This book follows on from the book Creative Mathematics in this series which began with three essays (on research into, on writing about and on presenting mathematics) and then continued with a series of problems, each of which was divided into three parts. Part I provided an introduction to the problem followed by some elementary questions about the problem. Part II contained an answer to these questions, as well as a deeper discussion and a generalisation of the problem. This led to more advanced questions which were discussed in Part III.

This book is a natural development of this approach, the main purpose of which is to give the reader experience in working on (as far as the reader is concerned) unsolved problems. The problems in this book are, generally speaking, more difficult than those in Creative Mathematics, and we assume a greater level of maturity of the reader.

One of the main purposes of this book is to show that mathematical problems are often solved using mathematics that is not, at first sight, connected to the problem, and readers are encouraged (and even urged) to consider as wide a variety of mathematical ideas as possible when trying to solve a problem. The reader will no doubt have met problems in what might be called ‘recreational mathematics’ where problems are solved for amusement, without necessarily understanding or investigating the key mathematical ideas that lie behind the solution. Here, by contrast, we focus on the important underlying ideas rather than on the solution itself.