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Preface ix

profile of the students who did well is interesting. They were not always the

final-year senior students, who were “busy” making sure they got grades

high enough to get into college, nor were they, on the whole, local contest

winners. Several were students who still had another year to spend at high

school (with some mathematics left to take); they struggled with the prob-

lem sets, but their work improved steadily during the year. One participant

gave the following assessment of her experience:

After innumerable years of “math enrichment” consisting of

pointless number games, I was prepared for another similar

course. Great was my surprise when I found this course to be

extremely challenging. Its difficulty was somewhat dismaying

at the start, but now I find that many doors have been opened

and that I have the confidence to tackle more complex ideas in

math.... I have gained a great deal of insight into a subject

I trivially used to discard as an easy school course. But most

important for me is that I have gained a vast amount of faith

in my ability to solve challenging problems.

It is assumed that the reader can manipulate simple algebraic expressions
and solve linear and quadratic equations as well as simple systems in two
variables. Some knowledge of trigonometry, exponentials and logarithms is
required, but a background in calculus is not generally needed. The few
places in which calculus intervenes can be passed over. While many of the
topics of this book will not appear in regular courses, they should be of
value through their historic importance, application or intrinsic interest
and as a backdrop to other college-level material.
Since this is not intended to be a comprehensive treatment, readers are
encouraged to delve into the often excellent publications that are recom-
mended. They will find that the boundary between elementary and deep
mathematics is often very thin, and that close to results known for centuries
one finds frontiers of modern research.
The book is organized along the following lines:
(a) Exercises: These introduce the basic ideas and advance the required
theory. Through examples, students should grasp the principal results and
techniques. The emphasis is on familiarity rather than proof; while readers
should get some sense of why a given result is true, it is expected that
they will have recourse to some other text for a formal treatment. Stu-
dents should work through the exercises in order, consulting the hints and
answers where necessary. However, if they feel that they have a general
understanding, they might skim through and work ahead, backtracking if
necessary to pick up a lost idea. Readers who find the last three sections
of Chapter 1 difficult may wish to proceed to Chapter 2 and 3 and return
to these sections later.