Solving the Circular Seating Arrangement Puzzle

This question asks us to determine who sits second to the left of a specific person, P, in a circular arrangement of six people (K, M, N, P, Q, R) all facing the center. We need to evaluate if the information provided in two separate statements (I and II) or both together is sufficient to answer this question.

Analyzing Statement I

Statement I provides the following clues:

• K sits to the immediate right of P.
• Only two people sit between P and Q.
• R sits second to the right of M.

Let's visualize a circular table with 6 positions. If all are facing the center, 'right' is clockwise and 'left' is anti-clockwise.

1. "K sits to the immediate right of P." This fixes the relative positions of P and K. Let's place P at a reference position. Then K is immediately next to P in the clockwise direction.

2. "Only two people sit between P and Q." In a circle of 6 people, if there are two people between two individuals, they are opposite to each other. So, P and Q are opposite.

Combining these two clues, we have P and Q opposite each other, and K immediately right of P. The arrangement so far looks like P - K (clockwise) and P opposite Q. Let's label positions 1 to 6 clockwise starting with P at position 1. K is at 2 (immediate right). Q is at 4 (opposite P).

Current state: Position 1: P, Position 2: K, Position 4: Q. Positions 3, 5, 6 are empty. The remaining people are M, N, R.

3. "R sits second to the right of M." This means there is one person between M and R in the clockwise direction.

Let's check the available empty positions (3, 5, 6) for M and R, keeping this rule in mind:

• If M is at position 3, second right of M is position 5. So R would be at position 5. The remaining person N would be at position 6. This gives the arrangement: P(1) - K(2) - M(3) - Q(4) - R(5) - N(6) (clockwise).
• If M is at position 5, second right of M is position 1, which is occupied by P. R cannot be P. So M cannot be at position 5.
• If M is at position 6, second right of M is position 2, which is occupied by K. R cannot be K. So M cannot be at position 6.

The only valid arrangement based on Statement I is: P - K - M - Q - R - N (clockwise around the circle).

Now, let's find who sits second to the left of P in this arrangement. Starting from P and moving in the anti-clockwise direction (left): The first person is N, and the second person is R.

Therefore, according to Statement I, R sits second to the left of P. Since we found a unique person, Statement I alone is sufficient to answer the question.

Analyzing Statement II

Statement II provides the following clues:

• Only two people sit between N and M.
• Q is an immediate neighbour of M.
• Only one person sits between Q and K.

1. "Only two people sit between N and M." Similar to clue 2 in Statement I, this means N and M are opposite each other in the circular arrangement of 6 people.

2. "Q is an immediate neighbour of M." Q sits next to M. Since M has two neighbours in a circle, Q is one of them. The other neighbour is not specified yet.

3. "Only one person sits between Q and K." This means K is second to the right or second to the left of Q.

Let's combine these clues. M and N are opposite. Q is next to M. So the segment looks like either Q - M or M - Q. The person next to M (other than Q) is opposite N. Let's call the position opposite N as X. So either Q M X N or X M Q N (clockwise/anti-clockwise segments, with M and N opposite).

Also, there is one person between Q and K. Consider the segment Q - M (where Q is to the left of M). The person to the right of M is opposite N. So we have ... Q M (Opposite of N) N ... If Q is to the left of M, then the sequence is ... ? Q M (Opposite of N) ... N ... One person between Q and K. Starting from Q, going clockwise: the person is M. Second person would be K. So sequence would be Q M K ... and M is opposite N. So the arrangement could be ... Q M K ... N ... Starting from Q, going anti-clockwise: one person is before Q, let's call them Y. Then K is next. So sequence would be ... K Y Q M ... N ...

Let's try a fixed position. Let M be at position 1, so N is at position 4.

• Q is next to M. Q can be at 2 or 6.
  • If Q is at 2: M(1), Q(2), N(4). One person between Q(2) and K. * Clockwise from Q(2), person at 3, then K at 4. Position 4 is N. K cannot be N. (This means K must be second left of Q). * Anti-clockwise from Q(2), person at 1 (which is M). Then K at 6. Arrangement: M(1), Q(2), ?, N(4), ?, K(6). Remaining are P and R at positions 3 and 5. * If P is at 3, second left of P(3) is Q(2) then M(1). Result is M. * If P is at 5, second left of P(5) is N(4) then the person at 3 (which is R). Result is R. Since the person second left of P could be M or R depending on P's position, this case is not sufficient.
  • If Q is at 6: M(1), Q(6), N(4). One person between Q(6) and K. * Clockwise from Q(6), person at 1 (which is M). Then K at 2. Arrangement: M(1), K(2), ?, N(4), ?, Q(6). Remaining are P and R at positions 3 and 5. * If P is at 3, second left of P(3) is K(2) then M(1). Result is M. * If P is at 5, second left of P(5) is N(4) then the person at 3 (which is R). Result is R. Since the person second left of P could be M or R depending on P's position, this case is not sufficient.

In all valid scenarios derived from Statement II, the position of P is not fixed relative to the others in a way that uniquely identifies the person second to its left. Statement II alone is not sufficient to answer the question.

Conclusion on Sufficiency

Based on the analysis:

• Statement I alone is sufficient because it leads to a unique seating arrangement where the person second to the left of P is identified as R.
• Statement II alone is not sufficient because it allows for multiple possible arrangements where the person second to the left of P could be different (e.g., M or R or K, depending on P's unknown relative position).

Therefore, the data in statement I alone is sufficient to answer the question, while the data in statement II alone is not sufficient.

Revision Table: Circular Seating Puzzles

Additional Information: Data Sufficiency in Logical Reasoning

Data Sufficiency questions test your ability to determine if the given information is adequate to answer a specific question, rather than finding the answer itself. Here's a breakdown:

• You are given a question and typically two statements (I and II) containing additional data.
• You must determine if Statement I alone is sufficient, Statement II alone is sufficient, both are sufficient together, or neither is sufficient.
• A statement is sufficient if the information within it leads to a single, definite answer to the question. If it leads to multiple possible answers, it is not sufficient.
• Analyze each statement independently first. Then, if neither is sufficient alone, combine them to see if they become sufficient together.
• Do not use information from one statement while analyzing the other statement independently.
• The final answer depends on the minimum information needed to answer the question definitively.