This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters.

From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition"

 “… It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. […] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.”

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. ... 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

"... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..."

LMS Newsletter, January 1999

"Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... "

SIGACT News, December 2011

Table of Contents

Number Theory

Chapter 1 Six proofs of the infinity of primes

Chapter 2 Bertrand's postulate

Chapter 3 Binomial coefficients are (almost) never powers

Chapter 4 Representing numbers as sums of two squares

Chapter 5 The law of quadratic reciprocity

Chapter 6 Every finite division ring is a field

Chapter 7 The spectral theorem and Hadamard's determinant problem

Chapter 8 Some irrational numbers

Chapter 9 Four times TT2/6

Geometry

Chapter 10 Hilbert's third problem: decomposing polyhedra

Chapter 11 Lines in the plane and decompositions of graphs

Chapter 12 The slope problem

Chapter 13 Three applications of Euler's formula

Chapter 14 Cauchy's rigidity theorem

Chapter 15 The Borromean rings don't exist

Chapter 16 Touching simplices

Chapter 17 Every large point set has an obtuse angle

Chapter 18 Borsuk’s conjecture 

Analysis

Chapter 19 Sets, functions, and the continuum hypothesis

Chapter 20 In praise of inequalities

Chapter 21 The fundamental theorem of algebra

Chapter 22 One square and an odd number of triangles

Chapter 23 A theorem of P61ya on polynomials

Chapter 24 Van der Waerden’s permanent conjecture

Chapter 25 On a lemma of Littlewood and Offord

Chapter 26 Cotangent and the Herglotz trick

Chapter 27 Buffon's needle problem

Combinatorics

Chapter 28 Pigeon-hole and double counting

Chapter 29 Tiling rectangles

Chapter 30 Three famous theorems on finite sets

Chapter 31 Shuffling cards

Chapter 32 Lattice paths and determinants

Chapter 33 Cayley's formula for the number of trees

Chapter 34 Identities versus bijections

Chapter 35 The finite Kakeya problem

Chapter 36 Completing Latin squares

Graph Theory

Chapter 37 Permanents and the power of entropy

Chapter 38 The Dinitz problem

Chapter 39 Five-coloring plane graphs

Chapter 40 How to guard a museum

Chapter 41 Turan's graph theorem

Chapter 42 Communicating without errors

Chapter 43 The chromatic number of Kneser graphs

Chapter 44 Of friends and politicians

Chapter 45 Probability makes counting (sometimes) easy

About the Author

Martin Aigner received his Ph.D. from the University of Vienna and has been professor of mathematics at the Freie Universität Berlin since 1974. He has published in various fields of combinatorics and graph theory and is the author of several monographs on discrete mathematics, among them the Springer books Combinatorial Theory and A Course on Enumeration. Martin Aigner is a recipient of the 1996 Lester R. Ford Award for mathematical exposition of the Mathematical Association of America MAA.

Günter M. Ziegler received his Ph.D. from M.I.T. and has been professor of mathematics in Berlin – first at TU Berlin, now at Freie Universität – since 1995. He has published in discrete mathematics, geometry, topology, and optimization, including the Lectures on Polytopes with Springer, as well as „Do I Count? Stories from Mathematics“. Günter M. Ziegler is a recipient of the 2006 Chauvenet Prize of the MAA for his expository writing and the 2008 Communicator award of the German Science Foundation.

Martin Aigner and Günter M. Ziegler have started their work on Proofs from THE BOOK in 1995 together with Paul Erdös. The first edition of this book appeared in 1998 – it has since been translated into 13 languages: Brazilian, Chinese, German, Farsi, French, Hungarian, Italian, Japanese, Korean, Polish, Russian, Spanish, and Turkish.