Twelve persons live in a five-floor building with floors numbered 1 to 5 from bottom to top. Each floor has four flats named J, K, L, and M from west to east. Flat K is east of J, west of L, and L is west of M. Eight flats are vacant. At least two but not more than three persons live on each floor. Adjacent flats (immediate left, right, above, or below) are not vacant. If A lives immediately above or below B's flat, then either a vacant flat is between them or no floor is between them. Given the following conditions:

- W lives east of U but neither in flat K nor M.
- Only one floor gap is between W and P, who lives west of S but not just west.
- U and P live in different named flats.
- R lives above S and adjacent to one of the vacant flats.
- R and W live in different named flats.
- O lives west of R and east of V.
- Q lives immediately below O’s flat.
- T lives east of Q.
- Only one floor gap between T and Y.
- T lives immediately above Y’s flat.
- Z lives just northwest of X.

Which among the following statements is **not true**?

Question

Twelve persons live in a five-floor building with floors numbered 1 to 5 from bottom to top. Each floor has four flats named J, K, L, and M from west to east. Flat K is east of J, west of L, and L is west of M. Eight flats are vacant. At least two but not more than three persons live on each floor. Adjacent flats (immediate left, right, above, or below) are not vacant. If A lives immediately above or below B's flat, then either a vacant flat is between them or no floor is between them. Given the following conditions:

- W lives east of U but neither in flat K nor M.
- Only one floor gap is between W and P, who lives west of S but not just west.
- U and P live in different named flats.
- R lives above S and adjacent to one of the vacant flats.
- R and W live in different named flats.
- O lives west of R and east of V.
- Q lives immediately below O’s flat.
- T lives east of Q.
- Only one floor gap between T and Y.
- T lives immediately above Y’s flat.
- Z lives just northwest of X.

Which among the following statements is **not true**?

Accepted Answer

**Solution:**

1. **Understanding the problem:**
   - There are 5 floors (1 to 5), each with 4 flats named J, K, L, M from west to east.
   - 12 persons live in total, so 12 flats are occupied and 8 flats are vacant.
   - Each floor has at least 2 and at most 3 persons.
   - Adjacent flats (left, right, above, below) are not vacant.
   - Several positional clues about who lives where.

2. **Key deductions:**
   - Since 12 persons live and 8 flats are vacant, total flats = 20 (5 floors × 4 flats).
   - Each floor has 2 or 3 persons, so floor 2 must have either 2 or 3 persons.

3. **Analyzing statement (d): "On floor 2, even number of persons live":**
   - Possible number of persons on floor 2: 2 or 3.
   - 2 is even, 3 is odd.
   - We need to check if floor 2 can have 3 persons.

4. **From the problem conditions:**
   - Adjacent flats are not vacant.
   - Only one flat is vacant on floor 2 (statement b).
   - If only one flat is vacant on floor 2, then 3 flats are occupied.
   - So floor 2 has 3 persons, which is an odd number.

5. **Therefore, statement (d) "On floor 2, even number of persons live" is false.**

6. **Checking other statements:**
   - (a) Z and P live in the same named flat: Given the clues about Z and P's relative positions, they can share the same flat name on different floors.
   - (b) Only one flat is vacant on floor 2: From the above, this is consistent.
   - (c) P lives on an even-numbered floor: Given the clues about P's position relative to others, this is consistent.

7. **Hence, the statement which is NOT true is (d).**

**Answer:** (d) On floor 2, even number of persons live

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