Solving the Circular Seating Arrangement Puzzle

This problem requires us to determine the seating arrangement of nine people (A, B, C, D, E, F, G, H, and K) around a circular table, all facing the center. By carefully following the given clues and using logical deduction, we can find the exact position of each person and then identify which group of three is sitting together.

Step-by-Step Deduction of the Seating Arrangement

Let's deduce the positions one clue at a time. We can visualize the circle with 9 positions or number the positions from 1 to 9 in a clockwise manner to keep track of the arrangement. Let's assume A is at position 1 to start.

• "A is third to the left of H": In a circular arrangement facing the center, 'left' is the counter-clockwise direction and 'right' is the clockwise direction. If A is third to the left of H, it means there are two people between H and A when counting in the counter-clockwise direction from H to A. This is equivalent to H being third to the right of A (counting clockwise from A). Starting with A at position 1 and counting 3 positions clockwise, we get position 1 → 2 → 3 → 4. So, H is at position 4.
• "H is fourth to the left of G": H is at position 4. Fourth to the left of G means there are three people between G and H when counting counter-clockwise from G to H. This means G is fourth to the right of H. Starting from H at position 4 and counting 4 positions clockwise: 4 → 5 → 6 → 7 → 8. So, G is at position 8.
• "D is second to the right of A": A is at position 1. Second to the right means there is one person between A and D when counting clockwise from A. Counting 2 positions clockwise from A at 1: 1 → 2 → 3. So, D is at position 3.
• "C is fourth to the right of D": D is at position 3. Counting 4 positions clockwise from D: 3 → 4 → 5 → 6 → 7. So, C is at position 7.
• "E is third to the left of G": G is at position 8. Third to the left means there are two people between G and E when counting counter-clockwise from G to E. Counting 3 positions counter-clockwise from G at 8: 8 → 7 → 6 → 5. So, E is at position 5.

At this point, we have placed the following people and their positions (assuming A is at 1, clockwise):

• A: Position 1
• D: Position 3
• H: Position 4
• E: Position 5
• C: Position 7
• G: Position 8

The occupied positions are 1, 3, 4, 5, 7, 8. The empty positions are 2, 6, 9. The remaining people to place are B, F, and K.

"B is fourth to the right of K": We need to place K and B in the remaining empty spots (2, 6, and 9) such that B is four positions clockwise away from K. Let's test the empty spots for K:

• If K is at position 2, counting 4 positions clockwise: 2 → 3 → 4 → 5 → 6. This means B would be at position 6. Both positions 2 and 6 are empty, so this is a valid placement. The remaining person, F, would then occupy the last empty position, 9.
• If K is at position 6, counting 4 positions clockwise: 6 → 7 → 8 → 9 → 1. This means B would be at position 1, but position 1 is already occupied by A. This doesn't work.
• If K is at position 9, counting 4 positions clockwise and wrapping around: 9 → 1 → 2 → 3 → 4. This means B would be at position 4, but position 4 is already occupied by H. This doesn't work.

Therefore, the only possible placement for K and B in the remaining spots is K at position 2 and B at position 6. F is at position 9.

Final Circular Arrangement

Combining all the positions, starting from position 1 and moving clockwise around the circle, the final seating arrangement is:

A (1) → K (2) → D (3) → H (4) → E (5) → B (6) → C (7) → G (8) → F (9)

Visualizing the circle sequence: A – K – D – H – E – B – C – G – F – (loops back to A)

Checking the Options for Persons Sitting Together

Now we need to check which of the given combinations consists of three people sitting consecutively in our final arrangement.

• Option 1: KEB – In the arrangement (A-K-D-H-E-B-C-G-F-A), K is at position 2, E is at position 5, and B is at position 6. These three are not sitting together.
• Option 2: DHE – In the arrangement, D is at position 3, H is at position 4, and E is at position 5. These three people (D, H, E) are sitting next to each other in sequence (D → H → E) in the circle.
• Option 3: BCK – In the arrangement, B is at position 6, C is at position 7, and K is at position 2. These three are not sitting together.
• Option 4: HCG – In the arrangement, H is at position 4, C is at position 7, and G is at position 8. These three are not sitting together.

Based on our final seating arrangement, the combination DHE consists of three persons sitting together consecutively.

Revision Table: Key Concepts in Seating Arrangements

Let's quickly review some essential terms used in seating arrangement problems.

Additional Information: Strategies for Seating Arrangement Problems

Solving seating arrangement puzzles effectively involves a systematic approach:

• Read All Clues First: Get an overview of the constraints.
• Start with a Fixed Position: If a person's position is given relative to the table (e.g., "A is at the North end"), start there. In circular problems facing the center, you can pick an arbitrary starting position for the first person mentioned (like A at position 1 here) as relative positions are key.
• Use Direct Clues: Place individuals based on clear relationships (e.g., "B is second to the right of A").
• Connect Clues: Use people who appear in multiple clues to link different parts of the arrangement.
• Handle Negative Clues: Clues like "X is not next to Y" help eliminate possibilities. (Not present in this specific problem).
• Deal with Unplaced People: Use the remaining clues to fit the remaining people into the empty spots. Often, a final clue will connect a few unplaced people.
• Verify the Final Arrangement: Once you have a complete arrangement, quickly check if it satisfies every single clue given in the problem statement.
• Check the Question: Ensure you answer exactly what is asked, whether it's relative positions, who is opposite whom, or who is sitting together.

Practice is key to mastering these types of logical reasoning questions, which are common in many competitive exams.