### Why do this problem?

This problem encourages visualisation of a three dimensional
idea in a two dimensional context. It offers a visual demonstration
of the sum of cubes which can lead to a proof. By extending the
picture, ideas of proof by induction could be introduced.

### Possible approach

Perhaps start by exploring the problem

Picturing Triangle Numbers, which develops the formula for the
nth triangular number using a pictorial approach.

Ask students to imagine that they are building cubes of
different sizes from smaller cubes.

How many small cubes will be required to make a 1 by 1 by 1
cube?

How many small cubes will be required to make a 1 by 1 by 1
cube and a 2 by 2 by 2
cube?

How many small cubes will be required to make a 1 by 1 by 1
cube and a 2 by 2 by 2 cube
and a 3 by 3 by 3
cube?

...

Discuss anything that they have noticed, and ask for
predictions for constructing a set of all the cubes up to 10 by 10
by 10.

Are they surprised by the patterns they have noticed?

One way of representing this result is using the image
provided in the problem. Hand out copies of

this
worksheet, and ask students to spend some time thinking about
how the image relates to the formula. Encourage them to use
multilink cubes or draw diagrams to show how the sum of cubes is
represented in the image.

Can they draw similar images or extend the existing image to
represent the sum of the first 7, 8, 9, 10 cubes? Does this support
their predictions above?

Can they now deduce the general formula for the sum of the first n
cube numbers? In small groups, they could develop pictures to
support a proof of their generalisation, which could be presented
to the rest of the class.

### Key questions

Where are the cube numbers in the picture?

How does the picture show $(1+2+3+4+5+6)^2$?

Could you draw similar pictures for other sums of cubes? Can you

always draw such a
picture?

### Possible extension

The problem

Summing Squares offers a similar visual proof idea in three
dimensions for the sum of the first n square numbers. Students
could read the article

Proof by Induction to find out more about this important method
of proof.

### Possible support

Try the problems

Picturing Triangle Numbers and

Picturing Square Numbers.