Directions (5-9): Study the following information carefully and answer the given questions.

In a building there are certain number of floors each floor has two flats i.e.; flat A is to the west of flat B. U lives on an odd numbered floor but below 7th floor. T lives two floors below U. X lives to the south west of T but not just south west. Three floors gap between S and T. X and S lives in the same named flat. The number of floors between S and X is two more than the numbers of floor between S and R. No one lives to the east of R. J lives to the immediate north west of R. The number of floors below X is same as the number of floors above O who lives just above R floors. K lives two floors below O’s flat. Only one floor gap between W and Q. W lives exactly between the S and U’s floor. S and T didn’t live on the adjacent floor. P lives to the north west of Q and lives on an even numbered floor. P and W live in different named flats. Q lives above S. W and U live in different named flat.

Who among the following persons live to the south east of P?

The correct answer is W.

Let's solve the puzzle step-by-step to determine the positions of all the persons in the building. There are two flats on each floor: Flat A (west) and Flat B (east).

Step 1: Determine the floors of U, T, S, and W
1. U lives on an odd-numbered floor below the 7th floor. Thus, the possible floors for U are 5, 3, or 1.
2. T lives two floors below U: floor(T) = floor(U) - 2.
3. There is a three-floor gap between S and T: |floor(S) - floor(T)| = 4. S and T do not live on adjacent floors.
4. W lives exactly between the floors of S and U: floor(W) = (floor(S) + floor(U)) / 2. This implies floor(S) and floor(U) must have the same parity.
5. Since U lives on an odd floor, S must also live on an odd floor.

Let's evaluate the cases:
• If floor(U) = 5:
Then floor(T) = 5 - 2 = 3.
For S, we have floor(S) = 3 + 4 = 7 (as floor 3 - 4 = -1 is impossible).
Floor of W is (5 + 7) / 2 = 6.
• If floor(U) = 3:
Then floor(T) = 3 - 2 = 1.
For S, we have floor(S) = 1 + 4 = 5.
Floor of W is (3 + 5) / 2 = 4.

Step 2: Use the position of X to eliminate invalid cases
1. We are told: "X lives to the south west of T but not just south west."
2. Since Flat A is to the west of Flat B, T must live in Flat B and X must live in Flat A.
3. For X to be south west of T, the floor of X must be less than the floor of T (floor(X) < floor(T)).
• If floor(T) = 1, floor(X) cannot be less than 1. Thus, the case floor(U) = 3 is invalid.
• Therefore, we must have: floor(U) = 5, floor(T) = 3, floor(S) = 7, and floor(W) = 6.
4. Since floor(X) < floor(T) (where floor(T) = 3) and X is "not just south west" of T (meaning it cannot be on floor 2), X must live on floor 1.
5. X and S live in the same named flat. Since flat(X) = Flat A, we have flat(S) = Flat A.

Step 3: Determine the floor of R, O, and the total floors
1. "The number of floors between S and X is two more than the number of floors between S and R."
The number of floors between S (floor 7) and X (floor 1) is 7 - 1 - 1 = 5 floors.
Thus, the number of floors between S and R is 5 - 2 = 3 floors.
This means |7 - floor(R)| - 1 = 3 ⇒ |7 - floor(R)| = 4.
So floor(R) can be either 3 or 11.
2. "No one lives to the east of R." This means R lives in Flat B.
3. "J lives to the immediate north west of R." This means J lives on the floor immediately above R in Flat A.
4. "The number of floors below X is same as the number of floors above O who lives just above R's floor."
Since floor(X) = 1, the number of floors below X is 0. Thus, there are 0 floors above O, making O's floor the topmost floor of the building.
Since O lives just above R, floor(O) = floor(R) + 1.
If floor(R) = 3, then floor(O) = 4, which is impossible because we already have floors up to 7 and 8.
Therefore, floor(R) = 11 and floor(O) = 12 (top floor). The total number of floors is 12.

Step 4: Place the remaining persons
1. R is on floor 11, Flat B (11B).
2. J is on floor 12, Flat A (12A).
3. O is on floor 12, Flat B (12B).
4. K lives two floors below O's flat, so K is on floor 10.
5. Q lives above S (floor 7). Also, there is a one-floor gap between W (floor 6) and Q. Thus, Q is on floor 8.
6. "P lives to the north west of Q and lives on an even numbered floor."
Since P is north west of Q, floor(P) > floor(Q) (floor 8) and P is in Flat A. Q must be in Flat B.
The even numbered floors above 8 are 10 and 12. Since floor 12 is fully occupied by J and O, P must be on floor 10, Flat A (10A).
7. "P and W live in different named flats." Since P is in Flat A, W lives in Flat B (6B).
8. "W and U live in different named flats." Since W is in Flat B, U lives in Flat A (5A).

Conclusion:
P lives on floor 10, Flat A. The person living to the south east of P must live on a lower floor in Flat B. Since W lives on floor 6, Flat B, W lives to the south east of P.