The problem requires determining who sits to the immediate right of K, given six people (K, M, N, P, Q, R) arranged around a circular table facing the centre. We need to evaluate if the provided statements I and II are sufficient individually or together to find the answer.

Statement I Analysis: Sufficiency for Seating Position

Let's analyze the information given in Statement I:

• Setup: Six people around a circular table, facing the centre.
• Condition 1: "K sits third to the left of M." In a circular arrangement of 6 people, being third to the left means K is directly opposite M. We can represent this mathematically. If we assign positions 1 to 6 clockwise, and M is at position 1, then the positions to the left are 6, 5, 4. So, K is at position 4. The arrangement looks like: M _ _ K _ _.
• Condition 2: "Only one person sits between K and N." This means N is two positions away from K (either clockwise or counter-clockwise).
• Condition 3: "R sits to the immediate left of N."

Combining Conditions:

1. From Condition 1, K is opposite M. Let's place M at position 1 and K at position 4 (clockwise): M(1), _(2), _(3), K(4), _(5), _(6).
2. Now consider Condition 2 and 3.
   • Scenario A: N is two positions clockwise from K. K is at 4, so N would be at 6. The arrangement is M(1), _(2), _(3), K(4), _(5), N(6). Condition 3 states "R sits to the immediate left of N". The immediate left of N(6) is position 5. So, R is at position 5. The arrangement becomes: M(1), _(2), _(3), K(4), R(5), N(6). The remaining people are P and Q, who must occupy positions 2 and 3. The arrangement is M, ?, ?, K, R, N.
   • Scenario B: N is two positions counter-clockwise from K. K is at 4, so N would be at 2. The arrangement is M(1), N(2), _(3), K(4), _(5), _(6). Condition 3 states "R sits to the immediate left of N". The immediate left of N(2) is position 1. But M is already at position 1. This scenario creates a contradiction. Therefore, Scenario B is not possible.
3. Revisiting Scenario A: M(1), _(2), _(3), K(4), R(5), N(6). The people remaining are P and Q. Let's check the positions. The arrangement clockwise is M, P, Q, K, R, N OR M, Q, P, K, R, N.
   • Test M, P, Q, K, R, N: M(1), P(2), Q(3), K(4), R(5), N(6).
     • K is third left of M? Yes (M->N->R->K).
     • One person between K and N? Yes (R is between K and N).
     • R is immediate left of N? Yes (N is at 6, R is at 5).
     • Who sits immediate right of K? R sits to the immediate right of K(4).
   • Test M, Q, P, K, R, N: M(1), Q(2), P(3), K(4), R(5), N(6).
     • K is third left of M? Yes (M->N->R->K).
     • One person between K and N? Yes (R is between K and N).
     • R is immediate left of N? Yes (N is at 6, R is at 5).
     • Who sits immediate right of K? R sits to the immediate right of K(4).
   In both valid sub-scenarios derived from A, R sits to the immediate right of K.

Conclusion for Statement I: Statement I alone provides enough information to determine that R sits to the immediate right of K. Thus, Statement I is sufficient.

Statement II Analysis: Sufficiency for Seating Position

Let's analyze the information given in Statement II:

• Setup: Six people around a circular table, facing the centre.
• Condition 1: "K sits second to the right of Q." This means there is one person between Q and K. If Q is at position 1, K is at position 3. Arrangement: Q _ K _ _ _.
• Condition 2: "Only two people sit between K and M." This implies K and M are opposite each other. If K is at position 3, M must be at position 6. Arrangement: Q _ K _ _ M.
• Condition 3: "P sits second to the left of R." This means there is one person between P and R.

Combining Conditions:

1. From Conditions 1 and 2, we have Q at position 1, K at position 3, and M at position 6 (clockwise): Q(1), _(2), K(3), _(4), _(5), M(6).
2. Now we need to place P, Q, N, and R using Condition 3: "P sits second to the left of R". The remaining seats are 2, 4, 5. The remaining people are P, N, R.
3. Let's try placing R:
   • If R is at 2: Q(1), R(2), K(3), _(4), _(5), M(6). P must be second to the left of R(2). Left of R(2) is Q(1), second left is M(6). So P should be at 6. But M is at 6. Contradiction.
   • If R is at 4: Q(1), _(2), K(3), R(4), _(5), M(6). P must be second to the left of R(4). Left of R(4) is K(3), second left is position 2. So P is at 2. Arrangement: Q(1), P(2), K(3), R(4), _(5), M(6). The only remaining person is N, who must be at position 5. Full arrangement: Q, P, K, R, N, M (clockwise).
   • Let's verify this arrangement: Q(1), P(2), K(3), R(4), N(5), M(6).
     • K sits second to the right of Q? Yes (Q->P->K).
     • Only two people sit between K and M? Yes (R and N are between K(3) and M(6)).
     • P sits second to the left of R? Yes (R(4) -> K(3) -> P(2)).
     This arrangement is valid.
   • If R is at 5: Q(1), _(2), K(3), _(4), R(5), M(6). P must be second to the left of R(5). Left of R(5) is position 4, second left is K(3). So P should be at 3. But K is already at 3. Contradiction.
4. The only valid arrangement is Q, P, K, R, N, M (clockwise).
5. In this arrangement, R sits to the immediate right of K.

Conclusion for Statement II: Statement II alone provides enough information to determine that R sits to the immediate right of K. Thus, Statement II is sufficient.

Final Conclusion on Sufficiency

Both Statement I and Statement II individually contain sufficient data to determine who sits to the immediate right of K. Statement I leads to the conclusion that R sits to the immediate right of K. Statement II also leads to the conclusion that R sits to the immediate right of K.

Therefore, either statement alone is sufficient to answer the question.

Answer

The correct option is the one stating that either statement I or statement II is sufficient.

Final Answer Choice: Either data in statement I or statement II is sufficient to answer the question.