What do we mean by ‘problem solving’, and what benefits should we expect from solving problems? Of course, by solving a given problem we obtain an answer, from which we may be able to draw a conclusion. There are practical problems which arise in industry where it is important, perhaps from financial considerations, to find an answer, and there is only a secondary interest in how the answer has been obtained. In such cases, the answer is the main objective. While I hope that the benefits that accrue from studying this text will improve the reader's skills in solving such problems, we shall focus here on the process of solving problems, rather than on the answers. In fact, the answers will be of little interest to us, except in that they illustrate a method, or suggest further investigations.

The primary object of our study is the problem itself, and its main roles are to show us how mathematics can be applied in a variety of ways, to provide motivation for us to learnmoremathematics, and to see and experience how simple cases lead to a greater understanding, and hence to further problems, generalisations, and so on. Solving problems in this sense is like a journey of exploration; we must constantly pay attention to the local details, but all the time be aware of how these details fit into a much larger, unknown, picture. It has been said that we do not understand a piece of mathematics unless we can generalise it, and a generalisation usually calls for different ideas. Thus we should see our attempt to solve a particular problem as a continuously evolving account of a wider problem.

Many educationalists favour this experimental approach, believing that one can only learn mathematics by doing mathematics oneself. This has much to recommend it but, by itself, it cannot be enough. There is no doubt that to succeed in mathematics one needs a vast supply of mathematical knowledge, and one cannot be expected to provide this entirely by one's own efforts. At some stage we must learn from others (Newton's phrase ‘standing on the shoulders of giants’ springs to mind here), so what is the best way to learn more mathematics? This text is an attempt to show how problems can motivate this learning.