problem solving


This session is slightly different - there's no particular topic; rather it's about trying to think creatively about mathematics. How do mathematicians go about solving problems? What do you do when you're stuck on a problem? What strategies should be in your toolkit for helping with probems?

When asked what they think mathematics is, many school pupils will reply along the lines of it being a series of techniques to answer 'sums'. I'd like to encourage you to think of it as much more than that - we use mathematics to solve all sorts of real-world problems, but it can also be a fascinating - and sometimes frustrating - pursuit in its own right.

1. How do we go about solving problems?

Have a look at these problems, and spend an hour or so trying (some of) them. Try to resist the temptation to look ahead to the solutions, and don't worry if you get tied up on just one of them!

Here are some video and written solutions to the problems, with some discussion of techniques used. Watch them through, and have a think about whether you can learn from the techniques employed. Are there any general ideas that hold true? What strategies might you employ with an unfamiliar problem?

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6 and written solution.

Question 7

2. A problem-solving strategy

Below is a list of strategies you might employ when solving problems. It's not the case that you will consciously use each of the every time you meet a new problem, but it's definitely worth asking yourself these questions (and even writing responses to them) when you do meet something unfamiliar. Read through the ideas, and bear it in mind for future problems; remember to come back to it when you're working through the problems later in the week.

Understanding the problem

Devising a plan

Carrying out the plan


Looking back

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

(Adapted from A Guide to Problem Solving on the NRICH website)

3. A set of problems

Work through (some of) these problems. You should try to do them on your own, and think about some of the strategies discussed above. Don't worry if you can't do them all; it's better - and more satisfying - to do one problem well than to have a cursory glance at many. Try also to focus on communicating your solutions clearly.

Solutions here.

Other sets of problems can be found on the UKMT website.