R, E, M, I, N and S live on six different floors of the same building. The…

Concept

In a floor-arrangement puzzle, place the absolute clues before the relative ones. In a building of n floors, a clue of the form "exactly k people live above X" pins X to floor n - k, and a clue naming exactly who lives above someone pins both of them at once.

Only after every absolute position is fixed do you fit the relative blocks: a clue of the form "P lives immediately above Q" needs a pair of consecutive floors that are both still free.

Applying it to this building

1. The six floors are numbered 1 at the bottom up to 6 at the top, and R, E, M, I, N and S occupy one floor each.

2. Four people live above I, so four floors lie above I. That pins I to floor 6 - 4 = 2.

3. Only S lives above M, so exactly one floor lies above M and S occupies it. That pins M to floor 5 and S to floor 6.

4. Floors 1, 3 and 4 are still free, and they must be taken by R, N and E.

5. N lives immediately above R, which needs two consecutive free floors. Among the free floors 1, 3 and 4 the consecutive pair is 3 and 4, so R is on floor 3 and N is on floor 4. The pair 1 and 2 is unusable because floor 2 already holds I.

6. E takes the single floor left over, floor 1.

7. The sum of the floor numbers of E and N is therefore 1 + 4 = 5.

Cross-check

• Above I on floor 2 live R, N, M and S, which is exactly four people.

• Above M on floor 5 lives S and nobody else.

• N on floor 4 sits immediately above R on floor 3.

• Every clue is satisfied and no other seating survives all three clues, so the sum 1 + 4 = 5 is forced.