Solving the Square Table Seating Arrangement Puzzle

This problem involves a seating arrangement puzzle around a square table. Eight people are seated with specific conditions regarding their position (corner or middle of a side) and facing direction (center or outside). We need to deduce the seating arrangement first to answer the question about the grouping.

Let's break down the information and build the arrangement step-by-step:

• There are 8 seats: 4 at corners and 4 in the middle of the sides.
• People at corners face the center.
• People in the middle of sides face outside.

Let's assign seat types and directions. We can label seats 1 through 8 around the table clockwise. Let's assume Corner seats are 1, 3, 5, 7 (facing In) and Middle seats are 2, 4, 6, 8 (facing Out). This setup ensures corners and middles alternate.

• A sits second to the right of C. If C is at seat X, A is at seat X+2 (moving clockwise).
• Two people sit between A and E. On an 8-seat arrangement, having two people between A and E in one direction means they are 3 positions apart. So if A is at seat Y, E is at seat Y+3 or Y-3 (adjusting for wrapping around the table).
• E's immediate neighbours are facing the centre of the table. People facing the center are at Corner seats. This means E's neighbours are at Corner seats. For E to have Corner neighbours, E must be at a Middle seat.

From the condition that E must be at a Middle seat, let's revisit A and C's positions.

• If E is at a Middle seat (2, 4, 6, or 8), and E is 3 positions away from A, A must also be at a position that is 3 positions away from a Middle seat. 3 positions away from a Middle seat (e.g., 2) is 2+3=5 (Corner) or 2-3=-1+8=7 (Corner). So A must be at a Corner seat.
• If A is at a Corner seat (1, 3, 5, or 7), and A is 2nd right of C, then C must be at a position from which A is 2nd right. If A is at seat Y (Corner), C must be at seat Y-2. Y-2 from a Corner seat will be a Corner seat (e.g., 3-2=1, if 3 & 1 are Corners). So C must also be at a Corner seat.

So, C and A are at Corner seats (facing In), and E is at a Middle seat (facing Out).

Let's place C at seat 1 (Corner, In). A sits second to the right of C: Seat 1 > Seat 2 (Middle) > Seat 3 (Corner). A is at seat 3 (Corner, In).

Two people sit between A (at 3) and E. E is 3 positions away from A. 3 + 3 = 6. 3 - 3 = 0 (or 8). So E is at seat 6 or seat 8. Both seats 6 and 8 are Middle seats (Out), which fits the condition that E is at a Middle seat.

• If E is at seat 6 (Middle, Out), its neighbours are seat 5 (Corner, In) and seat 7 (Corner, In). Both face Center. This works.
• If E is at seat 8 (Middle, Out), its neighbours are seat 7 (Corner, In) and seat 1 (Corner, In). Both face Center. This works.

We need to continue placing others to determine if E is at 6 or 8.

• G sits opposite to H. Opposite pairs on an 8-seat square table are (1,5), (2,6), (3,7), (4,8). G and H must occupy one of these pairs.
• H sits immediately to the left of F. Left means anti-clockwise. If F is at seat X, H is at seat X-1 (moving anti-clockwise). Equivalently, F is immediately to the right (clockwise) of H.
• B is neither a neighbour of A nor E.

Let's take Case 1: E is at seat 6 (Middle, Out). Current fixed positions: C(1, C, In), A(3, C, In), E(6, M, Out).

Opposite pairs remaining: (1,5), (2,6), (3,7), (4,8). Fixed people occupy: C(1) opp 5, A(3) opp 7, E(6) opp 2. The only remaining opposite pair is (4,8). Both seats 4 and 8 are Middle seats (Out).

G and H must occupy seats 4 and 8. So G and H are both at Middle seats.

H is immediately left (anti-clockwise) of F. The seats are 1-2-3-4-5-6-7-8. Anti-clockwise is 8-7-6-5-4-3-2-1. If H is at 4, F must be at 5. If H is at 8, F must be at 1.

• If H is at 4 (M, Out), F must be at 5 (C, In). This is possible. G is opposite H, so G is at 8 (M, Out).
• If H is at 8 (M, Out), F must be at 1 (C, In). But C is at seat 1. This is not possible.

So, we must have H at 4 (M, Out), F at 5 (C, In), and G at 8 (M, Out).

