I wonder how many different ways there are of combining white rods (1) and red rods (2) to make the same length as the orange rod (10) ...

There are going to be quite a few, so let's start with a simpler challenge...

Once you think you've found them all, click below to check.

There are five different ways to make the same length as 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.

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 the same length as the white rod, and only two ways of making the same length as the red rod.

Can you find all the ways of making the following lengths using just reds and whites:

- the light green rod (3)?
- the yellow rod (5)?
- the dark green rod (6)?

Can you explain the pattern?

Try making different lengths using combinations of white, red and light green rods.

Can you use the patterns you observe to make predictions about how many ways there will be?