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Sara has a house which she wants to convert to a hostel and rent it out to students of a nearby women’s college. The house is a two story building and each floor has eight rooms. When one looks from the outside, three rooms are found facing North, three found facing East, three found facing West and three found facing South. Expecting a certain number of students, Sara wanted to follow certain rules while giving the sixteen rooms on rent:

All sixteen rooms must be occupied.

No room can be occupied by more than three students.

Six rooms facing north is called north wing. Similarly six rooms facing east, west and south are called as east wing, west wing and south wing. Each corner room would be in more than one wing. Each of the wings must have exactly 11 students. The first floor must have twice as many students as the ground floor.

However Sara found that three fewer students have come to rent the rooms. Still, Sara could manage to allocate the rooms according to the rules.

How many students turned up for renting the rooms?

1) 24

2) None of these

3) 27

4) 30

5) 33

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First you say there are three rooms facing in each direction, then it's six. Perhaps the first time you intended to say that there are three rooms per floor facing in each direction? –  joriki Aug 25 '12 at 9:29
    
@DP: wonderful approach indeed. –  Rajesh K Singh Aug 25 '12 at 9:41

1 Answer 1

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The answer is c) $27$; this can be shown by obtaining bounds on the number of students.

There is at least one student in each room, and thus at least $8$ on the ground floor, $16$ on the first floor, and $6$ per wing. $8$ students on each floor are spread out to occupy all rooms, but on the first floor there are at least another $8$ students to be distributed.

If we want to have as few students as possible while fulfilling the requirement that each wing has $11$, we should put the extra students on the first floor in the corner rooms, where they count more towards the wing requirement. There's space for exactly $8$ students left, so we can fill the corner rooms, but that still only makes $10$ students per wing. Thus we have to add at least one more student in a ground-floor corner room and two more students in first-floor edge rooms. That fulfills all the requirements, so the least number of students is $27$.

If we want to have as many students as possible while fulfilling the requirement that each wing has $11$, we should put the extra students on the first floor in the edge rooms, where they count less towards the wing requirement. There's space for exactly $8$ students left, so we can fill the edge rooms, which gets us up to $8$ students per wing. That leaves a total of $4\cdot3=12$ of the wing requirement. If we add more students, one can go in an edge room on the ground floor, but two more have to go on the first floor, and these can now only go in corner rooms. Thus each group of $3$ students added uses up another $5$ of the wing requirement, and since there was only $12$ left, we can add at most two groups of $3$ students, which only gets us up to $30$ students.

Thus the minimum number of students is $27$ and the maximum is $30$.

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