Number Theory -- Six proofs of the infinity of primes -- Bertrand’s postulate -- Binomial coefficients are (almost) never powers -- Representing numbers as sums of two squares -- The law of quadratic reciprocity -- Every finite division ring is a field -- Some irrational numbers -- Three times ?²/6 -- Geometry -- Hilbert’s third problem: decomposing polyhedra -- Lines in the plane and decompositions of graphs -- The slope problem -- Three applications of Euler’s formula -- Cauchy’s rigidity theorem -- Touching simplices -- Every large point set has an obtuse angle -- Borsuk’s conjecture -- Analysis -- Sets, functions, and the continuum hypothesis -- In praise of inequalities -- The fundamental theorem of algebra -- One square and an odd number of triangles -- A theorem of Pólya on polynomials -- On a lemma of Littlewood and Offord -- Cotangent and the Herglotz trick -- Buffon’s needle problem -- Combinatorics -- Pigeon-hole and double counting -- Tiling rectangles -- Three famous theorems on finite sets -- Shuffling cards -- Lattice paths and determinants -- Cayley’s formula for the number of trees -- Identities versus bijections -- Completing Latin squares -- Graph Theory -- The Dinitz problem -- Five-coloring plane graphs -- How to guard a museum -- Turán’s graph theorem -- Communicating without errors -- The chromatic number of Kneser graphs -- Of friends and politicians -- Probability makes counting (sometimes) easy.

This revised and enlarged fourth edition of "Proofs from THE BOOK" features five new chapters, which treat classical results such as the "Fundamental Theorem of Algebra", problems about tilings, but also quite recent proofs, for example of the Kneser conjecture in graph theory. The new edition also presents further improvements and surprises, among them a new proof for "Hilbert's Third Problem". 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 "... 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. Moreover, the exposition makes them transparent. ..." LMS Newsletter, January 1999