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Mathematical Recreations

Dover Publications
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Maurice Kraitchik (1882–1957) was a Belgian mathematician who, fleeing the Nazis, came to the United States in 1939. He taught at the New School for Social Research (though what, aside from a course in recreational mathematics, how much, and for how long I do not know), joined the MAA in 1941 and attended at least two summer meetings in the 1940s. After that, except for a few publications, he seems to have disappeared, at least from easily accessible sources. He has left only light footprints in the sands of time.

While in Belguim he published four volumes on number theory (1922–1929) and others on other subjects. The ancestor of Mathematical Recreations is La mathématique des jeux ou Récréations mathématiques, a 566-page volume published in Brussels in 1930. From 1931 to 1939 he was the editor of Sphinx, a periodical devoted to recreational mathematics. He said that this allowed him to bring new material to MR but, because MR is only around 330 pages long, some things must have been omitted.

MR was first published by W. W. Norton in 1942. There was a British edition in 1955 and the Dover edition appeared in 1953, with further editions in 1981 and 2006. Except for a very few changes, discernable because of a different type style, the content now is the same as it was in 1942.

The twelve chapter titles give a good idea of the contents. The chapters usually contain problems followed by answers, often with no solution. On the other hand, some problems are followed by extensive discussion and the last three chapters, as well as the one on magic squares, consist almost entirely of exposition. So, this is not a problem book. Problem 8 in Chapter 1 is to find a square represented by four even digits, with just the answer given: the squares of 68, 78, 80, and 92. Problem 14 in Chapter 8 is Buffon’s needle problem, to find the probability that a needle tossed on a plane ruled with parallel lines will cross one of them, with just the answer given: 2l/aπ where l is the length of the needle and a is the distance between the lines. There is no indication that one problem might be harder than the other.

Kraitchik did this habitually. In his Introduction à la théorie des nombres (1952) a problem set on page 15 contained, among others, two problems that I paraphrase:

  • Show that the product of two integers is the product of their greatest common divisor and least common multiple.
  • Show that there are infinitely many twin primes.

There is no record that any readers of the book solved the second problem. Not knowing how difficult problems are can be frustrating, if you dislike being unable to solve problems, or liberating, letting you assume after failing to solve something that it is one of Kraitchik’s very hard problems, whether it is or not.

The book seems to be the original source of a problem (p. 134) that has spawned a fairly large literature. Kraitchik put in terms of neckties, but in a more common form it is about a simple game played by two players. Players A and B pull out their wallets and whoever has more money gives it to the player with less. Player A reasons that he stands to win more than he stands to lose, so his expected gain is positive and he willingly plays the game. Player B reasons likewise that his expected gain is positive. Because the game is zero-sum, something is wrong. Papers continue to be published on the seeming paradox.

The book is almost seventy years old or, if you count its predecessor, more than eighty. It shows. Its contents have dated. We no longer care as much as we did about magic squares, or knight’s tours on a chessboard, or five couples with jealous husbands crossing a river with an island in the middle. Nevertheless, the book contains many items of interest that will be new to the general mathematical reader. (Kraitchik does not hesitate to throw in a logarithm, an infinite series, or an integral sign where they are appropriate, so the book is not for beginners.) At Dover’s low price, it is well worth buying.

Woody Dudley has more items on MathSciNet than Kraitchik does, but he believes his footprints in the sands of time will be less discernible. 




Date Received: 
Thursday, January 1, 2009
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Maurice Kraitchik
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Underwood Dudley
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Mathematics without numbers
Ancient and curious problems
Numerical pastimes
Arithmetico-geometrical questions
The calendar
Magic squares
Geometrical recreations
Permutational problems
The problem of the queens
The problem of the knight

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Friday, January 14, 2011