Kitchen!

Geometry Level 1

Jim has a very big house.

It has 5 5 bedrooms, 5 5 bathrooms, 2 2 laundry rooms, 2 2 offices, 5 5 garages, a huge theater room, a living room, a dining room, and 2 2 kitchens -- amounting for a total of 20 20 rooms.

Assume all rooms are perfectly square shaped (even the garage), and are not necessarily the same size, is it possible for Jim to have a house in the shape of a square?

  • Jim has only a single storied house.
  • It is okay to place the garages in the interior
Yes No

Micah Gadbois by Micah Gadbois , 15, USA

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2 solutions

Azadali Jivani
Jul 20, 2017

As per given condition total rooms = 20
Suppose area of house = 10 * 10 = 100 units
size of one room = 9 * 9 = 81 units
size of each remaining 19 rooms = 1 * 1 = 1 therefore area of 19 rooms = 19 units
81 + 19 = 100 units
Note ; placed 19 rooms along the 2 sides of large room of size 9 * 9.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

That is a constructive and elegant solution.

Can we do this for any number of square rooms?

Agnishom Chattopadhyay - 3 years, 10 months ago
Micah Gadbois
Jul 19, 2017

Since the rooms don't have to be the same size, we can adjust them until they fit in a square. The answer is Yes

0 Helpful 0 Interesting 0 Brilliant 0 Confused

This is not a trivial claim. Can you show us how to adjust?

This problem is especially hard when it is told that all the rooms must be of integer length.

Agnishom Chattopadhyay - 3 years, 10 months ago

They don't have to be of integer length

Micah Gadbois - 3 years, 10 months ago

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That is okay, but it is still not clear how the adjusting should be done

Agnishom Chattopadhyay - 3 years, 10 months ago

Just put the rooms next to each other, (any size) than shrink to make them fit.

Micah Gadbois - 3 years, 10 months ago

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But how does that work? You'll need to make sure that the entire collection of rooms is a square too

Agnishom Chattopadhyay - 3 years, 10 months ago

More generally, what are all of the values of n n such that a square house can be divided into n n square rooms?

Calvin Lin Staff - 3 years, 10 months ago

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