Famous Puzzles of Great Mathematicians

Author(s): 
Miodrag S. Petkoviç
Publisher: 
American Mathematical Society
Year: 
2009
ISBN: 
ISBN-10: 0-8218-4814-2; ISBN-13: 978-0-8218-4814-2
Price (tentative): 
US36$ (List Price); US28.80$ (Member Price)
Short description: 

This book is an interesting work on recreational problems on mathematics in the history. These problems have survived, not because they were fostered, by the textbook writers, but because of their inherent appeal to our love of mystery. This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. About 65 intriguing problems, marked by * in the book, are given as exercises, to the readers. The selected problems do not require advanced mathematics, making this excellent book accessible to a variety of readers.
The history of mathematics is replete with examples of puzzles, games and entertaining problems that have fostered the development of new and emergent disciplines and sparked further research. The book is intended principally to amuse and entertain, incidentally to introduce the general reader to other intriguing mathematical topics and ideas. With this book, many stories and famous puzzles can be very useful to prepare teaching or lecture notes, to inspire and amuse students, and to instill affection for mathematics. In my opinion, it is a stimulating and excellent text will be for amuse and entertaining, and the teaching in recreational mathematics.

URL for publisher, author, or book: 
http://w.w.w.ams.org; http://w.w.w.ams.org/bookpages/mbk-63; order code: MBK/63.
MSC main category: 
00 General
MSC category: 
2000 MSC: 00A08, 97A20, 01A05, 01A70, 05A05, 05C45, 05C90
Other MSC categories: 
2000 MSC: 11D04, 11D09, 51E10, 51M16, 52C15, 52C22, 97D4O
Review: 

This book is an interesting work on recreational problems on mathematics in the history. This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. About 65 intriguing problems, marked by *, are given as exercises, to the readers. The selected problems do not require advanced mathematics, making this excellent book accessible to a variety of readers.
The book is intended principally to amuse and entertain, incidentally to introduce the general reader to other intriguing mathematical topics and ideas. Important relations and connections exist between those problems originally meant to amuse and entertain and mathematical concepts critical to combinatorial and chess, geometrical, and arithmetical puzzles, geometry, graph theory, optimization theory, probability, number theory, and related areas.
The book contains eleven chapters and four appendices.
The first six chapters are on: recreational mathematics (a brief and concise history of mathematics); arithmetics; number theory; geometry; tiling and packing and physics.

The chapter one is on Recreational Mathematics contains, before taking up the noteworthy mathematical thinkers and their memorable problems, a brief overview of the history of mathematical recreations. Perhaps the oldest known example is the magic square. Known as lo-shu to Chinese mathematicians around 2200 B.C., the magic square was supposedly constructed during the reign of the Emperor Yii. Chinese myth holds that Emperor Yii saw a tortoise of divine creation swimming in the Yellow River with the lo-shu, or magic square figure, adorning its shell. The Rhind (or Ahmes) papyrus dating to around 1650 B.C., suggests that the early Egyptians based their mathematics problems in puzzle form. Perhaps their main purpose was to provide intellectual pleasure. The ancient Greeks also delighted in the creation of problems strictly for amusement. The cattle problem is one of the most famous problems in number theory, whose complete solution was not found until 1965 by a digital computer. The Dido´s problem, cited by Virgil, and the elegant solution, established by Jacob Steiner, regarded as first problem in a new mathematical discipline, established 17 centuries later, as calculus of variations. Others interesting problems are included in this book, as Josephus problem; Alcuin of York´s problems and variants; Fibonacci´s amusing problems; IbnKallikan´s problem about the number of wheat grains on a standard 8x8 chessboard; and many instances interesting.
In chapter two, named Arithmetics, are related many instances of famous puzzles: Diophantus´ age; Mahavira: number of arrows; Fibonacci´s: square numbers problem, money in a pile, sequence, how many rabbits?; triangle with integral sides (Bachet); sides of two cubes (Viète), and others puzzles.
Chapter three on Number Theory deal with the following famous: cattle problem; dividing the square; wine problem; amicable numbers, Qorra formula; how many soldiers?; horses and bulls; the sailors, the coconuts and the monkey; stamp combinations, with a generalized problem of Frobenius and Sylvester.
In chapter four, on Geometry, are related some instances: Arbelos problem of Arquimedes, archimedean circles, perpendicular distance and two touching circles; minimal distance of Heron, a fly and a drop of honey, peninsula problem; dissection of three squares; dissection of four triangles; the minimal sum of distances in a triangle; volumes of cylinders and spheres of Kepler; Dido´s problem; the shortest bisecting arc of area de Polya, and other interesting problems.
Chapter five on Tiling and Packing deal interesting and amusing problems in the history of mathematics as: mosaics; non-periodic tiling; maximum area by pentaminoes; kissing spheres; the densest sphere packing, and the cube-packing puzzles.
The chapter six is related to famous problems on Physics as the gold crown of King Hiero; the length of traveled trip; meeting of ships; a girl and the bird, and the lion and the man.

There follows chapters on combinatorics; probability; graphs; chess and miscellany which contains problems from Alcuin de York, Abu´lWafa, Fibonacci, Bachet, Huygens, Newton and Euler.

The chapter seven on Combinatorics, deal the Josephus problem; rings puzzle; the problem of the misaddressed letters; eulerian squares, and the famous Kirkman´s schoolgirls problem. Others interesting problems as counting problem, the tower of Hanoi, the tree planting problem, etc., are included in this chapter.
In the chapter 8 on Probability are considered, the famous problem of the points, gambling game with dice, gambler´s ruin, Petersburg paradox, the probability problem with the misaddressed letters, and the match problem are all treated of elegant form.
The chapter nine deal on Graphs. Contains the famous problem of Königsberg´ bridges, Hamilton´game on a dodecahedron, some problems of Alcuin of York, Erdös, Poinsot, Poisson, Listing, and others problems of interest.
In the chapter ten on chess, many instances and problems are considered: classical knight, queen, rooks and the longest uncrossed knight´s tour.
The chapter eleven, titled Miscellany, contains problems from Alcuin de York, Abu´lWafa, Fibonacci, Bachet, Huygens, Newton and Euler.
Finally, the four appendices, for to help readers, on refer: method of continued fractions for solving Pell´s equation; geometrical inversion; some basic facts from graph theory; linear differences equations with constant coefficients.
The author includes bibliographical references and index, and sometimes amusing anecdotal material, with the objective that to underscore the informal and recreational character of the book.
The book is also high recommended also for individual study. In my opinion, it is a stimulating and excellent text will be for amuse and entertaining, and the teaching in recreational mathematics.

Reviewer: 
Francisco José Cano Sevilla
Affiliation: 
Universidad Complutense de Madrid

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