Copyright © University of Cambridge. All rights reserved.

This fun problem encouraged you to work systematically, and develop a way to record and organise your findings. For some numbers of sticks, there are several different possibilities, so it is important to approach the problem carefully, to ensure all of the options are explored.

Cong from St Peter's Roman Catholic School, Aberdeen was very thorough indeed, and sent in a fantastic solution to this problem. You must have worked very hard, Cong!

Cong sent us a table of all the different triangles you can make with 4 sticks up to 20 sticks. Making a table was a very good idea. However, it's a bit long to put on a webpage, so I've made a list of all the triangles in Cong's table so you won't have to scroll too far to read them all!

Four sticks - noneFive sticks -2, 2, 1

Six sticks - 2, 2, 2

Seven sticks -3, 3, 1 and 3, 2, 2

Eight sticks -3, 3, 2

Nine sticks - 4, 3, 2 and 4, 4, 1 and 3, 3, 3

Ten sticks - 4, 4, 2 and 4, 3, 3

Eleven sticks - 5, 5, 1 and 5, 4, 2 and 5, 3, 3 and 4, 4, 3

Twelve sticks - 5, 5, 2 and 5, 4, 3 and 4, 4, 4

Thirteen sticks - 6, 6, 1 and 6, 5, 2 and 6, 4, 3 and 5, 5, 3 and 5, 4, 4

Fourteen sticks - 6, 6,2 and 6, 5,3 and 6, 4,4 and 5, 5, 4

Fifteen sticks - 7, 7, 1 and 7, 6, 2 and 7, 5, 3 and 7, 4, 4 and 6, 6,3 and 6, 5, 4 and 5, 5, 5

Sixteen sticks - 7, 7, 2 and 7, 6, 3 and 7, 5, 4 and 6, 6, 4 and 6, 5, 5

Seventeen sticks - 8, 8, 1 and 8, 7, 2 and 8, 6, 3 and 8, 5, 4 and 7, 7, 3 and 7, 6, 4 and 7, 5, 5 and 6, 6, 5

Eighteen sticks - 8, 8, 2 and 8, 7, 3 and 8, 6,4 and 8, 5, 5 and 7, 7, 4 and 7, 6, 5 and 6, 6, 6

Nineteen sticks - 9, 9, 1 and 9, 8, 2 and 9, 7, 3 and 9, 6, 4 and 9, 5, 5 and 8, 8, 3 and 8, 7, 4 and 8, 6, 5 and 7, 7, 5 and 7, 6, 6

Twenty sticks - 9, 9, 2 and 9, 8, 3 and 9, 7, 4 and 9, 6, 5 and 8, 8, 4 and 8, 7, 5 and 8, 6, 6 and 7, 7, 6

Cong has made each triangle in turn in a very systematic way. Can you see how Cong has started with the longest side possible each time and worked down? He always lists the side lengths with the longest first as well. This makes it very easy to check. Well done!

Cong summarised the number of possible triangles with each number of sticks in another table:

Number of sticks | Number of triangles |

3 | 1 |

4 | 0 |

5 | 1 |

6 | 1 |

7 | 2 |

8 | 1 |

9 | 3 |

10 | 2 |

11 | 4 |

12 | 3 |

13 | 5 |

14 | 4 |

15 | 7 |

16 | 5 |

17 | 8 |

18 | 7 |

19 | 10 |

20 | 8 |

He also showed the same information on a bar chart. Notice that there are no possible triangles with four sticks. Holly from Anston Brook Primary School said:

If you try to make four it will end up as two lines.

Children from Riversdale Primary School examined the "rules" for the longest side of any triangle:

We discovered that the longest side of any triangle could be
found by taking one away and dividing by two for an odd number and
by dividing by two and then taking one away for an even
number.

Well done for spotting this. Can you see why this is the case? This is actually an example of the "triangle inequality". Do not worry, this sounds more difficult than it is! The triangle inequality applies to any triangle, and states that the length of a given side must be less than or equal to the sum of the two other sides. So, imagine that you have a triangle with sides of three, four and five. The longest side, five, is less than the sum of the two other sides ($3+4=7$). So, this is an example of the triangle inequality in action!

What does this mean for the problem with sticks? Well, firstly it provides a quick way to check if the triangles you have thought about are possible. Secondly, it can also save you some time when working out all of the possibilities. For example, you may be able to see that, for this task with sticks, the triangle inequality means that the length of any side cannot be greater than half of the total number of sticks. If the length of a side used is more than half of the total number of sticks, this would not agree with the triangle inequality, and a triangle could not be made. Why not have a go at this to see for yourself to check you understand why the triangle inequality is true?

Cong also spotted a general "rule":

The odd numbers of sticks get more triangles.

Yes, I see what you mean, Cong. Children from Riversdale Primary School also submitted a similar table. They also made a similar observation:

Odd numbers of seven and above always have more triangles than the even numbers adjacent to them. Odd numbers in the three times tables have even more as they have an equilateral triangle as well.

In addition, Ellie, Emma, Olivia and Ibrahim from Lakeside looked at "rules" for being able to make equilateral triangles:

If the number of sticks is a multiple of three, an equilateral
triangle can be made because if the number is divisible by $3$ the
number of sticks making each side will be the same.

Well done for these fantastic solutions and observations!