# Making Cuboids

## Problem

Let's say you can only use two different lengths - $2$ units and $4$ units.

If you are using these lengths to make the sides of rectangles, how many different ones can you make? (Squares are just special rectangles!)

There are three because two are the same - just rotated.

But we are not making $2$-dimensional rectangles!

We are going to make $3$-dimensional cuboids.

Using just these two lengths as the edges of the cuboids how many different ones can you make?

Here are two of them. How many more are there?

It is a good idea to make them from squared paper and sticky tape, but you also need to find a way to record your results.

This is one way to show the results above, making a list.

The smallest is | $2 \times 2 \times2$ |

The middle sized one is | $2 \times4 \times4$ |

You should find a way to record your results that makes sense to you.

How can you make sure that your cuboids are not the same if rotated?

How many different cuboids did you get?

But what if we have three different lengths: $2$ units, $3$ units and $4$ units?

How many different cuboids can you make now?

Make them from squared paper and sticky tape.

Remember to record your results.

Find a way of showing what they look like.

Once they are made perhaps you can arrange them in order of size - but how will you work this out?

## Getting Started

Using some squared paper and having sticky tape and scissors would be a good idea.

How will you know that all your cuboids are different from each other?

What can you make just using the $2$ unit lengths?

What can you make using just the $3$ unit lengths?

What can you make using just the $4$ unit lengths etc ...?

## Student Solutions

There are lots of answers to this problem, depending on what questions you choose to ask.

Here is some work that Lydia from St Mary's Catholic High School in Chesterfield sent to us:

There must be ten cuboids:

$6\times6\times6$ |
$4\times4\times4$ |
$8\times8\times8$ |
$6\times8\times8$ |
$6\times8\times6$ |
$4\times8\times4$ |

$3\times3\times3$ | $2\times2\times2$ | $2\times4\times4$ | $3\times4\times4$ | $3\times4\times3$ | $2\times4\times2$ |

$4\times8\times8$ |
$4\times6\times6$ |
$4\times6\times4$ |
$4\times6\times8$ |

$2\times4\times4$ | $2\times3\times3$ | $2\times3\times2$ | $2\times3\times4$ |

These are the only possible shapes using the three lines (edges) with $2, 3$ and $4$ units of length.

To make these cuboids I had to double the length of each edge so I ended up with $4, 6$ and $8$ units of length, this made it easier to construct them.

The workings out are on the pictures below.

Thank you, Lydia. You seem to have worked hard at this challenge.