# problem solving

## Introduction

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?

## 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

What area of mathematics is this?

What exactly am I being asked to do?

What do I know?

What do I need to find out?

What am I uncertain about?

Can I put the problem into my own words?

### Devising a plan

Work out the first few steps before leaping in!

Have I seen something like it before?

Is there a diagram I could draw to help?

Is there another way of representing?

Would it be useful to try some suitable numbers first?

Is there some notation that will help?

### Carrying out the plan

STUCK?

Try special cases or a simpler problem

Work backwards

Guess and check

Be systematic

Work towards subgoals

Imagine your way through the problem

Has the plan failed? Know when it’s time to abandon the plan and move on.

### Looking back

Have I answered the question?

Sanity check for sense and consistency

Check the problem has been fully solved

Read through the solution and check the flow of the logic.

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

Am I getting stressed?

Is my plan working?

Am I spending too long on this?

Could I move on to something else and come back to this later?

Am I focusing on the problem?

Is my work becoming chaotic, do I need to slow down, go back and tidy up?

Do I need to STOP, PEN DOWN, THINK?

(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.