# Puzzle of renting rooms?

I have a doubt in this question :- How many students turned up for renting the rooms? , for convenience I have posted the same question below also

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

I am not able to visualize the whole setup of building, there are 8 rooms on each floor , and there are 2 floors which means 16 rooms.

Now the orientation part is confusing me a lot, the question says "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. " which gives a total of 12 rooms , but we know that we have 16 rooms in the building.

Also how is the wing system being designed, it would be really nice if someone could help me out with a 2-D or a 3-D figure for the setting , so that I can give a try to this question

This is what I have been able to imagine for a floor ?:-

Here R1, R2, R3 are facing north; R4, R5, R6 facing east; R7,R8,R9 facing south and R10,R11,R12 facing west, this counts to 12 rooms alone per floor , but there are only 8 rooms as per question

• Presumably the count is per floor. And don't forget, the corner rooms face in two directions. Convince yourself that the given count corresponds to $8$ rooms per floor, hence $16$ altogether.
– lulu
Dec 14, 2021 at 22:36
• The bolded statement is meant to be "there are 3 rooms facing ... per floor", or six on each wall, as noted by the author in a comment. But this still doesn't give 16 rooms when summed, because some rooms are on a corner, and are facing two directions. Dec 14, 2021 at 22:36
• A $3$x$3$ square missing the $1$x$1$ square in the center is eight sub-squares... Dec 14, 2021 at 22:37
• To break it down more simply, consider that on a given floor, 3 rooms face east, 3 rooms face north, but 5 rooms face east and/or north. Dec 14, 2021 at 22:38
• if it is per floor also then the total count should be 8, right , why 12 ? Dec 14, 2021 at 22:41

This is how the rooms on floor $$1$$ are arranged:

+--------+--------+--------+         N
| Room 1 | Room 2 | Room 3 |         ^
+--------+--------+--------+         |
| Room 4 |        | Room 5 |   W <---+---> E
+--------+--------+--------+         |
| Room 6 | Room 7 | Room 8 |         v
+--------+--------+--------+         S


There are three rooms facing north: rooms 1, 2 and 3.
There are three rooms facing east: rooms 3, 5 and 8.
There are three rooms facing south: rooms 6, 7 and 8.
There are three rooms facing west: rooms 1, 4 and 6.

• Thank you so much @Mike Earnest , that clears some doubts for me, my line of thought was that when it said room faced north meant that the door which leads entry to that room faces to north, but room 1 is facing west also , which means we have to consider the windows also ? does the window of room 1 faces west as per your figure ? Dec 14, 2021 at 23:25
• Also the rooms just above Room1 , Room 2, Room 3 , lets call them Room 9, Room 10, Room 11 ; so do these 6 rooms then constitute the north wing i.e. (Room1, Room 2, Room 3, Room 9, Room 10, Room 11) ? Dec 14, 2021 at 23:27
• The direction a room is facing is determined by the direction its windows are facing (in this context). Yes, the north wing consists of rooms numbered {1, 2, 3, 9, 10, 11}. Dec 14, 2021 at 23:32
• The corner rooms on floor 1 are rooms numbered 1, 3, 6 and 8. Room 1 is in the north wing AND the west wing. Room 3 is in the north wing AND the east wing. etc @fin27 Dec 15, 2021 at 22:35
• @Fin27 Exactly. Fresh post, where you clearly explain what you have already figured out and what you are still confused on. Best of luck. Dec 15, 2021 at 23:53

This is similar (if not identical) to a recent TedEd riddle called the human cannonball riddle. It explains the answer thoroughly.

As for your question concerning the wings, the corner rooms have two windows, and are therefore in two wings. With no room in the center, it makes a 3*3 square minus 1, and so 8 rooms per floor.

• Isn't one the door and other the window ? also do the 3 rooms just above 3 rooms on a floor constitute a wing ? Dec 14, 2021 at 23:28
• While getting into the corner rooms would be impractical at best, for the purpose of the puzzle, they exist. And yes, the rooms on both floors facing a given direction constitute a wing. Dec 15, 2021 at 12:37