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Amusements in Mathematics

Henry E. Dudeney
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Underwood Dudley
, on

Times change, whether we like it or not, sometimes for the better and sometimes for the worse. There was a time before Facebook, before World of Warcraft, before the internet, before television, movies, radio: a time when people had to turn elsewhere to satisfy the universal and eternal human need for entertainment. Some turned to recreational mathematics — trying to solve problems such as

A gentleman who recently died left the sum of ₤8,000 to be divided among his widow, five sons and four daughters. He directed that every son should receive three times as much as a daughter, and that every daughter should have twice as much as their mother. What was the widow’s share?

That is the seventh problem in Amusements in Mathematics, a collection of problems and solutions, first published in 1917. (The answer is ₤205 2s. 6d. and 10/13 of a penny.)

Recreational mathematics, mathematics of the fun of it, goes back a long way. The first book devoted to it was Problèmes Plaisants & Délectables by Claude-Gaspar Bachet (1581–1638), first published in 1612 and still in print, with a new edition being published this year. It contains the problem of getting four pints of water given containers holding 8, 5, and 3 pints and many other recreational problems still being posed today.

Recreational mathematics had its greatest flowering in the second half of the nineteenth century and into the twentieth, when many periodicals for general audiences had puzzle sections. In his preface, Dudeney gives credit to The Strand Magazine, Cassell's Magazine, The Queen, Tit-Bits, and The Weekly Dispatch, where some of his problems originally appeared. Lewis Carroll’s A Tangled Tale first appeared in 1880 as a serial in The Monthly Packet. In the 1890s the Brooklyn Daily Eagle had a puzzle column.

Dudeney and Sam Loyd (1841–1911) were the puzzle giants of their day, Dudeney in England and Loyd in the United States. For a time they collaborated but Loyd, who tried to make a living as a recreational mathematician, allegedly stole some problems from Dudeney, who broke off the relationship. Dudeney wisely kept his day job.

Other recreational mathematicians of the day included W. W. Rouse Ball (1850–1925) and, in France, Édouard Lucas (1842–1891) who originated the Tower of Hanoi puzzle that lives on as a first exercise in recursion. Hubert Phillips (1891–1964), whose nom de problème was “Caliban”, created original puzzles in the 1930s in England.

In spite of radio, movies, television, and everything else, recreational mathematics carries on. The work of Martin Gardner (1914–2010) does not need to be summarized here. A Canadian, J. A. H. Hunter, had puzzle columns in newspapers in the middle of the twentieth century. In 1961, Joseph Madachy started, all by himself, Recreational Mathematics Magazine, which was taken over by Baywood Publishers and renamed the Journal of Recreational Mathematics in 1968. It continues to this day. Besides Amusements in Mathematics, Dover Publications offers many other recreational mathematics books.

The book contains 430 problems. Its sections are labeled

  • Arithmetical and algebraical problems
  • Geometric problems
  • Points and lines problems
  • Moving counter problems
  • Unicursal and route problems
  • Combination and group problems
  • Chessboard problems
  • Measuring, weighing, and packing puzzles
  • Crossing river problems
  • Problems concerning games
  • Puzzle games
  • Magic square problems
  • Mazes and how to thread them
  • The paradox party
  • Unclassified problems

Statements of problems consume about 150 pages and 100 pages are given over to answers and solutions. The solutions section contains mini-essays on mazes and on magic squares. Some problems have answers only but many have interesting comments, at various lengths, on the solutions. Though Dudeney does not indicate which is which, many problems are new while others are not. (The problem of cutting out corners from a sheet of material and folding to make a box of maximum volume is here, but with just an answer. How Dudeney expected his non-calculus readers, who must have been the vast majority, to find it he does not say.)

The problems range in difficulty from simple, as the inheritance problem quoted above, to those that are difficult indeed, such as bisecting a line segment using compasses alone. Though readers will probably not want to try to solve many of the problems, it is fun to read their solutions. And that, fun, is the point of recreational mathematics.

Woody Dudley retired from teaching in 2004 and retired from doing recreational mathematics problems some years before that. Life contains enough problems.


  Arithmetical and Algebraical Problems
    Money Puzzles
    Age and Kinship Puzzles
    Clock Puzzles
    Locomotion and Speed Puzzles
    Digital Puzzles
    Various Arithmetical and Algebraical Problems
  Geometrical Problems
    Dissection Puzzles
    Greek Cross Puzzles
    Various Dissection Puzzles
    Patchwork Puzzles
    Various Geometrical Puzzles
  Points and Lines Problems
  Moving Counter Problems
  Unicursal and Route Problems
  Combination and Group Problems
  Chessboard Problems
    The Chessboard
    Statical Chess Puzzles
    The Guarded Chessboard
    Dynamical Chess Puzzles
    Various Chess Puzzles
  "Measuring, Weighing, and Packing Puzzles"
  Crossing River Problems
  Problems concerning Games
  Puzzle Games
  Magic Square Problems
    "Subtracting, Multiplying, and Dividing Magics"
    Magic Squares of Primes
  Mazes and how to thread Them
  The Paradox Party
  Unclassified Problems