Colour Building

My friends and I are all very excited because our new sets of Cuisenaire rods arrived in the post today. Sadly, I'd placed my order much later than all my friends did and so when it came to making up my set, the suppliers had only red and white rods in stock.

My friends have very kindly lent me one of each of the other coloured rods, so I know how long they are. I'm going to duplicate these rods by sticking some of my red and white rods together.

There are five different ways for me to make the pink rod:



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 the interactivity below, can you work out how many different ways there are, using only the red and white rods, to make up:

• The white rod?
• The red rod?
• The light green rod?
• The yellow rod?
• The dark green rod?

In each case, what are all the different ways?

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 - are you right?

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!