Logical Puzzles: Seating and Arrangement

Seating and spatial arrangement puzzles are a distinctive variant of logical puzzles in the UCAT Decision Making subtest. They differ from linear ordering problems because they involve circular tables or two-dimensional layouts where adjacency, opposition, and relative position work differently. This lesson teaches you the specific techniques needed for these spatial puzzles.

---

Circular vs Linear Arrangements

The key differences between circular and linear arrangements are:

Feature	Linear	Circular
Endpoints	Has a first and last position	No endpoints — every seat is equivalent
Adjacency	End seats have 1 neighbour; middle seats have 2	Every seat has exactly 2 neighbours
"Opposite"	Not applicable	The seat directly across the table
Rotational equivalence	Position 1 is different from position 2	Rotating everyone by one seat gives an equivalent arrangement

Rotational Equivalence

In a circular arrangement, there is no "position 1." Arrangements that are rotations of each other are considered identical. To handle this, always fix one person's position as an anchor and then determine everyone else's position relative to them.

UCAT Tip: When you see a circular arrangement question, immediately fix one person (usually the first one mentioned or the most constrained) in a position. This eliminates the confusion of rotational equivalence and makes the problem equivalent to a linear arrangement with wraparound.

---

Circular Table Setup

For a table with N seats, label the seats 1 through N going clockwise. Key relationships:

For 6 seats:

Seat	Opposite seat	Left neighbour	Right neighbour
1	4	6	2
2	5	1	3
3	6	2	4
4	1	3	5
5	2	4	6
6	3	5	1

For 4 seats:

Seat	Opposite seat	Left neighbour	Right neighbour
1	3	4	2
2	4	1	3
3	1	2	4
4	2	3	1

---

Worked Example 1: Six People at a Round Table

Six colleagues — Anna, Ben, Clara, David, Emma, and Faisal — sit around a circular table with six seats. The following constraints apply:

• Anna sits directly opposite David
• Ben sits next to Anna
• Clara does not sit next to Emma
• Faisal sits next to David

Question: Who sits directly opposite Ben?

Step 1: Fix Anna's position.

Let Anna = Seat 1. Then David = Seat 4 (opposite).

1	2	3	4	5	6
Anna	_	_	David	_	_

Step 2: Ben sits next to Anna.

Ben = Seat 2 or Seat 6.

Step 3: Faisal sits next to David.

Faisal = Seat 3 or Seat 5.

Step 4: Try Ben = Seat 2.

1	2	3	4	5	6
Anna	Ben	_	David	_	_

Faisal = Seat 3 or Seat 5. Remaining people: Clara, Emma, Faisal.

Sub-case 4a: Faisal = Seat 3. Remaining: Clara, Emma for Seats 5, 6.

1	2	3	4	5	6
Anna	Ben	Faisal	David	?C/E	?E/C

Check: Clara does not sit next to Emma. Seats 5 and 6 are adjacent. So if Clara = 5 and Emma = 6, they are next to each other → violates the constraint. If Clara = 6 and Emma = 5, they are still next to each other (seat 5 and 6 are adjacent) → violates the constraint.

Sub-case 4a fails. ✗

Sub-case 4b: Faisal = Seat 5. Remaining: Clara, Emma for Seats 3, 6.

1	2	3	4	5	6
Anna	Ben	?C/E	David	Faisal	?E/C

Check: Clara does not sit next to Emma.

• If Clara = 3, Emma = 6: Clara's neighbours are Ben (2) and David (4). Emma's neighbours are Faisal (5) and Anna (1). Clara and Emma are not adjacent. ✓
• If Clara = 6, Emma = 3: Clara's neighbours are Faisal (5) and Anna (1). Emma's neighbours are Ben (2) and David (4). Clara and Emma are not adjacent. ✓

Both work! But the question asks who sits opposite Ben. Ben = Seat 2. Opposite of Seat 2 = Seat 5 = Faisal.

Step 5: Verify with Ben = Seat 6 (the other possibility).

1	2	3	4	5	6
Anna	_	_	David	_	Ben