Twelve people live in a five-floor building such that the lowest floor is numbered 1, the floor immediately above it is numbered 2, and so on up to the topmost floor numbered 5. On each floor, four flats are arranged from west to east as J, K, L, and M. Flat J is to the east of flat K and

Question

Twelve people live in a five-floor building such that the lowest floor is numbered 1, the floor immediately above it is numbered 2, and so on up to the topmost floor numbered 5. On each floor, four flats are arranged from west to east as J, K, L, and M. Flat J is to the east of flat K and to the west of flat L, and flat L is to the west of flat M. Eight flats are empty. Note: (1) At least two but not more than three people live on each floor. Adjacent flats are not empty. Adjacent flats are immediately to the left, immediately to the right, immediately above, and immediately below. (2) If it is given that A lives immediately above/below B's flat, then either there is one empty flat between A and B or there is no empty flat between A and B. U lives on an even-numbered floor below the third floor. W lives to the east of U but neither in flat M nor in flat K. There is only one floor difference between W and P, and P lives to the west of S but not immediately to the west. U and P live in flats with different names. R lives above S and next to an empty flat. R and W live in flats with different names. O lives to the west of R and to the east of V. Q lives immediately below O's flat. T lives to the east of Q. There is only one floor difference between T and Y. T lives immediately above Y's flat. Z lives immediately to the northwest of X. In which flat do the least number of people live?

Accepted Answer

Either flat K or flat L. The arrangement constraints force several people into specific relative positions, while adjacent flats cannot be empty. After satisfying all conditions across the five floors, flats K and L emerge as the sparsest options. No other flat can have fewer occupants than these two, so the answer is either flat K or flat L.

Solution

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