This book is a guide to discovering mathematics.
Every mathematic s textboo k i s filled with result s an d technique s whic h
once were unknown. Th e results were discovered by mathematicians who exper-
imented, conjectured, discussed their work with others, and then experimented
some more. Man y promising ideas turned out to be dead-ends, and lots of hard
work resulted i n little output. Ofte n th e first progress was the understandin g
of some special cases. Continue d work led to greater understanding, and some-
times a complex picture began to be seen as simple and familiar. B y the time
the work reaches a textbook, it bears no resemblance to its early form, and the
details of its birth and adolescence have been lost. Th e precise and methodical
exposition of a typical textbook is often the first contac t one has with the topic,
and this leads many people to mistakenly think that mathematics is a dry, rigid,
and unchanging subject.
We believe tha t th e mos t excitin g par t o f mathematic s i s th e proces s of
invention an d discovery. Th e aim of this boo k is to introduc e tha t proces s to
you, th e reader. B y mean s of a wide variety of tasks, thi s book will lea d you
to discover some real mathematics. Ther e are no formulas to memorize. Ther e
are no procedures to follow. B y looking at examples, searching for patterns in
those examples, an d then searching for the reasons behind those patterns, you
will develop your own mathematical ideas . Th e book is only a guide; its job is
to start you in the right direction , an d to bring you back if you stray too far.
The discovery is left to you.
This book is suitable for a one semester course at the beginning undergrad-
uate level. Ther e are no prerequisites. An y college student interested in discov-
ering the beauty of mathematics can enjoy a course taught from this book. A n
interested high school student will find this book to be a pleasant introductio n
to some modern areas of mathematics.
While preparin g thi s boo k w e were fortunat e t o hav e acces s t o excellen t
notes taken by Hui-Chun Lee. We thank Klaus Peters and Gretchen Wright for
helpful comments on an early version of this book.
David W. Farmer
Theodore B. Stanford