I’m sure that many of you will have organised your students’ participation in the Junior, Intermediate or Senior Maths Challenges that the United Kingdom Mathematics Trust (UKMT) run every year. They’ve always been in the ‘armoury’ of strategies that I’ve employed to raise the profile of maths within school, as well as providing a challenge for the most enthusiastic and able young mathematicians.

As well as being a repository of past questions and solutions, the Challenges section of the UKMT website also contains sets of solutions and investigations for each of the Junior, Intermediate, and Senior Maths Challenges set over the last decade or so. Not every question has ideas for further investigation attached to it, but there’s plenty of inspiration to be found if you spend a few minutes browsing an individual paper and its solutions.

I particularly enjoy thinking about some of the questions that don’t have investigations attached, trying to work out ways in which I might extend this problem with a group of students. A recent favourite has been this question from the 2011 Intermediate Maths Challenge paper.

What are the key features of this problem? What would your students come up with?

- “
*It starts with an angle of 150° on the right-hand side and you have to work towards the left.*” - “
*The apex angles at the tops of the triangles half each time as you work right to left.*” - “
*Two angles ‘rest’ onto the first triangle on the right.*”

What could you change about this problem?

- Start with a different angle other than 150° on the right-hand side. How does this effect the value of angle

- on the left? Can you predict whether it will get bigger or smaller? Are there any limits on the range of values that the 150° angle can take?
- What happens if you change the 80° angle and/or the rule that the other angles follow?
- Is it possible to introduce another triangle leaning in on the left?

I hope you feel inspired to take a closer look at the UKMT website and the wealth of ideas it has amongst its solutions and investigations. I also think that there’s much to be gained by exposing some of these individual Maths Challenge questions to students and getting them to think mathematically about the key features of a question and what happens if some of these features are changed or varied.