- The general aim is to help bridge the gap between school and university mathematics.
You might wonder why such a gap exists. The reason is that mathematics is taught at school for various purposes: to improve numeracy; to hone problem-solving skills; as a service for students going on to study subjects that require some mathematical skills (economics, biology, engineering, chemistry — the list is long); and, ﬁnally, to provide a foundation for the small number of students who will continue to a specialist mathematics degree. It is a very rare school that can achieve all this, and almost inevitably the course is least successful for its smallest constituency, the real mathematicians.

- The more speciﬁc aim is to help you to prepare for STEP or other examinations required for university entrance in mathematics. To ﬁnd out more about STEP, read the next section.

It used to be said that mathematics and cricket were not spectator sports; and this is still true of mathematics. To progress as a mathematician, you have to strengthen your mathematical muscles. It is not enough just to read books or attend lectures. You have to work on problems yourself.

One way of achieving the ﬁrst of the aims set out above is to work on the second, and that is how this book is structured. It consists almost entirely of problems for you to work on.

The problems are all based on STEP questions. I chose the questions either because they are ‘nice’ — in the sense that you should get a lot of pleasure from tackling them (I did), or because I felt I had something interesting to say about them.

The ﬁrst two problems (the ‘worked problems’) are in a stream of consciousness format. They are intended to give you an idea how a trained mathematician would think when tackling them. This approach is much too long-winded to sustain for the remainder of the book, but it should help you to see what sort of questions you should be asking yourself as you work on the later problems.

Each subsequent problem occupies two pages. On the ﬁrst page is the STEP question, followed by a comment. The comments may contain hints, they may direct your attention to key points, and they may include more general discussions. On the next page is a solution; you have to turn over, so that your eye cannot accidentally fall on a key line of working. The solutions give enough working for you to be able to read them through and pick up at least the gist of the method; they may not give all the details of the calculations. For each problem, the given solution is of course just one way of producing the required result: there may be many other equally good or better ways. Finally, if there is space on the page after the solution (which is sometimes not the case, especially if diagrams have to be ﬁtted in), there is a postmortem. The postmortems may indicate what aspects of the solution you should be reviewing and they may tell you about the ideas behind the problems.

I hope that you will use the comments and solutions as springboards rather than feather beds. You will only really beneﬁt from this book if you have a good go at each problem before looking at the comment and certainly before looking at the solution. The problems are chosen so that there is something for you to learn from each one, and this will be lost to you for ever if you simply read the solution without thinking about the problem on your own.

I have given each problem a difficulty rating ranging from $\text{}\u2713\text{}$ to $\text{}\u2713\text{}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\text{}\u2713\text{}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\text{}\u2713\text{}$. Difficulty in mathematics is in the eye of the beholder: you might ﬁnd a question difficult simply because you overlooked some key step, which on another day you would not have hesitated over. You should not therefore be discouraged if you are stuck on a $\u2713$-question; though you should probably be encouraged if you get through one of the rare $\u2713$$\u2713$$\u2713$-questions without mishap.

This book is about depth not breadth. I have not tried to teach you any new topics. Instead, I want to lead you towards a deeper understanding of the material you already know. I therefore restricted myself to problems requiring knowledge of the speciﬁc and rather limited syllabus that is laid out at the end of the book. The pure mathematics section corresponds to the syllabus for STEP papers I and II. If you are studying British A-levels, Scottish Highers, or the International Baccalaureate, for example, you will be familiar with most of this material. For the mechanics and statistics/probability sections, there is less agreement about what a core syllabus should be (in the IB there is no mechanics at all), so I gave myself a freer hand.

Calculators are not required for any of the problems in this book and calculators are not permitted in STEP examinations. In the early days of STEP, calculators were permitted but they were not required for any question. It was found that candidates who tried to use calculators sometimes ended up missing the point of the question or getting a silly answer. My advice is to remove the battery so that you are not tempted.

I started this section by listing the aims of the book. You may have noticed that teaching you mathematics is not
an aim. I can’t remember where I heard the following rather nice analogy. In 1464, a huge block of Carrara
marble was carefully chosen from a quarry in Tuscany and transported to Florence, where it lay almost
untouched for many years. In 1501 it was given to the sculptor Michelangelo. He worked hard on it, chipping
away and chipping away for three years, until at last, inside the block, he found a beautiful statue of David. You
can see a picture at:

http://www.accademia.org/explore-museum/artworks/michelangelos-david/

And the analogy? I can’t teach you mathematics with this book, but I believe that much hard work on your part, chipping away at the problems, will eventually reveal the mathematician that is within you.

I hope you enjoy using this book as much as I have enjoyed putting it together.