Nine persons – A, B, C, D, E, F, G, H, and I – are sitting around a circular table, all facing towards the center. But not necessarily in the same order.

Only one person sits between A and F. C sits third to the right B, who is an immediate neighbor of A. Only three people sit between A and G. G sits second to the left of H. I is an immediate neighbor of both A and H. F is not an immediate neighbor of D.

Four of the following five are alike in a certain way and thus form a group. Find the one which doesn’t belong to the group.

To find the odd one out, let us first determine the seating arrangement of the nine persons (A, B, C, D, E, F, G, H, and I) sitting around a circular table facing the center.

Let us analyze the clues step-by-step:
1. "Only three people sit between A and G."
In a circular table with 9 people, having 3 people between A and G means G can be either 4 positions to the left of A or 4 positions to the right of A.
Let us place A at position 1 (numbered 1 to 9 clockwise). Since there are 3 people between A and G, G must be at position 5 (going clockwise: 2, 3, 4 are the 3 people, so G is 5; or going counter-clockwise: 9, 8, 7 are the 3 people, so G is 6). Let's trace both possibilities.

2. "I is an immediate neighbor of both A and H."
Since I is adjacent to both A and H, the sequence must be either H-I-A or A-I-H. This means H and A are separated by exactly one person, who is I.

3. "G sits second to the left of H."
Since everyone faces the center, left is clockwise and right is counter-clockwise (or vice versa depending on orientation). Let's define clockwise as left and counter-clockwise as right.
If G is second to the left of H, then H sits second to the right of G.
If we place G at position 5 (with A at 1):
H, being 2 positions to the left of G (so G is 2 positions to the left of H, which means H is 2 positions to the right of G, i.e., counter-clockwise): H would be at position 3.
If H is at position 3, since I must be between A (at 1) and H (at 3), I must be at position 2. This fits perfectly!
Let us verify the other direction. If G was at position 6 (3 people between A at 1 and G at 6 going counter-clockwise: 9, 8, 7 are between them):
If G is second to the left of H, H is second to the right of G, so H would be at position 4. But then I cannot be adjacent to both A (at 1) and H (at 4) because they are not separated by exactly one position. Thus, our first case is correct:
- A is at position 1
- I is at position 2
- H is at position 3
- G is at position 5

4. "C sits third to the right of B, who is an immediate neighbor of A."
Since B is an immediate neighbor of A, and A's neighbor at position 2 is already occupied by I, B must be A's other neighbor at position 9.
Since C sits third to the right of B (counter-clockwise is right): starting from B at 9, the first to the right is A (1), second is I (2), and third is H (3). But position 3 is already occupied by H. This means "right" must refer to the clockwise direction in this orientation, or we need to reverse our direction mapping.
Let's define the directions consistently:
Let's place the positions 1 to 9 in clockwise order:
- A is at 1.
- Since "I is an immediate neighbor of both A and H", we have two options for the placement of H and I:
- Case A: I at 2, H at 3.
- Case B: I at 9, H at 8.
Let's test Case A (A at 1, I at 2, H at 3):
- "G sits second to the left of H." Since they face the center, left is clockwise. So second to the left of H (3) is 5. Thus, G is at 5. This matches our previous placement.
- "B is an immediate neighbor of A." The neighbors of A (1) are 2 and 9. Since 2 is occupied by I, B must be at 9.
- "C sits third to the right of B." Facing the center, right is counter-clockwise. From B (9), going counter-clockwise (decreasing numbers): 1st right is 8, 2nd right is 7, 3rd right is 6. So C is at 6.
- Let's check: "Only three people sit between A and G." A is at 1, G is at 5. Between them clockwise are 2, 3, 4 (3 people: I, H, and the person at 4). Between them counter-clockwise are 9, 8, 7, 6 (4 people). This satisfies the condition "Only three people sit between A and G" (in one of the directions).

5. "Only one person sits between A and F."
The positions adjacent to the neighbors of A are 3 and 8. Position 3 is occupied by H. So F must be at position 8 (one person, B at 9, sits between A at 1 and F at 8).

6. "F is not an immediate neighbor of D."
The remaining positions to fill are 4 and 7. The remaining people are D and E.
F is at 8, so its neighbors are 7 and 9 (B). Since F is not an immediate neighbor of D, D cannot be at 7. Thus, D must be at 4, and E must be at 7.

Let's write down the final circular arrangement (clockwise):
1 - A
2 - I
3 - H
4 - D
5 - G
6 - C
7 - E
8 - F
9 - B

Now let's examine the options to find the relationship between the pairs of letters:
- B-E: B is at 9, E is at 7. There is exactly 1 person (F at 8) sitting between them.
- F-C: F is at 8, C is at 6. There is exactly 1 person (E at 7) sitting between them.
- C-D: C is at 6, D is at 4. There is exactly 1 person (G at 5) sitting between them.
- D-I: D is at 4, I is at 2. There is exactly 1 person (H at 3) sitting between them.
- H-B: H is at 3, B is at 9. There are 3 people between them in one direction (I, A, B is not possible, let's see: 4, 5, 6, 7, 8 are between 3 and 9; and 2, 1, 9: between 3 and 9 clockwise are 4, 5, 6, 7, 8 (5 people) and counter-clockwise are 2, 1 (2 people)). Therefore, H and B do not have exactly 1 person sitting between them.

In all the first four pairs, the two persons have exactly one person sitting between them. H-B is the only pair that does not follow this pattern.