Seating Arrangement

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

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Round-table arrangements

⏱ 5 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

Around a table a circle has no first seat, so fix one person first. Facing the centre, your left neighbour is the next person clockwise. Opposite seats exist only for an even number of people, and they fix a seat at once.

Fixing a circle

A circle can be rotated, so it has no first seat. Remove that freedom at once. Put the first named person at the top. Now every other seat is fixed by that person.

Mark the direction on your drawing.

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

Tip: Stand at the bottom of the circle, facing the centre. Your left hand points west. Going from the bottom to the west is clockwise from above.

Opposite seats

With $n$ seats and $n$ even, the person opposite X is $\dfrac{n}{2}$ places away. That is 3 places in a circle of 6, and 4 places in a circle of 8.

Opposite clues are the strongest clues in a circle, because they fix a seat exactly. Place them before any left or right clue.

An odd table has no opposite. If a question asks about one, read the puzzle again.

Counting both ways

Two paths join any two people around a circle. So "how many sit between A and D" needs care. It usually means the shorter side, or the puzzle names the direction.

If D is $k$ -th to the left of A, D is also $(n - k)$ -th to the right of A.

A worked example

Eight people A to H sit around a table facing the centre. A sits opposite E. B is second to the right of A. F sits opposite B. C is immediate left of E. D sits opposite C. G is not a neighbour of A.

A = seat 0. Number clockwise from 0 to 7. E = seat 4.
Right is anticlockwise. B = seat 6. F = seat 2.
C = immediate left of E = seat 5. D = seat 1.
Seats 3 and 7 remain for G and H. Seat 7 is next to A, so G = 3 and H = 7.
Third to the left of H: $7 \to 0 \to 1 \to 2$ . That is F.

Outward and mixed facing

If everyone faces outward, swap the rule. Left is anticlockwise. If some face the centre and some face outward, draw a small arrow at each person when you place them.

Watch: "Q is immediate right of P" depends on how P faces, not on how Q faces.

Neighbour clues

"X is next to Y" does not say which side. Keep both until another clue decides.
"X sits between Y and Z" in a circle usually means immediately between.
"Y and Z are not neighbours" is the best clue for the last two seats.

Couples and attributes

Some puzzles add couples, colours or jobs. Fix the seats first, using the position clues. Then fill an attribute grid, one row per person.

Check with a loop

When the circle is full, walk the left neighbours from any person. You should return to the start after exactly $n$ steps. If you do not, one arrow was drawn backwards. Fix it before you read the questions.

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

Chains around a circle facing the centre

How to spot it:

Clues link neighbours, such as 'A is immediate left of B' and 'D is second right of B'.

\text{left} = \text{clockwise (facing centre)}

Method

Put the first named person at the top.
Follow each 'immediate left' clue clockwise.
Follow each 'right' clue anticlockwise.
Check that the string closes into a loop.

Why it works:

Left chains join into one clockwise string, so the circle builds itself.

Try this

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

Show solution

B = seat 0. D = seat 1 (left of B). C = seat 2. E = seat 3.
A is immediate right of B, so A = seat 4. The loop closes at E.
Second to the right of D is seat $1 - 2 = 4$ .

Answer

Type 2very common3 practice Q

Opposite pairs

How to spot it:

An even number of seats with 'sits opposite' clues.

\text{opposite} = \dfrac{n}{2}\text{ places}

Method

Place every opposite pair as two seats across the table.
Add the left and right clues from a placed person.
Fill the last seats with the 'not' clue.

Why it works:

An opposite clue fixes a seat exactly, so it needs no guessing.

Try this

Eight people A to H sit around a table facing the centre. A sits opposite E. B is second to the right of A. F sits opposite B. C is immediate left of E. D sits opposite C. G is not a neighbour of A. Who is third to the left of H?

Show solution

A = seat 0, E = seat 4. B = seat 6, F = seat 2.
C = seat 5, D = seat 1.
G and H take seats 3 and 7. Seat 7 is next to A, so H = 7.
Third to the left of H: $7 + 3 = 10$ , which is seat 2.

Answer

Type 3common2 practice Q

Facing outward

How to spot it:

Everyone faces away from the centre.

\text{outward: left} = \text{anticlockwise}

Method

Swap the rule: left is anticlockwise and right is clockwise.
Place the first person at the top.
Follow each clue with the swapped rule.

Why it works:

Turning around turns the hands around, so clockwise and anticlockwise swap.

Try this

Six people A to F sit around a table facing outward. 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. Facing outward, left is $-1$ and right is $+1$ .
B = seat 5. C = seat 2. D = seat 3. E = seat 4.
F takes seat 1. Immediate right of A is seat 1.

Answer

Type 4occasional2 practice Q

Mixed facing

How to spot it:

Some people face the centre and some face outward.

Method

Write each person's facing when you place them.
Read each clue from the named person's own facing.
Place the next person only after that.

Why it works:

A clue is read from the person it names, so the facing of that person decides the side.

Try this

Four people sit around a table. A faces the centre. B faces outward. B sits immediate left of A. C sits immediate right of B. Who is immediate right of A?

Show solution

A = seat 0. A faces the centre, so left is clockwise. B = seat 1.
B faces outward, so right is clockwise. C = seat 2.
D takes seat 3. A's right (facing the centre) is seat 3.

Answer

Type 5common

Neighbour clues with two sides

How to spot it:

Clues say 'X and Y are neighbours of A' without a side.

Method

Put the two neighbours on both sides of A.
Try both orders.
Use the next clue to kill the order that breaks a seat.

Why it works:

'Next to' allows two sides, and the later clue picks the right one.

Try this

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

Show solution

A = seat 0. B and C take seats 1 and 5.
If B = 5, E would be seat 0. That is A's seat, so B = 1, E = 2, C = 5.
D = seat 3. Opposite C is seat $5 + 3 = 8$ , which is seat 2.

Answer

Shortcuts that save time

⚡ Walk the circle in one direction

Take the longest chain of 'immediate left' clues and lay it out clockwise. Close the loop with the last clue.

Example

Facing the centre, D is immediate left of B. C is immediate left of D. E is immediate left of C. Write the clockwise string.

Show solution

Immediate left means the next seat clockwise.
Start at B: B, D, C, E.

Answer

B, D, C, E clockwise

⚡ Convert left to right around a circle

In a circle of $n$ , the $k$ -th to the left is the same person as the $(n - k)$ -th to the right. Use the smaller number to save steps.

Example

In a circle of 8, who is seventh to the left of A in terms of a shorter walk to the right?

Show solution

$8 - 7 = 1$ .
Seventh to the left equals first to the right.

Answer

The immediate right neighbour of A

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Fixing every seat number before placing anyone.

Fix only the first person. Number the rest from there.

Mistake 02

Reading left as anticlockwise while facing the centre.

Facing the centre, left is clockwise.

Mistake 03

Looking for an opposite person at an odd table.

Opposite exists only for an even number of seats.

Mistake 04

Reading a clue from the wrong person's facing in mixed rows.

Read each clue from the named person's own facing.

Mistake 05

Fixing which side a neighbour sits on too early.

'Next to' allows two sides. Keep both until a clue decides.

Quick revision

Read this the night before the exam.

Fix the first person at the top of the circle.
Facing the centre: left is clockwise. Facing outward: left is anticlockwise.
Opposite is $\dfrac{n}{2}$ places away, and only for even $n$ .
$k$ -th left is the same as the $(n - k)$ -th right.
Mixed facing: read each clue from the named person.

Practice: 14 questions

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

Topic test · 10 questions

Suggested time 11 min · wrong answers go to your mistake notebook automatically.

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