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.

Which among the following statement(s) is/are true?

I. J lives on the topmost floor
II. 12 floors are there in the building
III. Even number of floors are between K and T

To determine which statements are true, let us analyze the given clues step-by-step to reconstruct the floor and flat arrangements of the building. Each floor has two flats: Flat A (West) and Flat B (East).

Step 1: Determine the positions of U, T, W, and S
- U lives on an odd-numbered floor below the 7th floor. This means U can live on floor 5, 3, or 1.
- T lives two floors below U (i.e., Floor of T = Floor of U - 2). Thus, U cannot be on floor 1. If U is on floor 3, T is on floor 1. If U is on floor 5, T is on floor 3.
- S and T do not live on adjacent floors.
- There is a three-floor gap between S and T. Therefore:
- If T is on floor 1, S must be on floor 1 + 3 + 1 = 5.
- If T is on floor 3, S must be on floor 3 + 3 + 1 = 7.
- W lives exactly between the S and U’s floor. Let us check the options:
- Case 1: If U is on floor 5, T is on floor 3, and S is on floor 7. The floor exactly between S (7) and U (5) is floor 6. So W lives on floor 6. This case is consistent.
- Case 2: If U is on floor 3, T is on floor 1, and S is on floor 5. The floor exactly between S (5) and U (3) is floor 4. So W lives on floor 4. This is also consistent. Let us proceed with Case 1 first.

Step 2: Place X and determine flat associations
- X lives to the south-west of T, but not just south-west (not adjacent). Since X is south-west of T, T must live in Flat B, and X must live in Flat A on a floor below T.
- For Case 1 (T is on floor 3, in Flat B): X is in Flat A and lives on a floor below T. The floors below T are 1 and 2. Since X is not just south-west, X cannot be on floor 2 (which is immediately south-west of floor 3, Flat B). Thus, X must live on floor 1, Flat A.
- S and X live in the same named flat. Since X lives in Flat A, S must also live in Flat A (floor 7, Flat A).
- W and U live in different named flats. W and P live in different named flats.

Step 3: Analyze R, J, and O
- The number of floors between S and X is:
Floor of S (7) - Floor of X (1) - 1 = 5 floors.
- The number of floors between S and X is two more than the number of floors between S and R. Thus, the number of floors between S and R is 5 - 2 = 3 floors.
- Since S is on floor 7, R must be on either floor 7 - 3 - 1 = 3 or floor 7 + 3 + 1 = 11.
- No one lives to the east of R. This means R lives in Flat B.
- J lives to the immediate north-west of R. This means J lives on the floor immediately above R, in Flat A. (Floor of J = Floor of R + 1, Flat A).
- The number of floors below X (1) is 0 (since X is on floor 1).
- The number of floors below X is the same as the number of floors above O, who lives just above R's floor. So, O lives on the floor just above R's floor (Floor of O = Floor of R + 1). Since the number of floors above O is 0, O must live on the topmost floor of the building. Thus, O is on the topmost floor, and the floor of O is the total number of floors. Therefore, Floor of R must be below the top floor.
- Since O is just above R's floor (Floor of O = Floor of R + 1) and there are 0 floors above O, the total number of floors in the building is Floor of R + 1.
- If R is on floor 3: the top floor would be 4, but S lives on floor 7, which is a contradiction. Thus, R must be on floor 11. This means the top floor O is on floor 12, making the total number of floors in the building 12.
- Since J lives to the immediate north-west of R (who is on floor 11), J lives on floor 12, Flat A. This means J lives on the topmost floor (floor 12).

Step 4: Place K and verify statements
- K lives two floors below O's flat. Since O is on floor 12, K lives on floor 10.
- Let us check the number of floors between K and T:
- Floor of K = 10
- Floor of T = 3
- The floors between them are floors 4, 5, 6, 7, 8, and 9 (total of 6 floors, which is an even number).
- Let us evaluate the statements:
- I. J lives on the topmost floor (Floor 12): True.
- II. 12 floors are there in the building: True.
- III. Even number of floors are between K and T (6 floors): True.
- Therefore, all statements I, II, and III are true.