Seating Arrangement Puzzles: How to Solve Round and Row Tables

To solve a seating arrangement puzzle, draw the seats first, place the person with the firmest clue, and then test each remaining clue as a short list of cases until only one seating survives. On a round table, fix one person in a seat before you start, because turning the whole table does not create a new answer.

That is the core of the method. This guide covers how round and straight tables differ, how to read clues about neighbours and opposite seats, what to do with the rule that nobody may sit in their own seat, a full worked puzzle and a practice puzzle with its answer.

What is a seating arrangement puzzle?

A seating arrangement puzzle gives you a set of people, a set of seats and a short list of clues. Your job is to find the one way to seat everyone so that every clue is true.

The people are usually labelled with letters, such as A to F. The seats are either around a round table or along a straight row. The clues talk about position: who sits next to whom, who sits opposite whom, who sits at an end, and who sits to the left or right of someone else.

A good puzzle of this kind has exactly one answer. If you reach two seatings that both fit, a clue was misread or the puzzle is faulty.

How do round tables and straight rows differ?

The two layouts look alike when drawn, but they behave differently. A straight row has two ends and a clear first seat. A round table has no ends, and every seat has two neighbours.

The rows Turning the layout and Ways to seat six people matter most. In a row, the first seat can take any of the six people, the next any of the five left, and so on, which gives 6 x 5 x 4 x 3 x 2 x 1 = 720 seatings.

At a round table, fixing one person in seat 1 leaves the other five to arrange, which gives 5 x 4 x 3 x 2 x 1 = 120.

Because turning a round table changes nothing, you can always put one chosen person in seat 1 without losing any answers. That single step cuts the work by a large factor before you read a single clue.

How do you read left, right and opposite clues?

Most mistakes in these puzzles come from direction words. This guide uses one rule throughout: left and right always mean the seated person's own left and right.

• Straight row. Picture the row from behind, with everyone facing the same way as you, so their left and right match yours. Number the seats from left to right. Then to the right of means a higher seat number, and immediately to the right means a seat number exactly one higher.
• Round table. Number the seats clockwise as seen from above. When everyone faces the middle, the next seat clockwise is on a person's left, and the next seat anticlockwise is on their right.
• Opposite. At a round table with an even number of seats, the seat directly opposite seat 1 is the one halfway around. With six seats, seat 1 faces seat 4, seat 2 faces seat 5 and seat 3 faces seat 6.
• Next to. This always means either side. A clue that says next to without a direction leaves two cases, and you should write both.

A clue such as A is not next to B is a negative clue. It removes options but rarely places anyone, so keep it until the end and use it to break a tie.

What is the step-by-step method?

1. Draw the seats and number them. Use the direction rules above, and draw a round table as a circle of small boxes.
2. Fix one person. On a round table, pick someone named in a clue that ties them to another person, and put them in seat 1. In a row, start with any clue that names an end or a seat number.
3. Place pairs that are locked together. Clues such as opposite or immediately to the left fix a second person as soon as the first is placed.
4. Split into cases when a clue allows two places. Write each case as its own small diagram. Two or three cases are easy to keep track of in a diagram.
5. Test the remaining clues against each case. Cross out any case that breaks a clue. Negative clues are most useful here.
6. Fill the last seats. When only one person and one seat are left, they belong together.
7. Check every clue once more. Read the clues in order against your final seating.

How does the method solve a six-seat round table?

Six people, A, B, C, D, E and F, sit around a round table with six equally spaced seats. Find where everyone sits.

1. A sits directly opposite D.
2. B sits next to A.
3. E sits directly opposite B.
4. F sits in the next seat clockwise from D.

Step one: fix A. Put A in seat 1 and number the other seats clockwise. Turning the table would not change the answer, so no possibilities are lost.

Step two: place D. Clue 1 puts D opposite A, so D is in seat 4.

Step three: split on B. Clue 2 says B is next to A, which allows seat 2 or seat 6. Draw both cases.

• Case one: B in seat 2. Clue 3 puts E opposite B, in seat 5. Seats 3 and 6 are still free.
• Case two: B in seat 6. Clue 3 puts E opposite B, in seat 3. Seats 2 and 5 are still free.

Step four: test the last clue. Clue 4 puts F in the seat just clockwise from D, which is seat 5. In case one, seat 5 already holds E, so case one fails. In case two, seat 5 is free, so F takes it.

Step five: fill the last seat. Only C and seat 2 remain, so C sits in seat 2.

Answer: going clockwise from A, the order is A, C, E, D, F, B. A faces D, C faces F, and E faces B.

Notice that every clue did real work. Without clue 4 there would be four different seatings, and without any one of the other clues the answer would not be unique either. A careful setter checks this, and so can you: if a clue never changed your diagram, reread it.

