he problem with most mathematics textbooks is that they don’t teach much mathematics. They teach you how to apply some procedures and formulas, but that’s not what mathematicians do; that’s what computers do. In a world where you can look up any formula in seconds, and computers can be programmed to apply them, we don’t need people who have memorised how to solve a bunch of problems; we need people who can tackle problems that no-one has solved before, problems that require creativity and cunning. That’s what mathematics is about. That’s what this book is about.

Mathematical problems make for delightful puzzles which you might solve just for the fun of it, but there is also a practical side to mathematics. Why not practise mathematical problem solving by deriving all the important mathematical facts that you’d normally learn by rote, by answering questions asked throughout history by mathematicians trying to understand the heavens, help people through better technology, manage their family’s finances, or get to the fundamental truth and beauty of the universe?

This book leads you to discover all the mathematical rules and formulae normally taught in schools. Not only does this make you a better problem solver, but also satisfies your curiosity about why mathematical formulae work, and how people figured them out.

I’m not claiming that discovery is the most efficient way to learn how to apply a formula though. If I ask you to figure out how to find the area of a triangle, it isn’t because that will make you better at calculating areas of triangles; it’s to give you practice in mathematical problem solving. If you just want to know the formula you can skip over the discovery questions entirely. Any important conclusions will be written in a box so you can easily find them.

I’ve tried to give you as much choice as possible in what you learn and in what order. You can let your curiosity and individual needs guide you. You don’t have to read this book in the order it’s written. I’ve been frustrated by other textbooks that don’t let you do this – they don’t tell you which parts you need to read before other parts, and which parts you can skip. If you skip ahead in this book and come across a symbol you don’t recognise, you can look it up in Appendix B: Symbols.

You can also choose how hard the problems should be. I’ve made them very challenging, but you can use the hints to make them easier. Sometimes you might need to read the start of the answer to get some ideas, or just treat it as a worked example and read through the whole answer.

If you’re new to mathematical problem solving then the best advice I can give you is to give yourself permission to be wrong. In solving a problem, you’ll come up with ideas that don’t get you anywhere, and that’s perfectly normal. You just need to keep playing around. If you put pressure on yourself to get to the perfect solution as quickly as possible then it stifles creativity. Instead, treat it as a game and let your imagination run wild.

Practice Questions

There are many facts and procedures that need to be memorised in mathematics – everything from times tables to how to differentiate trigonometric functions. Fortunately there has been a lot of research on the most effective and efficient ways to memorise information, and we now know that one of the best ways is to try to recall the information, check the answer, and then repeat that process some time later depending on how hard the fact was to remember.

Many students use ‘spaced repetition software’ to schedule revisions. I myself use it to learn foreign language vocabulary. You’re supposed to open the program every day and it will ask you the questions that you need to revise that day.

There is a strict distinction in this book between the regular questions and practice questions. The regular questions are usually quite complicated and require a lot of creative problem solving, so they have hints and full solutions. Practice questions on the other hand should be as simple as possible – ideally they should only require recalling a single fact.

You would normally look at a regular question until you understand it, then you don’t need to go over it again. But you need to return to practice questions over and over again, for many weeks, until you know them by heart. This process is made so much easier by spaced repetition software because it can figure out the ideal day to revise a fact so that you don’t waste your time by revising too early, or forget the fact completely by revising too late.

It’s a bit difficult to use spaced repetition software to practise mathematics, because you often need a slightly different variation of the question each time. For example, if you want to remember that $(a+b)^2 = a^2 + 2ab + b^2$, and you only practise with the question ‘Expand $(a+b)^2$’ then you probably won’t be able to expand $(x+y)^2$ very quickly. You really need to practise the procedure on different variables. Or to take a more basic example, if you only practise remembering $8\times 3$, you might be stumped when you come across $3\times 8$. It’s the same fact (that the numbers $8$ and $3$ in any order have a product of $24$), but you need to practise each variation of it.

At the moment, I don’t know of any spaced repetition software that allows you to make variations of a question without a custom plugin. Making different ‘cards’ for each variation isn’t a good idea because then you’d be practising the fact too often. Rather than getting people to download the software and the plugin and work out how to install it, I decided to make a web-based spaced repetition program.

If you log in to, it decides what questions to ask you that day, and cycles through the different variations. You add questions by clicking on the ‘Add these questions to my cards’ link whenever you come across a new fact on the website.

Of course you can do your revision any way you like; you don’t have to use spaced repetition software. But I recommend doing so, and practising for a few minutes every day. It’s a good idea to decide in advance when you’ll be practising, for example ‘while I’m waiting for the bus in the morning’, or ‘after dinner’. Or you can use some sort of alarm to remind you.