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Famous Puzzles of Great Mathematicians

Miodrag S. Petković
American Mathematical Society
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Underwood Dudley
, on

The book’s title is a bit overstated, because few would put Alcuin of York or Bachet de Méziriac on their lists of great mathematicians, nor would they call the isoperimetric problem a “puzzle.” In the other direction, Gauss appears when he shouldn’t — Gauss was much too serious to pose puzzles and never did. However, titles are constructed so as to maximize sales and Selected Topics in Recreational Mathematics and Some Extensions probably did not receive much consideration.

The book’s contents are many of the classics of recreational mathematics: the Josephus problem of arranging people in a circle so that the undesirable ones get thrown overboard, the wolf, goat, and cabbage crossing the river, and so on, with solutions. There are also many problems not usually thought of as being recreational, such as determining the diameter of a solid sphere with a straightedge and compass construction, also with solutions. Extensive historical information is included and the bibliography includes one hundred and ninety-four entries.

The author says that his book is intended “to amuse and entertain (and only incidentally to introduce the general reader to other intriguing mathematical topics).” There are more than sixty problems for the reader to attempt, with answers at the end of chapters. There are many illustrations and the book is unusually free of misprints (except for names — the author gets seven of them wrong in one place or another).

The author has done an admirably accurate and thorough job in presenting his material, though somewhere he has picked up the odd notion that Isaac Newton was “of Jewish origin.”

To give more of an idea of what the book contains, here is a list of every ninth entry in the subheadings of the chapters in the table of contents:

  • Diophantus’ age
  • Wine and water
  • How many soldiers?
  • Height of a suspended string
  • Division of space by planes
  • Cube-packing puzzles
  • Rings puzzle
  • Problem of married couples
  • The problem of Königsberg’s bridges
  • Non-attacking rooks

Petković keeps the mathematics simple, but does not hesitate to give the recursion relations for the convergents of a continued fraction and explain how the reader can use them to solve x2 – 61y2 =1, the equation that arises in the “How many soldiers?” problem. There are many summation signs and the double-angle formula is used without comment.

The book contains little that is new; connoisseurs of recreational mathematics, with the works of Dudeney, Loyd, and Gardner at their fingertips, may pass the volume by. But for those not familiar with the field, I can think of no better introduction. The author has put much labor into it, to good effect. The problems are here, their histories are here, the mathematics needed to solve them is here. The book would be the ideal graduation present for a mathematics major, an ideal prize for the winner of an integration contest, an ideal book to have lying around a mathematics department (if properly chained down, that is). I hope that it is successful.

Woody Dudley, who retired from teaching in 2004, first saw the wolf, goat, and cabbage crossing the river in approximately 1946 and is glad to know that they are still at it.

  • Recreational mathematics
  • Arithmetics
  • Number theory
  • Geometry
  • Tiling and packing
  • Physics
  • Combinatorics
  • Probability
  • Graphs
  • Chess
  • Miscellany
  • Appendices A-D
  • Biographies
  • Bibliography
  • Name index