Eight persons A, B, C, D, E, F, G and H live on different floors (but not necessarily in the same order) of a four-floor building where ground floor is numbered as 1 and the topmost floor is numbered as 4. Each floor has two flats i.e., flat 1 and flat 2. Flat 1 is in the west of Flat 2. Flat 1 of floor 2 is just above the flat 1 of floor 1 and just below the flat 1 of floor 3. Similarly, flat 2 of floor 2 is just above the flat 2 of floor 1 and just below the flat 2 of floor 3.

Two floors gap between B and G. Both B and G lives in odd numbered flat. One floor gap between G and A. G and A live in the same flat. H lives in flat 2 but not on the floor of A and B. D lives just below the floor of H but in different flats. C lives on an even numbered floor but above the floor of F.

What is the direction of A with respect to F?

Correct Answer: North-west

Let us step-by-step analyze the given floor puzzle constraints to determine the exact positions of all eight persons (A, B, C, D, E, F, G, and H) and then find the direction of A with respect to F.

Building Structure:
The building has 4 floors (numbered 1 to 4 from bottom to top).
Each floor has 2 flats: Flat 1 (West) and Flat 2 (East).

Step 1: Determine the positions of B and G
- "Two floors gap between B and G."
This means one of them lives on Floor 4 and the other lives on Floor 1 (since 4 - 1 - 1 = 2 floors gap).
- "Both B and G live in an odd numbered flat."
The only odd-numbered flat is Flat 1. So both B and G live in Flat 1 on Floors 1 and 4.

Step 2: Determine the position of A
- "One floor gap between G and A. G and A live in the same flat."
Since G is in Flat 1 on either Floor 4 or Floor 1:
Case 1: If G is on Floor 4 (Flat 1), then A must be on Floor 2 (Flat 1). Then B is on Floor 1 (Flat 1).
Case 2: If G is on Floor 1 (Flat 1), then A must be on Floor 3 (Flat 1). Then B is on Floor 4 (Flat 1).

Step 3: Analyze the positions of H and D
- "H lives in flat 2 but not on the floor of A and B."
- "D lives just below the floor of H but in different flats."
Since H is in Flat 2 and D is in a different flat just below H's floor, D must be in Flat 1.
Let us test the cases:

Testing Case 1: G = Floor 4, Flat 1; A = Floor 2, Flat 1; B = Floor 1, Flat 1.
- A is on Floor 2, B is on Floor 1, G is on Floor 4.
- H cannot be on the floor of A (Floor 2) or B (Floor 1). So H must be on Floor 3, Flat 2.
- D lives just below H's floor (Floor 3), so D must be on Floor 2, in a different flat than H (i.e., Flat 1).
- But Floor 2, Flat 1 is already occupied by A! So Case 1 leads to a conflict.

Testing Case 2: G = Floor 1, Flat 1; A = Floor 3, Flat 1; B = Floor 4, Flat 1.
- Floors of A (Floor 3) and B (Floor 4) mean H cannot be on Floor 3 or Floor 4.
- H lives in Flat 2, so H must be on Floor 2, Flat 2.
- D lives just below H's floor (Floor 2) in a different flat (Flat 1), so D must be on Floor 1, Flat 1.
- But Floor 1, Flat 1 is occupied by G! This means G must be on Floor 4 and B on Floor 1.
Wait, let's re-evaluate:

Let's carefully check D's position:
If B is on Floor 4, Flat 1 and G is on Floor 1, Flat 1:
- A is on Floor 3, Flat 1 (1 floor gap from G at Floor 1? No, 1 floor gap between Floor 1 and Floor 3 is correct: Floor 2 is between them).
- H lives on Flat 2, not on floor of A (3) or B (4). So H can be on Floor 2, Flat 2 or Floor 1, Flat 2.
If H is on Floor 2, Flat 2: D is just below H's floor (so D is on Floor 1) in a different flat (Flat 1). But G is at Floor 1, Flat 1. Conflict!
If H is on Floor 4, Flat 2: Not allowed (H not on floor of B).
If H is on Floor 3, Flat 2: Not allowed (H not on floor of A).

Let's re-check Case 1 with B on Floor 4, Flat 1 and G on Floor 1, Flat 1 vs B on Floor 1, Flat 1 and G on Floor 4, Flat 1:
If B = Floor 1, Flat 1 and G = Floor 4, Flat 1:
- A is 1 floor gap from G in same flat (Flat 1) -> A is on Floor 2, Flat 1.
- Floors of A and B are 2 and 1. Thus H (in Flat 2) cannot be on Floor 1 or 2.
- So H can be on Floor 3, Flat 2 or Floor 4, Flat 2.
- But H cannot be on floor of B (Floor 1), so Floor 3 or 4 are open. But wait! Can H be on Floor 4? Is B on Floor 4? No, in this setup B is on Floor 1.
Wait, G is on Floor 4, so is H allowed on Floor 4? Yes! The rule says "H lives in flat 2 but not on the floor of A and B." It doesn't restrict H from G's floor!

Let's check H on Floor 3, Flat 2:
- D lives just below H's floor (Floor 3), so D is on Floor 2, in Flat 1 (different flat from H).
- But A is on Floor 2, Flat 1. Conflict!

Let's check H on Floor 4, Flat 2:
- D lives just below H's floor (Floor 4), so D is on Floor 3, in Flat 1 (different flat from H).
- Floor 3, Flat 1 is vacant! So D = Floor 3, Flat 1.

Now let's place C and F:
- "C lives on an even numbered floor but above the floor of F."
The even floors are Floor 4 and Floor 2.
Vacant positions right now:
- Floor 4, Flat 1: G
- Floor 4, Flat 2: H
- Floor 3, Flat 1: D
- Floor 3, Flat 2: Vacant
- Floor 2, Flat 1: A
- Floor 2, Flat 2: Vacant
- Floor 1, Flat 1: B
- Floor 1, Flat 2: Vacant

Even numbered floors with vacant spots:
- Floor 2, Flat 2 is vacant! So C must be on Floor 2, Flat 2.
- C is above the floor of F, so F must be on Floor 1, Flat 2.
- The remaining person E takes Floor 3, Flat 2.

Final Layout Table:

Floor 4: Flat 1 = G, Flat 2 = H
Floor 3: Flat 1 = D, Flat 2 = E
Floor 2: Flat 1 = A, Flat 2 = C
Floor 1: Flat 1 = B, Flat 2 = F

Step 4: Determine the direction of A with respect to F
- A is located on Floor 2, Flat 1.
- F is located on Floor 1, Flat 2.
- Floor 2 is North of Floor 1.
- Flat 1 is West of Flat 2.
Therefore, relative to F (Floor 1, Flat 2), A (Floor 2, Flat 1) is to the North-west direction.