# Numbers as Shapes

## Problem

Here are the numbers $1$ to $6$ drawn using coloured squares:

We can call these numbers squares, rectangles or sticks.

$1$ is a square but just a single one.

We can't make $2$ into a square so $2$ is a stick.

We can't make $3$ into a square or a rectangle so $3$ can only be a stick.

We can make $4$ into a square.

We can't make $5$ into a square or a rectangle, so $5$ can only be a stick.

We can't make $6$ into a square, but we can make $6$ into a rectangle.

If you had $7$ yellow squares, what could you make them into? A square, a rectangle or a stick? How about $8$ squares?

Try this with the numbers up to and including $20$.

Which ones can only be sticks?

Which ones can make rectangles?

Which ones are squares?

What do you notice?

*This problem was inspired by a session at an ATM conference, led by Tandi Clausen-May.*

## Getting Started

It might be a good idea to find some cubes or tiles so you can make the numbers. Or you could use some squared paper and colour the squares.

## Student Solutions

Susie sent in a table to show her solution to this problem:

Numbers that can be squares | Numbers that can be sticks | Numbers that can be rectangles | Numbers that can only be sticks |

1 | 2 | 6 | 2 |

4 | 3 | 8 | 3 |

9 | 4 | 10 | 5 |

16 | 5 | 12 | 7 |

6 | 14 | 11 | |

7 | 15 | 13 | |

8 | 16 | 17 | |

9 | 18 | 19 | |

10 | 20 | ||

11 | |||

12 | |||

13 | |||

14 | |||

15 | |||

16 | |||

17 | |||

18 | |||

19 | |||

20 |

Susie also thought:

Children from Rampart School in the US sent a very full solution. They said:

The prime numbers from $1$ to $20$ ($2, 3, 5, 7, 11, 13, 17$, and $19$) can only be sticks. Each prime number has only two factors, $1$ and itself, so none of them can make rectangles. They can only make sticks of dimension $1 \times $ the prime number.

The numbers that are neither prime nor square ($6, 8, 10, 12, 14, 15, 18, 20$) can make rectangles because they all have factors other than $1$ and themselves. For example, $20$ has the factors $1$ and $20$, $2$ and $10$, and $4$ and $5$.

This leaves the square numbers that, subsequently, are the only numbers that can form the squares, for obvious reasons. Every square number can have the form $n \times n$, which also relates to the dimensions of the square.

So, we notice that only square numbers can form squares; prime numbers form sticks; and the composite, non-square numbers form rectangles.

Thank you also to Jack from Allerton Grange, Sophie from Belgium and Nathan from Rushmore Primary who sent in well-explained solutions.