What changes when nobody may sit in their own seat?

Some puzzles add a rule about assigned seats. Each person has a seat with their own letter on it, but after a break nobody goes back to their own seat. An arrangement like this, where nothing stays in its original place, is called a derangement.

A paper published on 4 March 2009 counted derangements, defined there as arrangements with no fixed point, under extra conditions on where the order is allowed to go down ^[1].

The same paper notes that if an arrangement is picked at random, the chance that it leaves nothing in its original place is about 1 in e, a little over a third ^[1]. In puzzle terms, the own-seat rule on its own rules out a large share of seatings before any other clue is read.

The counts grow quickly. The paper also gives the well-known rule that the count for n items equals n minus 1 times the sum of the counts for the two smaller sizes ^[1]. The table below was worked out by listing every arrangement directly, and it agrees with that rule.

For a solver, the useful habit is simple. When the own-seat rule is in play, write each person's forbidden seat inside its box before you begin. It works like a pre-filled X on a grid, and it often settles a case split on its own.

How do several sittings around round tables work?

A harder kind of seating problem asks for a whole plan rather than a single seating. A group meets for several sittings at round tables, and the aim is that every pair of people sits side by side exactly once over the whole plan.

For five people at one table, two sittings are enough. Seat them A, B, C, D, E around the table the first time, and A, C, E, B, D the second time. The first sitting makes the neighbour pairs AB, BC, CD, DE and EA, and the second makes AC, CE, EB, BD and DA. Those ten pairs are every possible pair, each used once.

For seven people at one table, three sittings do the same job: A B C D E F G, then A C E G B D F, then A D G C F B E.

The number of people must be odd. At every sitting each person gains two neighbours, so the number of other people each one must meet has to be even.

A harder version uses several tables of fixed sizes. In graph language, the general problem asks how to split the complete network on an odd number of points into identical sets of cycles ^[2]. Read each point as a person and each cycle as one round table: one split is one sitting, and a line between two points means those two sat side by side.

A paper published on 15 November 2005 looked at the three-table case in which two of the tables are the same size ^[2]. It notes that tables of three, three and five seats can never work ^[2]. With those tables, eleven people would need five sittings to meet every pair once, and the paper records that no plan exists for that case ^[2].

The paper shows that, apart from that case, a cyclic solution exists whenever the two equal tables each seat from three to thirteen people and the third table seats an odd number that is at least five ^[2].

Cyclic here means the whole plan comes from one starting seating. Every later sitting is made by adding the same number to every label, while one special person stays in place.

What are the most common mistakes?

• Forgetting that turns do not count. On a round table, two diagrams that differ only by a turn are the same answer. Fixing one person at the start avoids this trap.
• Mixing up left and right. Always read them from the seated person's point of view, and write the rule beside the diagram.
• Reading next to as on the left. Next to allows both sides unless the clue says otherwise.
• Dropping a case too early. Keep every case until a clue breaks it.
• Forgetting the own-seat rule. When it applies, it is a clue like any other and should be marked on the diagram.

What should you do when you get stuck?

• Find the person mentioned in the most clues and place them first.
• Look for opposite and immediately clues, since they lock two seats together.
• Write out every case for the next clue, even if there are three or four.
• If a clue has not been used yet, ask what it would forbid in each case.
• If everything seems to fit two ways, look for a negative clue you have not applied.

Can you solve this practice puzzle?

Five people, A, B, C, D and E, have assigned seats in a straight row, all facing the same way, numbered 1 to 5 from their left to their right. A owns seat 1, B owns seat 2, C owns seat 3, D owns seat 4 and E owns seat 5. After a break, they all sit down again, and nobody sits in their own seat.

1. E sits at one end of the row.
2. C sits in the seat numbered exactly one higher than D's seat.

Where does everyone sit? Draw five boxes, mark each forbidden seat, and try it before reading the answer.

Answer: seat 1 holds E, seat 2 holds A, seat 3 holds D, seat 4 holds C and seat 5 holds B.

Here is why. Clue 2 means D and C sit in two touching seats, with D in the lower-numbered one. D may not use seat 4 and C may not use seat 3, so the pair can only be seats 1 and 2 or seats 3 and 4.

If D and C took seats 1 and 2, E would need the other end, seat 5, which is E's own seat. So D sits in seat 3 and C in seat 4. E then takes the free end, seat 1.

Seats 2 and 5 are left for A and B, and B may not use seat 2, so B sits in seat 5 and A in seat 2.

Only two clues were needed, because the own-seat rule did most of the work. Now try a live riddle that asks which fruit one person has. Draw a small table and rule out what cannot be true.