Proofs from THE BOOK

From the Reviews:

"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999

"... the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." LMS Newsletter, January 1999

This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals".

The proofs are almost all magnificent (although I wonder how Buffon and his needles got in there) and even the well-known and time-honoured ones have a new twist or new extension.

The level of mathematics required to follow the proofs is reasonably low (high-school 'A' levels in the British system, no idea about other countries) although the book gives a deeper explanation in some areas (e.g. trans-finite arithmetic) than in others (e.g. number theory). I wonder if this unevenness reflects the interests of the authors.

But these are tiny nit-pickings. This is a wonderful and inspiring book and reading it should be made compulsory by the government in all high-school mathematics classes.


A fitting tribute to the great Paul Erdos   March 21, 2001
KARTIK KRISHNAN S. (Hamilton, Ontario Canada)
12 out of 14 found this review helpful

Paul Erdos once remarked that you need not believe in God, but you certainly have to believe in the book in which God maintains the "perfect" mathematical proofs. Martin Aigner and Gunter Ziegler have certainly done a great job with this book, a fitting tribute to the great Erdos himself.

I had purchased a copy of the 1st edition of this book and was plesantly surprised that the authors had come up with a 2nd edition, with a few more "perfect" proofs.

My personal favorites are "The Shannon capacity of a graph". where the Lovasz theta number would eventually lead to semidefinite programming, Erdos' probabilistic method where probability makes counting sometimes easy, computing the number of trees in a graph, how many guards it takes to guard a museum, and the section on Turan's theorem.

This book deserves to be on the bookshelves of both amateur and professional mathematicians.


A Nice Survey of some Interesting Propositions with (pretty much) Elementary Solutions   December 29, 2005
Giovanni Dimatteo (Padova, Italy)
8 out of 9 found this review helpful

Most of this book could be read by an undergraduate with only background in calculus and basic discrete math. The proofs are mostly self-contained, and there are helpful appendices to each chapter when prerequisite material is needed. I think this would be a good book for undergraduates or enthusiastic high school students to read for fun, or for lecturers to draw interesting examples from.

The proofs, while "elementary," are sometimes quite involved, however, and do require some maturity to be able to appreciate the "big picture" of each proof, and it's tempting to "island-hop" (i.e., just check that each if... then... works out in the proof, without looking at the whole thing put together). This would be especially nice material for an undergraduate math club or for math enthusiasts to read at liesure. The problems are also highly "integrated," in that they make use of ideas from several different types of math, and are not usually straightforward, so they are good examples of creative solutions to challenging problems.