Seating Arrangement Logic on a Square Table

This solution explains the step-by-step process to determine the seating arrangement of eight students (U, V, W, X, Y, P, R, S) around a square table, solving the logic puzzle based on the provided clues.

Table Setup and Facing Directions

• The puzzle involves 8 students seated around a square table.
• 4 students are positioned at the corners, facing away from the table's center.
• 4 students are positioned in the middle of each side, facing towards the table's center.
• We can label the seats 1 through 8 clockwise. Seats 1, 3, 5, 7 are the corners (Facing Away - A). Seats 2, 4, 6, 8 are the middle sides (Facing Towards - T).

Deducing the Seating Arrangement

1. Initial Placement (Y and X): The clue "X sits third to the right of Y" is key. Let's assume Y is at corner seat 1 (Facing Away). Counting three seats clockwise (seat 2, seat 3, seat 4), X is at seat 4 (middle, Facing Towards).

   Current Arrangement: Y is at seat 1(A), X is at seat 4(T). Seats: Y _ _ X _ _ _ _

2. Placing P: "Two students sit between P and X." With X at seat 4, we count two seats forward (seat 5, seat 6). The next seat is 7. Placing P at seat 7 (corner, Facing Away) fulfills this condition.

   Current Arrangement: Y is at 1(A), X is at 4(T), P is at 7(A). Seats: Y _ _ X _ _ P _

3. Positioning U and V: "U and V sit opposite each other." The opposite pairs of seats are (1,5), (2,6), (3,7), (4,8). Given Y is at 1, X at 4, and P at 7, U and V cannot be in seats 1, 4, 5, 7. This leaves seats 2 and 6 as the only possible opposite pair for U and V.
4. Applying U's Adjacency Rule: "U does not sit adjacent to Y." Y is at seat 1. Its adjacent seats are 2 and 8. Since U cannot be at seat 2, V must be at seat 2 (middle, Facing Towards), and U must be at seat 6 (middle, Facing Towards).

   Current Arrangement: Y(1,A), V(2,T), X(4,T), U(6,T), P(7,A).

5. Placing W: "W faces the centre." This means W must be in a middle seat (2, 4, 6, or 8). Seats 2, 4, 6 are already taken by V, X, and U. Thus, W must be in seat 8 (middle, Facing Towards).

   Current Arrangement: Y(1,A), V(2,T), X(4,T), U(6,T), P(7,A), W(8,T).

6. Placing R and S: The remaining students are R and S. The available seats are 3 and 5 (both corners, Facing Away).
7. Final Constraint Check (R, S, Y relation): "The same number of people sit between R and S as that of between Y and S." Let's check the two possibilities for R and S in seats 3 and 5.
   • Case 1: S at 3(A), R at 5(A).
     • Between R(seat 5) and S(seat 3): Seat 4(X). Count = 1 person.
     • Between Y(seat 1) and S(seat 3): Seat 2(V). Count = 1 person.
     • Condition satisfied ($1 = 1$). This configuration works.
   • Case 2: R at 3(A), S at 5(A).
     • Between R(seat 3) and S(seat 5): Seat 4(X). Count = 1 person.
     • Between Y(seat 1) and S(seat 5): Seats 2(V), 3(R), 4(X). Count = 3 people.
     • Condition not satisfied ($1 \neq 3$). This configuration is incorrect.

   The deduced arrangement (clockwise) is: Y, V, S, X, R, U, P, W.

   Positions & Facing: Y(1,A), V(2,T), S(3,A), X(4,T), R(5,A), U(6,T), P(7,A), W(8,T).

Solving the Final Question

The question is: "The same number of students are sitting between Y and ____ as that of between U and W."

First, calculate the number of students between U and W:

• U is at seat 6. W is at seat 8.
• Counting clockwise between U(6) and W(8): Seat 7(P). The count is 1 student.
• We need to find which student (let's call them Z) has exactly 1 student between Y(1) and Z.

Check the options against the final arrangement (Y V S X R U P W):

• R (seat 5): Between Y(1) and R(5) are V, S, X. Count = 3 students. (Not 1)
• X (seat 4): Between Y(1) and X(4) are V, S. Count = 2 students. (Not 1)
• Y (seat 1): Between Y(1) and Y(1). Count = 0 students. (Not 1)
• S (seat 3): Between Y(1) and S(3) is V. Count = 1 student. (Matches the count between U and W!)
• W (seat 8): Between Y(1) and W(8) are V, S, X, R, U, P. Count = 6 students. (Not 1)

The student S fits the condition because the number of people ($N$) between Y and S ($N_{Y,S}$) is 1, which equals the number of people between U and W ($N_{U,W}$), which is also 1. Thus, $N_{Y,S} = N_{U,W}$.

The correct option is S.