X Preface(b) Explorations: While these are inserted near related material in the
exercises, readers should not feel obliged to work at them right away. In
general, they are not needed to follow the main thread. Their purpose is
to raise questions and encourage investigation; some explorations involve
new theory, some are straightforward problems and others involve questions
which have deep ramifications. All are intended only as starting points. The
investigations should be revisited as more experience is gained.
(c) Problems: Each chapter concludes with problems drawn from a vari-
ety of sources: journals such as the American Mathematical Month/y and
Crux Mathematicorum, contests and Olympiads, examination and scholar-
ship papers. The first ten or so of each set are moderately difficult, but after
that they are not arranged in any particular order. Some are tough. Stu-
dents who get blocked should return to the problem intermittently. Hints
are provided.
In referring to exercises and problems, I will use a single number to refer
to a question in the same section, and the section number with the question
number separated by a dot to refer to a question in a different section of
the same chapter. A triple designation will refer to a question in a different
chapter; for example, 2.3.4 refers to Exercise 4 of Section 3 of Chapter 2.
One source of problems is worth special mention. Until the mid 1960s
students in Ontario wrote Grade 13 examinations set by the provincial
Department of Education. Besides the regular papers (Algebra, Analytic
Geometry, Trigonometry and Statistics), students vying for a university
scholarship had the opportunity to write a Mathematical Problems Paper.
Through Jeff Martin of the Etobicoke Board of Education, I have acquired
copies of these papers. In many of the problems, I have been struck by
the emphasis on mathematical competence; they could be done, not by a
leap of ingenuity, but rather through a thorough grasp of standard but
somewhat sophisticated techniques. These problem papers (and I am sure
they had their counterpart in other jurisdictions) should not be lost to our
collective memories; they are indicative of the skills which were expected
of a previous generation of students who planned to do university level
mathematics. I believe that students still need to be skillful, and indeed
should not be denied the pleasure of feeling competent in what they do.
I would like to acknowledge the assistance and advice of various organiza-
tions and individuals.‘In particular, I am indebted to the Ontario Ministry
of Education and the Queen’s Printer of Ontario for permission to use
problems from the Ontario Problems Papers, the Canadian Mathematical
Society for permission to use problems appearing in Crux Mathematicorum,
the Canadian Mathematical Olympiad and its other publications, and the
Mathematical Association of America for permission to use problems from
the Putnam Competition, the Monthly and the Magazine.
I am grateful to the Samuel Beatty Fund, administered by a board rep-
resenting the graduates of Mathematics and Physics at the University of