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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 ?:-

enter image description here

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

Fin27
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    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 '21 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. – Angelica Dec 14 '21 at 22:36
  • A $3$x$3$ square missing the $1$x$1$ square in the center is eight sub-squares... – abiessu Dec 14 '21 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. – Angelica Dec 14 '21 at 22:38
  • if it is per floor also then the total count should be 8, right , why 12 ? – Fin27 Dec 14 '21 at 22:41
  • It's not $12$, you counted wrong. As has been remarked, you doubled counted each of the corner rooms. – lulu Dec 14 '21 at 22:43
  • which corner room am I double counting ? sorry, I cant figure it out when you said a room faces 2 directions – Fin27 Dec 14 '21 at 22:49

2 Answers2

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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.

Mike Earnest
  • 75,930
  • 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 ? – Fin27 Dec 14 '21 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) ? – Fin27 Dec 14 '21 at 23:27
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    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}. – Mike Earnest Dec 14 '21 at 23:32
  • what is the meaning of this line "Each corner room would be in more than one wing" , can't interpret it , could you help out please – Fin27 Dec 15 '21 at 22:34
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    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 – Mike Earnest Dec 15 '21 at 22:35
  • Thanks you so much :) @Mike Earnest, I was able to visualize the thing and tried attempting the questions following this puzzle. I was eventually able to answer the question given in the post, however there are 2 more questions attached to it, which I gave a try, maybe the answers are wrong or am I going wrong, please let me know, I'll post the questions in the comments below along my arrangement which I landed up with , and the official answer given – Fin27 Dec 15 '21 at 23:30
  • Question 2:- If Sara allocates the north -west corner room on the ground floor to 2 students, then the number of students in the corresponding room on the first floor, and the number of students in the middle room in the first floor of the east wing are:

    so my arrangement for this question was:- R1=2, R2=R3=R4=R5=R6=R7=R8=1 , R9=3, R10=1, R11=3, R12=1, R13=2, R14=3, R15=2 , R16=3 , which gives me an answer as 3,2 respectively, but the answer given is 3,1 respectively

     – Fin27 Dec 15 '21 at 23:35
  • Question 3:- If all the students that Sara had expected initially had come to rent the rooms, and if Sara had allocated the north west corner room in the ground floor to 1 student, then the number of students in the corresponding room on the first floor, and the number of students in the middle room in the first of the east wing would have been:- My arrangement for this question was:- R1=1, R2=1, R3=2, R4=1, R5=1, R6=2, R7=1, R8=1, R9=2, R10=3, R11=2, R12=3, R13=3, R14=2, R15=3, R16=2, which gives me answer as 2,3 respectively but the answer is given as 1 and 2 respectively – Fin27 Dec 15 '21 at 23:40
  • Just so you know, I will no longer be responding to any comments about this question. – Mike Earnest Dec 15 '21 at 23:44
  • do I need to open a fresh post for these 2 questions then ? Please guide – Fin27 Dec 15 '21 at 23:47
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    @Fin27 Exactly. Fresh post, where you clearly explain what you have already figured out and what you are still confused on. Best of luck. – Mike Earnest Dec 15 '21 at 23:53
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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.

PiGuy314
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  • Isn't one the door and other the window ? also do the 3 rooms just above 3 rooms on a floor constitute a wing ? – Fin27 Dec 14 '21 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. – PiGuy314 Dec 15 '21 at 12:37