Current arrangement:

• Seat 1 (Corner, In): C
• Seat 2 (Middle, Out): ?
• Seat 3 (Corner, In): A
• Seat 4 (Middle, Out): H
• Seat 5 (Corner, In): F
• Seat 6 (Middle, Out): E
• Seat 7 (Corner, In): ?
• Seat 8 (Middle, Out): G

Remaining seats: 2 (Middle, Out) and 7 (Corner, In). Remaining people: B and D.

B is neither a neighbour of A nor E.

• A is at seat 3. Its neighbours are seat 2 and seat 4 (H). So B cannot be at seat 2.
• E is at seat 6. Its neighbours are seat 5 (F) and seat 7. So B cannot be at seat 7.

The remaining seats are 2 and 7. B cannot be at 2 and cannot be at 7. This means B cannot be placed in this arrangement. So Case 1 (E at 6) is incorrect.

Let's take Case 2: E is at seat 8 (Middle, Out). Current fixed positions: C(1, C, In), A(3, C, In), E(8, M, Out).

Opposite pairs remaining: (1,5), (2,6), (3,7), (4,8). Fixed people occupy: C(1) opp 5, A(3) opp 7, E(8) opp 4. The only remaining opposite pair is (2,6). Both seats 2 and 6 are Middle seats (Out).

G and H must occupy seats 2 and 6. So G and H are both at Middle seats.

H is immediately left (anti-clockwise) of F. If H is at 2, F must be at 3. If H is at 6, F must be at 7.

• If H is at 2 (M, Out), F must be at 3 (C, In). But A is at seat 3. This is not possible.
• If H is at 6 (M, Out), F must be at 7 (C, In). This is possible. G is opposite H, so G is at 2 (M, Out).

So, we must have H at 6 (M, Out), F at 7 (C, In), and G at 2 (M, Out).

Current arrangement:

• Seat 1 (Corner, In): C
• Seat 2 (Middle, Out): G
• Seat 3 (Corner, In): A
• Seat 4 (Middle, Out): ?
• Seat 5 (Corner, In): ?
• Seat 6 (Middle, Out): H
• Seat 7 (Corner, In): F
• Seat 8 (Middle, Out): E

Remaining seats: 4 (Middle, Out) and 5 (Corner, In). Remaining people: B and D.

B is neither a neighbour of A nor E.

• A is at seat 3. Its neighbours are seat 2 (G) and seat 4. So B cannot be at seat 4.
• E is at seat 8. Its neighbours are seat 7 (F) and seat 1 (C). So B cannot be at seat 1 or seat 7 (which are already taken). The remaining seats 4 and 5 are not neighbours of E.

B cannot be at seat 4. The remaining seats are 4 and 5. Therefore, B must be at seat 5.

If B is at seat 5 (Corner, In), then the last remaining person, D, must be at seat 4 (Middle, Out).

Let's verify B's condition with the final placement: B is at 5. Neighbours are 4 (D) and 6 (H). A is at 3. E is at 8. B at 5 is not a neighbour of A (at 3) or E (at 8). This final arrangement works.

Final Seating Arrangement Summary:

• Seat 1 (Corner, In): C
• Seat 2 (Middle, Out): G
• Seat 3 (Corner, In): A
• Seat 4 (Middle, Out): D
• Seat 5 (Corner, In): B
• Seat 6 (Middle, Out): H
• Seat 7 (Corner, In): F
• Seat 8 (Middle, Out): E

People at Corner seats (facing In): C, A, B, F

People at Middle seats (facing Out): G, D, H, E

Analyzing the Groups

The question asks which pair among the options does not belong to the group formed by four of the options. Let's look at the seat types of the people in each pair:

• G - E: G is at a Middle seat, E is at a Middle seat. (Middle - Middle)
• F - E: F is at a Corner seat, E is at a Middle seat. (Corner - Middle)
• G - C: G is at a Middle seat, C is at a Corner seat. (Middle - Corner)
• A - D: A is at a Corner seat, D is at a Middle seat. (Corner - Middle)
• B - D: B is at a Corner seat, D is at a Middle seat. (Corner - Middle)

Options 2, 3, 4, and 5 all consist of a pair where one person is seated at a Corner and the other is seated at a Middle. This forms the group.

Option 1 consists of a pair where both people are seated at Middle seats. This pair does not fit the pattern of the other four pairs.

Therefore, G - E does not belong to the group.