, B = seat 5. Free seats: 1, 2, 4.
D is not next to A, so D = seat 1.
E is not next to B, so E is not seat 4. E = seat 2.
C is left with seat 4, which is not an end.

Answer

Type 5common

Blocks and exact gaps

How to spot it:

Clues say 'exactly two people between A and B' or name a pair that sits together.

|a - b| = k + 1

Method

Turn 'k people between' into a seat difference of $k + 1$ .
Try both sides if the clue does not name one.
Place the block and strike out the clashes.

Why it works:

A gap clue fixes the distance but not the side, so test both.

Try this

Six people A to F sit in a row facing north. A is at the left end. Exactly two people sit between A and B. C is immediate right of B. E is immediate left of B. D is immediate right of A. Who sits at the right end?

Show solution

A = seat 1. Two between A and B gives B = seat 4.
C = seat 5. E = seat 3. D = seat 2.
The last person, F, takes seat 6.

Answer

Shortcuts that save time

⚡ Chain links, do not guess gaps

'Second to the right of A' pins B exactly two seats from A. Mark it with an arrow as soon as you read it. Most puzzles have four such arrows, and the picture then completes itself.

Example

In a row facing north, C sits third from the left. A sits second to the right of C. B sits next to both C and A. Where does B sit?

Show solution

C = seat 3. A = seat 5.
B touches both, so B = seat 4.

Answer

Seat 4

⚡ Turn the row to face up

If the row faces south, redraw it from east to west. Now every person 'faces up' and left is your left. Use this drawing for the whole puzzle.

Example

Three people sit in a row facing south. X is at the west end. Y is immediate left of X. Where is Y in the west-to-east numbering?

Show solution

South-facing left is east.
Y = seat $1 + 1 = 2$ .

Answer

Seat 2

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Mixing the flipped signs with a redrawn row.

Pick one method for south-facing rows and use it to the end.

Mistake 02

Guessing one case when a clue allows two.

Draw both cases and carry them until a clue kills one.

Mistake 03

Using $|a - b|$ as the number of people between.

People between = $|a - b| - 1$ .

Mistake 04

Fixing Y and Z in order for 'immediately between'.

Y and Z can be on either side until another clue decides.

Mistake 05

Ignoring the negative clues at the end.

In a nearly full row, one 'not' clue decides the last seats.

Quick revision

Read this the night before the exam.

North-facing: right is $+k$ , left is $-k$ . South-facing: the signs flip.
Order: absolute clues, distances, floating blocks, negative clues.
Two possible places means two drawn rows.
'Immediately between' is a 3-block with the ends swappable.
$k$ people between means a seat gap of $k + 1$ .

Practice: 16 questions

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

Topic test · 10 questions

Suggested time 10 min · wrong answers go to your mistake notebook automatically.

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