Colour Building

Using only the red and white rods, how many different ways are there to make up the other rods?

Age

11 to 14

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

In this activity, you will need to use Cuisenaire rods, or the interactive Cuisenaire environment which you can find below. If you have not used the Cuisenaire Environment before, you might find it helpful to look at the instructions and video.

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...

In how many ways can you combine white rods (1) and red rods (2) to make the same length as the pink rod (4)?

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.

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 spot a pattern that will help you to predict how many different ways there are to make up the black rod (7)? Use the interactivity to check your prediction.

Without using the interactivity, can you work out how many different ways there are to make up the orange rod (10)?

Can you explain the pattern?

Extension

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?

Student Solutions

Mahdi from Mahatma Gandhi International School in India sent a full solution. This is the first part of Mahdi's solution, with some diagrams edited to show the rods being built:

Let $T_n$ define the number of ways of making a rod of length $n$ just by using red and white rods. We already know the following:

$T_1 = 1$ (there is $1$ way to make a white rod from red and white rods)

$T_2 = 2$ (there are $2$ ways to make a red rod from red and white rods)

$T_3 = 3$ (there are $3$ ways to make a light green rod)

$T_4 = 5$ (there are $5$ ways to make a pink rod)

We can count all of the possibilities systematically making sure none is left. We observe that we start to make the $n$ length rod using one white at the left most place and cover the $n-1$ rods using red and whites (for example, put one white rod at the beginning and cover the length of the green rod using red and white rods).

The ways to cover $n-1$ length rod as defined in the beginning is $T_{n-1}$ (there were $3$ ways to make a light green rod).

Next we introduce a red at the left most side and try to cover the rest of the $n-2$ remaining units with the red and the whites (in the example, put one red rod at the beginning and cover the length of the second red rod using red and white rods). The total ways of making a rod of length $n-2$ is as defined in the beginning $T_{n-2}$ (there were $2$ ways to make a red rod).

So there are $5$ ways to make a pink rod using red and white rods.

Thus we get the explicit formula for combining a rod of length $n$ using red and white as: $$T_n = T_{n+1} + T_{n+2}$$

Using this formula we can calculate $T_5, T_6, T_7, ...$

$T_5 = T_4 + T_3 = 5+3=8$

$T_6 = T_5+T_4=8+5=13$

$T_7 = T_6+T_5=13+8=21$

$T_8=T_7+T_6=21+13=34$

$T_9 = T_8 + T_7 = 34+21=55$

$T_{10} = T_9+T_8=55+34=89$ (orange rod)

You can see that $T_n=F{n-1}$ where $F_n$ is the $n^{th}$ Fibonacci number.

Mahdi's solution continues to find the definition of $T_n$ in terms of $n.$ Mahdi then considers finding lengths using white, red and green rods, or using white, red, green and pink rods, or using the first $k$ rods available. Click here to see Mahdi's full solution.

Teachers' Resources

Why do this problem ?

This problem provides an opportunity for students to work systematically, and to discover the powerful technique of using smaller cases to predict and explain what will happen with larger cases.

Possible approach

The problem assumes some familiarity with Cuisenaire rods, so if students are not familiar with the different colours and lengths, it may be worth spending some time first exploring a set of rods or using the online environment.

"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) ..."

Give students some time to think about the challenge, and then share their thoughts. The following might emerge:

"There are going to be loads of different ways"

"How are we going to be able to make sure we don't miss any?"

If no-one has suggested it: "Perhaps we could work on a simpler version of the problem to see if that helps. Let's see how many ways we could make the pink rod (4) out of whites and reds."

Once students have agreed on the five ways that a pink can be made from whites and reds, invite them to find the number of ways to make light green (3), yellow (5), and dark green (6). Then encourage them to make a prediction for black (7) and then test it out.

It is important to set aside enough time for students to think about and appreciate why each answer is the sum of the previous two. To draw out this insight, you might suggest that students organise their work into two categories: solutions that start with a red (what's left?) and solutions that start with a white (what's left now?)

Key questions

Is there a way to work systematically to make sure you have found all the possibilities?

When we make the length of the pink rod, how many possibilities have a red on the left?

How many possibilities have a white on the left?

Can you use this idea of thinking separately about the solutions that begin with a red, and the ones that begin with a white, to help you explain the patterns that you find?

Possible extension

Students could try 1 Step 2 Step, which has the same mathematical structure but set in a different context.

Possible support

Focus on how to work systematically to find all the solutions for smaller cases before introducing the orange rod challenge.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners