Solving the Circular Seating Arrangement Puzzle

This problem involves arranging seven people (D, E, F, G, I, J, K) around a circular table facing the centre and determining a specific position based on given clues.

Analyzing the Clues

We represent the seating arrangement clockwise. Since they face the centre, 'left' corresponds to the clockwise direction.

• Clue 1: E sits to the immediate left of I. This means the clockwise order is I followed by E. (Sequence: \(I E\))
• Clue 2: F sits to the immediate left of D. This means the clockwise order is D followed by F. (Sequence: \(D F\))
• Clue 3: D sits second to the left of K. This means K is followed by one person, then D, in clockwise order. (Sequence: \(K _ D\))
• Clue 4: G sits third to the left of I. This means I is followed by two people, then G, in clockwise order. (Sequence: \(I _ _ G\))

Step-by-Step Arrangement

1. Combine Clues 2 and 3: We have \(K _ D\) and \(D F\). Combining these gives the sequence \(K _ D F\) (clockwise). This sequence involves 4 people.
2. Combine Clues 1 and 4: We have \(I E\) and \(I _ _ G\). Since E is immediately to the left (clockwise) of I, E must be the first person after I. The sequence becomes \(I E _ G\) (clockwise). This sequence also involves 4 people.
3. Identify the Missing Person: The people involved in the sequences are D, E, F, G, I, K. The missing person is J.
4. Fit the Sequences: We need to arrange all 7 people. Let's use the sequence \(I E _ G\) and place I at a starting point. Let the positions be numbered 1 to 7 clockwise.
   • Place I at position 1.
   • E is at position 2 (immediate left).
   • G is at position 4 (third left).
   • The sequence is: \(I E _ G _ _ _\) (Positions 1, 2, 3, 4, 5, 6, 7)
   • The person at position 3 is the blank in \(I E _ G\). This must be J. Arrangement: \(I E J G _ _ _\)
5. Place Remaining People: We need to place K, D, F in positions 5, 6, 7 using the sequence \(K _ D F\).
   • Test K at position 5: \(I E J G K _ _\). D must be second left of K. Clockwise from K(5): Seat 6, Seat 7. D would be at Seat 7. \(I E J G K _ D\). F must be immediate left of D, so F is at Seat 5. But K is there. This placement fails.
   • Test K at position 6: \(I E J G _ K _\). D must be second left of K. Clockwise from K(6): Seat 7, Seat 1. D would be at Seat 1. But I is there. This placement fails.
   • Test K at position 7: \(I E J G _ _ K\). D must be second left of K. Clockwise from K(7): Seat 1, Seat 2. D would be at Seat 2. But E is there. This placement fails.
6. Re-evaluate Placement: Let's try placing K first using \(K _ D F\).
   • Place K at position 1.
   • Sequence \(K _ D F\) means D is at position 3, F is at position 4. Position 2 is unknown. Arrangement: \(K _ D F _ _ _\)
7. Integrate the second sequence: We need to fit \(I E _ G\) using the remaining seats (2, 5, 6, 7) and people (E, G, I, J).
   • Place I at position 6: \(K _ D F _ I _\).
   • E is immediate left of I, so E is at position 7: \(K _ D F _ I E\).
   • G is third left of I. Clockwise from I(6): Seat 7 (E), Seat 1 (K), Seat 2. G is at position 2. Arrangement: \(K G D F _ I E\).
   • Now check the remaining constraint: D is second left of K. Clockwise from K(1): Seat 2 (G), Seat 3 (D). This matches: \(K G D\).
   • F is immediate left of D: Clockwise from D(3): F is at Seat 4. This matches: \(D F\).
   • The only person left is J, and the only seat left is position 5.
   • Final Clockwise Arrangement: \(K G D F J I E\)

Finding the Final Answer

The arrangement (clockwise) is K, G, D, F, J, I, E.

The question asks: Who sits third to the right of J?

Since they face the centre, 'right' means anti-clockwise.

• J is at position 5.
• The person first to the right (anti-clockwise) of J (pos 5) is F (pos 4).
• The person second to the right of J (pos 5) is D (pos 3).
• The person third to the right of J (pos 5) is G (pos 2).

Therefore, G sits third to the right of J.