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Colour Building

Age 11 to 14 Challenge Level:
In this activity, you will need to use Cuisenaire rods, or the interactive Cuisenaire environment which you can find below.

I wonder how many different ways we could combine white rods (1) and red rods (2) to make the same length as the orange rod (10) ...
There are going to be quite a lot, so let's start with a smaller challenge...

In how many ways can you combine red and white rods to make the same length as the pink rod?
Once you think you've found them all, click below to check.

There are five different ways for me to make the pink rod:
rr, wwr, wrw, rww, wwww
Note that I count white, white, red and white, red, white as different, even though they both use two white rods and one red rod.

Using just red and white rods, there's only one way of making a white rod, and only two ways of making a red rod.
Can you find all the ways of making:
  • the light green rod?
  • the yellow rod?
  • the dark green rod?
Can you spot a pattern that will help you to predict how many different ways there are to make up the black rod using only red and white rods?

Check your answer for the black rod.

Without using the interactivity, how many different ways are there to make up the orange rod (equivalent to 10 white rods)?

Can you explain the pattern?

Now suppose I'd placed my order a little earlier, and the suppliers had still had some light green rods in stock. What patterns would I observe if I tried to make up all the different rods using white, red and light green rods? Can you use your results from the previous investigation to help you?

As an extension, why not pick some different colours of rod and see what patterns you observe when you use them to try to make up all the different rods. Beware, with some combinations of rods there are lengths you won't be able to make at all!