Circular Seating: Facing Centre and Outside

A circular seating puzzle seats a group around a round table and gives clues about who sits where. Unlike a row, a circle has no ends and no first seat: if everyone moves one place round, the arrangement is the same. Eight people can therefore sit around a table in (8 − 1)! = 5040 different ways, not 8!.

That gives the first step of every circular puzzle. Draw the circle with its seats, put the person named in the most clues at the top, and place everyone else relative to that person. Then use the clues that name a definite seat relative to someone already placed, such as "opposite" or "immediately to the left of", before clues like "not a neighbour of".

Left and right always mean the seated person's own left and right, and around a circle that depends on which way the person faces. Picture the table from above:

Person faces	Their left is	Their right is
The centre	Clockwise	Anticlockwise
Outside (away from the centre)	Anticlockwise	Clockwise

Facing the centre, left is clockwise. Facing outside, everything flips.

Two more facts help. With n seats, the person opposite is n/2 seats away in either direction. In a circle of 8, opposite is 4 seats away, and 3 people sit between two opposite people on each side. Also, k places to the left is the same seat as n − k places to the right.

For a person facing the centre in a circle of 8, third to the left is the same seat as 8 − 3 = 5th to the right.

Worked example 1: eight people

A, B, C, D, E, F, G and H sit around a circular table facing the centre.

C sits immediately to the left of A.
B sits opposite A.
F sits second to the left of A.
D sits immediately to the left of B.
G sits immediately to the right of A.
H is not a neighbour of D.

Solution. Put A at the top and number the seats 0 to 7 clockwise from A. Everyone faces the centre, so left is clockwise. Clue 1 puts C in seat 1 and clue 3 puts F in seat 2. Clue 2 puts B in seat 4, opposite A. Clue 4 puts D in seat 5, one place clockwise from B. Clue 5 puts G in seat 7, one place anticlockwise from A. Seats 3 and 6 remain for E and H. Seat 6 is next to D, so clue 6 puts H in seat 3 and E in seat 6.

Clockwise from A, the order is A, C, F, H, B, D, E, G.

Without clue 6, E and H could swap and the puzzle would have two answers. That is the usual job of a "not a neighbour" clue: it settles the last open choice.

When directions are mixed, mark an arrow on each person as soon as you learn their direction, and read every left or right from that arrow. A clue such as "T sits second to the right of R" can also tell you R's direction, if only one direction leaves a free seat.

Worked example 2: six people, two directions

P, Q, R, S, T and U sit around a circular table. Some face the centre and some face outside.

P faces the centre, and S sits opposite P.
Q sits immediately to the left of P.
Q faces outside, and R sits immediately to the right of Q.
T sits second to the right of R.
U sits immediately to the left of T.
S and U face the same direction as Q.

Solution. Put P in seat 0 and number the seats clockwise. With 6 seats, opposite is 6 ÷ 2 = 3 seats away, so S is in seat 3. P faces the centre, so P's left is clockwise and Q is in seat 1. Q faces outside, so Q's right is clockwise and R is in seat 2. Second to the right of R must be a free seat: if R faced the centre it would be seat 0, which is P's, so R faces outside and T is in seat 4. The only free seat left is 5, for U. U is to T's immediate left, which is clockwise from T, so T faces the centre. Clue 6 makes S and U face outside.

Clockwise from P: P (centre), Q (out), R (out), S (out), T (centre), U (out).

✘ Reading left as clockwise for a person who faces outside.
✘ Using your own left on the page instead of the seated person's left.
✘ Counting arrangements with n! instead of (n − 1)!.
✘ Forgetting that a direction clue can be forced by which seats are still free.

Rows instead of tables are covered in linear seating arrangement, and counting circular arrangements is part of permutations and combinations. The full list of reasoning topics is in the study hub.

Q1. In puzzle 1 on this page (eight people A to H facing the centre), who sits opposite C? medium

(a) B (b) E (c) D (d) H

✅ Answer: (c) D

💡 Explanation: Clockwise from A the seats are A, C, F, H, B, D, E, G. C is in seat 1 and the seat opposite is four places on, seat 5, where D sits.

Q2. In puzzle 1 on this page, who sits second to the right of B? medium

(a) F (b) H (c) D (d) C

✅ Answer: (a) F

💡 Explanation: Everyone faces the centre, so right means anticlockwise. Two seats anticlockwise from B, in seat 4, is seat 2, where F sits.

Q3. In puzzle 2 on this page (P to U, some facing the centre and some outside), who sits immediately to the right of S? hard

(a) R (b) P (c) U (d) T

✅ Answer: (d) T

💡 Explanation: S faces outside, so S's right is clockwise. The next seat clockwise from S is the one T sits in.

Q4. In puzzle 2 on this page, how many people face the centre? hard

(a) One (b) Two (c) Three (d) Four

✅ Answer: (b) Two

💡 Explanation: P faces the centre by clue 1, and clue 5 forces T to face the centre. Q, R, S and U all face outside, so exactly two face the centre.

Q5. Ten people sit around a circular table facing the centre. X is third to the left of Y. What is X's position to the right of Y? medium

(a) 3rd (b) 6th (c) 7th (d) 8th

✅ Answer: (c) 7th

💡 Explanation: Going around a circle of 10, three places one way is the same seat as 10 − 3 = 7 places the other way. So X is 7th to the right of Y.

Q6. Twelve people sit equally spaced around a circular table. Two of them sit exactly opposite each other. How many people sit between them on one side? easy

(a) 5 (b) 6 (c) 4 (d) 10

✅ Answer: (a) 5

💡 Explanation: Leaving out the two opposite people, the other 12 − 2 = 10 split equally between the two sides, so 5 sit on each side.

Why can I fix the first person anywhere?

Around a round table there is no first seat, so rotating everyone gives the same arrangement. Placing the first person at the top of your drawing loses nothing; it only removes the copies that differ by rotation.

How do I remember which way left is?

Imagine yourself sitting at the table facing the centre. Your left hand points clockwise around the table. Turn round to face outside and your left hand now points anticlockwise. Everything else follows from that one picture.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 7 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.

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