Eight persons – P, Q, R, S, T, U, V and W sit around a circular table facing the centre, but not necessarily in the same order. Q sits two places away from S. Two persons sit between S and P who does not sit adjacent to Q. W sits second to the right of P. T and the one who sits opposite to W are immediate neighbours. As many persons sit between Q and W as between T and V. R sits third to the left of V.

Which of the following statement(s) is/are true?
I. T faces S
II. P doesn’t sit next to W
III. V doesn’t sit next to Q

The correct answer is Both I and II.

Step-by-step Explanation:

Let us arrange the eight persons (P, Q, R, S, T, U, V, W) around a circular table facing the centre in 8 numbered positions (1 to 8 in clockwise order).

1. Position of Q and S:
Q sits two places away from S. This means there is one person between Q and S. Let us place S at position 1. Then Q can be at position 3 or position 7.

2. Position of P:
Two persons sit between S and P. Since S is at position 1, P must be at position 4 or position 6.
We are given that P does not sit adjacent to Q.
If Q is at position 3, P cannot be at position 4 (since 4 is adjacent to 3). Therefore, P must be at position 6.
If Q is at position 7, P cannot be at position 6 (since 6 is adjacent to 7). Therefore, P must be at position 4.

3. Position of W:
W sits second to the right of P.
Facing the centre means right is in the counter-clockwise direction.
Let us analyze Case 1: S at 1, Q at 3, P at 6.
Second to the right of P (position 6) counter-clockwise is position 4. So W is at position 4.
Let us check: opposite of W (position 4) is position 8.
T and the one who sits opposite to W (position 8) are immediate neighbours. So T could be at position 7 or position 1. But S is at 1, so T could be at position 7 or position 2.

4. Analyzing the distance between Q and W, and T and V:
In Case 1 (S=1, Q=3, P=6, W=4):
The number of persons sitting between Q (position 3) and W (position 4) along the shortest path is 0.
So, as many persons sit between Q and W (which is 0) as between T and V. This means T and V must be immediate neighbours.
We know R sits third to the left of V.
Third to the left means 3 steps in the clockwise direction.
Let us test positions for T, V, R:
If T is at position 5 (opposite W is 8, neighbour is 7 or 1), wait, opposite of W (4) is 8. Neighbours of 8 are 7 and 1.
Since S is at 1, T must be at position 7.
Since T and V are immediate neighbours (0 persons between them), V must be at position 8 (since 6 is occupied by P).
If V is at position 8, third to the left of V (clockwise: 8 → 1 → 2 → 3) is position 3.
Position 3 is occupied by Q, but R must sit third to the left of V. This creates a contradiction (R and Q would be at position 3).

Let us try Case 2: S at position 1, Q at position 7, P at position 4.
- W sits second to the right of P (counter-clockwise from 4): position 2. So W is at position 2.
- Opposite of W (position 2) is position 6.
- T is an immediate neighbour of the person opposite to W (position 6). So T can be at position 5 or position 7.
- But position 7 is occupied by Q. Therefore, T must be at position 5.
- Persons between Q (position 7) and W (position 2) is 2 persons (positions 8 and 1 or 6, 5, 4, 3). So the number of persons between Q and W is 2.
- Thus, 2 persons must sit between T (position 5) and V. Moving 3 steps from 5: V can be at position 2 or position 8. Since position 2 is W, V must be at position 8.
- Now check R: R sits third to the left of V (position 8). Third to the left (clockwise 8 → 1 → 2 → 3) means R is at position 3.
- Positions filled so far: S=1, W=2, R=3, P=4, T=5, Q=7, V=8.
- The only remaining position is 6, which must be occupied by U.

Final Seating Arrangement (Clockwise from position 1):
1: S
2: W
3: R
4: P
5: T
6: U
7: Q
8: V

Evaluating the Statements:
- Statement I: T faces S
T is at position 5 and S is at position 1. Opposite positions in an 8-person circle differ by 4 places (1 and 5 are directly opposite). Thus, T faces S. (True)
- Statement II: P doesn’t sit next to W
P is at position 4 and W is at position 2. They are separated by R at position 3. So P does not sit next to W. (True)
- Statement III: V doesn’t sit next to Q
V is at position 8 and Q is at position 7. They sit directly next to each other. (False)

Therefore, both statements I and II are true.