1. The figure shows one side of a five-story building, in some of the windows the light is on (they are drawn in white). In this building everything is as usual: all apartments are one-story, the layout of apartments on each floor is the same, each apartment has at least one window on this side, windows from one apartment can only be consecutive.

(a) What is the maximum number of apartments that can be in this building? (1 point)

(b) Can there be 13 apartments in this building? (3 points)

(c) How many apartments can be in the building if the light is on in exactly two apartments on the first floor? (6 points)

(d) Suppose it is known that the light is on in exactly 9 apartments. List all possible apartment layouts on the floors and prove that there are no other options. (10 points)

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# Solution.

(a) 25. This number of apartments is obtained when each apartment has exactly one window facing the given side of the house. There cannot be more apartments than windows.

(b) No, it cannot, because the number of apartments must be divisible by the number of floors.

(c) If the dark window on the first floor is a separate apartment, then the fifth window is also a separate apartment, and the first three windows must belong to one apartment. In this case, there will be only three apartments on the first floor, and 15 apartments in the house. If the dark window on the first floor is not a separate apartment, then it belongs to one of the two apartments where the light is on. In this case, there will be only two apartments on the first floor, and 10 apartments in the entire house.

(d) If two adjacent windows on a floor belong to different apartments, we will say that there is a partition between them. There may or may not be a partition between two adjacent windows. Therefore, there are only 16 possible arrangements of partitions. By going through all these 16 options, we find that the light will be on in exactly 9 apartments only in the case where partitions are between the first and second window and between the fourth and fifth window on the floor. Of course, in the complete solution, all 16 options should be provided.