Room Doubling

Investigate the different ways you could split up these rooms so that you have double the number.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative


A good friend of mine was telling me about how his house had been changed, part of it was just as simple as this: four square rooms looking a bit like

Image
Room Doubling
which could be represented simply as
Image
Room Doubling


and two walls were taken down so as to make two "double" rooms:

Image
Room Doubling
Image
Room Doubling


Another way in which it could have been done would look like:-

Image
Room Doubling
Image
Room Doubling


You could make a "model" by using two blocks that are $2$ by $1$ by $1$.

Image
Room Doubling


There is nothing particularly exciting about this! But wait ...

We could imagine something a bit larger.

A plan of a building that has six rooms:

Image
Room Doubling
Image
Room Doubling


Knocking down walls could make it look like:-

Image
Room Doubling
Image
Room Doubling
Image
Room Doubling


if you were using blocks you might just record it like:-

Image
Room Doubling
Image
Room Doubling


Maybe we could go to $8$ rooms.

Image
Room Doubling
Image
Room Doubling


Now we have a lot more examples to explore:-

Image
Room Doubling

Image
Room Doubling
Image
Room Doubling
Image
Room Doubling
Image
Room Doubling


You may also discover:-

Image
Room Doubling
Image
Room Doubling


You might say it is the same as the first one. Well it depends what rules you want to make!

The last thing now is to look at making the building have $10$ rooms:-

Image
Room Doubling


Some of the results you get might be :-

Image
Room Doubling
Image
Room Doubling

Image
Room Doubling



Well now it is time for you to explore more of your own. You might be using squared paper, blocks that are "two" long, or drawings in which you "rub out" the walls that get knocked down.

The only rules are

  1. that you start with a rectangular building that is two rooms wide;
  2. that you only make rooms that are two of the small square rooms put together.

#set var="roll-text" value="Be a detective!" #set var="roll-text" value=""

So you probably need to make some record of the ones that you have found. Maybe you do that by drawing, models or numbers.

#set var="roll-text" value="Be a detective!" #set var="roll-text" value=""

Well that's all very well.

I hope you got to a $12$-room house.

But you should of course ask "I wonder what would happen if the $12$ rooms were $3$ by $4$ instead of $6$ by $2$?

Image
Room Doubling


You might then gets results like :-

Image
Room Doubling
Image
Room Doubling

Image
Room Doubling



but of course there are lots more to find.

Explore and have fun.

#set var="roll-text" value="Be a detective!" #set var="roll-text" value=""

The last one I suggest to look at is

Image
Room Doubling


with results like :-

Image
Room Doubling
Image
Room Doubling


#set var="roll-text" value="Be a detective!" #set var="roll-text" value=""

It's now up to you to decide what further "I wonder what would happen if....?" questions.

If you draw your results or put down numbers in a table to show many you can find for each number of rooms then you should have something to spend time thinking about. You might then predict how many examples could be found for larger numbers of rooms.