Five persons, A, B, C, D and E, are sitting at each corner of a pentagonal table, facing the centre. C is to the right of A and is second to the left of D. E is not between A and D. Who is sitting third to the left of E?

Question

Prepp Team · Accepted Answer

Solving the Pentagonal Seating Arrangement Puzzle

This problem requires us to determine the seating arrangement of five persons, A, B, C, D, and E, around a pentagonal table and then identify the person sitting in a specific position relative to another.

The five persons are sitting at the corners of a pentagonal table, all facing the center. In a circular arrangement like this, "right" typically means the next person in the clockwise direction, and "left" means the next person in the counter-clockwise direction.

Step-by-Step Analysis of Clues:

C is to the right of A: This means if we start at A and move clockwise, the next person is C.
- Clockwise sequence starts: A → C → ...
C is second to the left of D: "Left" means counter-clockwise. If C is second to the left of D, then moving counter-clockwise from D, the first person is someone, and the second person is C. This is equivalent to saying C is second to the left of D, which means D is second to the right of C. Moving clockwise from C, the first person is someone, and the second person is D.
- Clockwise sequence fragment: C → (someone) → D → ...

Combining the first two clues, we can start building the clockwise sequence around the table. Since A → C and C → (someone) → D, the sequence must start like this:

A → C → (Position 3) → D → (Position 5) → A

There are five positions. A, C, and D are placed relative to each other. The people filling Position 3 and Position 5 must be the remaining two persons, B and E.

The clockwise arrangement so far is A, C, (B or E), D, (E or B).

E is not between A and D: This is a crucial constraint in a circular arrangement. "Between A and D" can refer to the person(s) on the path connecting A to D. In a pentagon, there are two paths between any two people. Let's look at the paths between A and D in our potential arrangement A → C → (Pos 3) → D → (Pos 5) → A:
- Clockwise path from A to D: A → C → (Pos 3) → D. The people "between" A and D are C and the person in Pos 3.
- Counter-clockwise path from A to D: A → (Pos 5) → D. The person "between" A and D is the person in Pos 5.

In a pentagon (5 positions), the clockwise path from A to D involves 3 steps (A to C, C to Pos 3, Pos 3 to D). The counter-clockwise path from A to D involves 2 steps (A to Pos 5, Pos 5 to D). The counter-clockwise path is shorter.

A common interpretation of "between A and D" in circular puzzles, especially when paths are of different lengths, refers to the person(s) on the shorter path. The shorter path here is A → (Pos 5) → D, and the person between A and D on this path is the one in Pos 5.

The constraint states "E is not between A and D". Under the interpretation that "between A and D" refers to the person on the shorter path (Pos 5), this means E is not in Position 5.

If E is not in Position 5, then Position 5 must be occupied by B.
Since Position 3 and Position 5 must be B and E, and Position 5 is B, then Position 3 must be E.

So, the clockwise arrangement is: A → C → E → D → B → A.

Let's verify this arrangement against all clues:

C right of A: Yes (A → C is clockwise).
C second left of D: Yes (From D, counter-clockwise: B is 1st left, C is 2nd left).
E is not between A and D: The shorter path A → D is A → B → D. The person between is B. E is not B. This is consistent.

Final Seating Arrangement (Clockwise):

Position	Person
1 (Start)	A
2 (Right of A)	C
3	E
4	D
5	B

Finding the Person Third to the Left of E:

We need to find the person sitting third in the counter-clockwise direction starting from E.

Starting from E in the counter-clockwise direction:

1st person left of E is C.
2nd person left of E is A.
3rd person left of E is B.

Therefore, B is sitting third to the left of E.

Revision Table: Key Clues and Deductions

Clue	Deduction / Implication
5 persons, pentagonal table, facing center	Circular arrangement, 5 positions. Right = Clockwise, Left = Counter-clockwise.
C is to the right of A	A → C (Clockwise)
C is second to the left of D	C → (someone) → D (Clockwise)
Combined 1 & 2	A → C → (Pos 3) → D → (Pos 5) → A (Clockwise). Pos 3, Pos 5 are B and E.
E is not between A and D	Interpreted as E is not on the shorter path A → D (which is A → Pos 5 → D). So, E is not in Pos 5.
Pos 5 is not E	Pos 5 must be B. Pos 3 must be E.
Final Arrangement (Clockwise)	A → C → E → D → B → A
Question: Who is third to the left of E?	From E, 3 steps counter-clockwise: E → C → A → B.

Additional Information: Understanding Circular Arrangements

Circular seating arrangement problems are common in logical reasoning. Key points to remember:

Direction: "Right" and "Left" are relative to the direction the people are facing (usually the center). Right means clockwise, left means counter-clockwise, or vice-versa, depending on the context. Always establish this first.
Relative Positioning: Clues define positions relative to others (e.g., "to the right of", "second to the left of"). Build the arrangement step-by-step using these relative positions.
"Between": In circular arrangements with more than 3 people, there are two paths between any two individuals. The interpretation of "between" can be crucial. It might mean:
- The person(s) on the shorter path.
- The person(s) on the longer path.
- The person(s) immediately adjacent to both, if possible (usually only for 3 people).
The context of the puzzle usually implies the intended meaning, often the shorter path unless specified otherwise.
Fixed vs. Relative: In these problems, the absolute positions don't matter, only the relative positions of people to each other. Once one person is placed, all others are fixed relative to them.