Understanding the Seating Arrangement Problem

The problem describes eight people sitting around a square table. Key details are:

• Four people at the corners of the table are facing outside.
• Four people in the middle of each of the four sides are facing inside.
• A critical rule: "Two people who are facing the same direction (either inside or outside) are not sitting with each other." This implies an alternating pattern of facing directions around the table (e.g., Corner (Outside) - Middle (Inside) - Corner (Outside) - Middle (Inside), and so on). This alternating pattern is inherently satisfied by the table's structure where corner positions alternate with middle positions.

To solve this seating arrangement puzzle, we will use a numbered position system (1 to 8) around the table, moving clockwise. This helps in consistently tracking relative positions and directions.

Deducing Seating Positions Step-by-Step

Let's place each individual based on the given clues through logical deduction:

1. D's Position: The first clue states, "D is not facing inside." Given the table setup, people facing outside are at the corners. Therefore, D must be at a corner position and faces outside. We can arbitrarily place D at Position 3 (Top-Right Corner, Facing Outside) for our deductions. The final arrangement will be consistent regardless of the starting corner.
2. G's Position: "G is sitting fifth to the right of D."
   D is at Position 3, and faces Outside. For a person facing outside, "right" means moving in the clockwise direction around the table. Counting 5 positions clockwise from D (Position 3):
   \[ \text{Position 3 (D)} \xrightarrow{\text{1st right}} \text{Position 4} \xrightarrow{\text{2nd right}} \text{Position 5} \xrightarrow{\text{3rd right}} \text{Position 6} \xrightarrow{\text{4th right}} \text{Position 7} \xrightarrow{\text{5th right}} \text{Position 8} \] So, G is at Position 8. This is a Middle position, and consistent with the rule of facing Inside.
3. A's Position: "A is sitting just opposite to G."
   G is at Position 8 (Middle, Left). On a square table with 8 people, the person directly opposite Position 8 is Position 4 (Middle, Right).
   So, A is at Position 4. This is also a Middle position, consistent with facing Inside.
4. E's Position: "E is sitting third to the right of A."
   A is at Position 4, and faces Inside. For a person facing inside, "right" means moving in the anti-clockwise direction around the table. Counting 3 positions anti-clockwise from A (Position 4):
   \[ \text{Position 4 (A)} \xrightarrow{\text{1st right}} \text{Position 3} \xrightarrow{\text{2nd right}} \text{Position 2} \xrightarrow{\text{3rd right}} \text{Position 1} \] So, E is at Position 1. This is a Corner position, consistent with facing Outside.
5. B's Position: "B is sitting second to the left of E."
   E is at Position 1, and faces Outside. For a person facing outside, "left" means moving in the anti-clockwise direction. Counting 2 positions anti-clockwise from E (Position 1):
   \[ \text{Position 1 (E)} \xrightarrow{\text{1st left}} \text{Position 8} \xrightarrow{\text{2nd left}} \text{Position 7} \] So, B is at Position 7. This is a Corner position, consistent with facing Outside.

Finalizing the Seating Arrangement

At this stage, we have placed five individuals. The filled positions are:

• D: Position 3 (Corner, Out)
• G: Position 8 (Middle, In)
• A: Position 4 (Middle, In)
• E: Position 1 (Corner, Out)
• B: Position 7 (Corner, Out)

The remaining people to place are C, F, and H. The remaining vacant positions are Position 2 (Middle, In), Position 5 (Corner, Out), and Position 6 (Middle, In).

6. C's Position: "Neither A nor D is an immediate neighbour of C."
   
   • A is at Position 4. Its immediate neighbours are Position 3 (D) and Position 5 (Vacant Corner). So, C cannot be at Position 5.
   • D is at Position 3. Its immediate neighbours are Position 2 (Vacant Middle) and Position 4 (A). So, C cannot be at Position 2.
   Considering the vacant positions (2, 5, 6), and knowing C cannot be at 2 or 5, C must be at Position 6 (Middle, Facing Inside).
7. F's Position: "F is not sitting just opposite to C."
   C is at Position 6 (Middle, Bottom). The person opposite Position 6 is Position 2 (Middle, Top). Therefore, F cannot be at Position 2.
   The remaining vacant positions (after C is placed at 6) are Position 2 (Middle, In) and Position 5 (Corner, Out). Since F cannot be at Position 2, F must be at Position 5 (Corner, Facing Outside).
8. H's Position: The last remaining person is H. H must fill the last remaining vacant position.
   H is at Position 2 (Middle, Facing Inside).

Complete Seating Arrangement Summary

The final and complete seating arrangement is:

This arrangement satisfies all the given conditions:

• The corner positions (1, 3, 5, 7) are occupied by E, D, F, B, all facing outside.
• The middle positions (2, 4, 6, 8) are occupied by H, A, C, G, all facing inside.
• The alternating sequence of facing directions (Outside-Inside-Outside-Inside...) is maintained around the table.

Answering the Question: People Between B and F

The question asks: "How many people are sitting between B and F, when counted from the left of F?"

From our final arrangement, F is at Position 5 (Bottom-Right Corner, Facing Outside).

To count from the left of F, we move in the anti-clockwise direction from F, listing the people until we reach B:

• Starting from F (Position 5).
• Moving left (anti-clockwise): The person immediately to F's left is at Position 4, which is A.
• The next person to the left is at Position 3, which is D.
• The next person to the left is at Position 2, which is H.
• The next person to the left is at Position 1, which is E.
• The next person to the left is at Position 8, which is G.
• The next person to the left is at Position 7, which is B.

So, the people sitting between F and B when counted from the left of F are: A, D, H, E, G.

Counting these individuals, we find:

• A: 1st person
• D: 2nd person
• H: 3rd person
• E: 4th person
• G: 5th person

Therefore, there are Five people sitting between B and F when counted from the left of F.