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The Unfinished Game: Pascal, Fermat and the Seventeenth-Century Letter that Made the World Modern

Keith Devlin
Basic Books
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Fabio Mainardi
, on

In 1494, in his book Summa de Arithmetica, Geometrica, Proportioni et Proportionalita, Luca Pacioli asked the following question: how should the stakes be divided when a game of several rounds must be abandoned unfinished? Two centuries later, this problem was the object of a correspondence between Fermat and Pascal, which eventually changed our lives. In fact, in the attempt to solve this problem, the two great French mathematicians laid the foundations of what is now known as probability theory (the word “probability”, from Latin “probare” only came into use later).

Probability theory permeates our lives today. It has found applications to pricing theory (via the Black-Scholes equation), gambling, and even to credit cards (many cryptographic systems are based on the difficulty of testing whether a given integer is prime, and some of the most powerful primality tests are probabilistic in nature). Needless to say, probability theory is at the heart of the insurance industry.

Devlin’s book tells the fascinating story of this ground-breaking discovery, through a careful analysis of the correspondence held between Fermat and Pascal around 1654. As the author correctly points out, at the time it was by no means obvious that risk could be measured at all. Nowadays, the problem of the unfinished game can be solved by anyone who has taken a few hours of an introductory course in probability; this is symptomatic of the huge change that occurred in our vision of the world during the last three centuries. Today we are accustomed to the idea of assigning numbers to nearly everything, including our own expected lifetime.

What made the problem of the unfinished game such a challenge before Pascal and Fermat solved it was the fact that one cannot know how the players would have actually played if the game had not been abandoned. Psychologically, it was hard to assign a precise number to a random event. In fact, Pascal and Fermat had to perform their calculations within a worldview that considered what they were doing impossible. For a philosopher, probability theory can have theological implications. In fact Pascal, a fervent Catholic, later proposed a probabilistic argument to prove the ‘utility’ of faith!

This book is not only about mathematics. It is also a tale of how mathematics, and science in general, is really done. Pascal is described as “a man of slight build” (he died indeed at the age of thirty-nine), who struggles with a conceptual difficulty about the way the problem should be correctly modelled. We learn, also, how hard was for the two men to meet each other. In fact, it never happened: at the time. it was quite dangerous and time-consuming to travel, even within the same country, and especially for two unhealthy men.

Throughout this book, we meet so many brilliant minds. Among them, Girolamo Cardano, another pioneer of probability theory, who lived an unconventional, and often embittered, life. And the amazing Bernoulli family: eight members of the family distinguished themselves in some branch of mathematics, especially probability, something that one might argue defies the laws of probability…

Summing up: Devlin’s book is very well written and accessible to everyone, it doesn’t require any special expertise in any area of mathematics, and the correct solution of the unfinished game is thoroughly explained. His analysis of the Fermat-Pascal correspondence is penetrating and never technical.

This is highly recommended reading. I think it should find a place in every mathematician’s library.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at

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akirak's picture

I really enjoyed reading this book. It is an excellent account of the history of probability theory.

The story begins with a correspondence between the two great French mathematicians, Pierre de Fermat and Blaise Pascal. The book has an interesting hook, with the opening paragraph being a letter sent by Pascal to Fermat. The correspondence asks ‘how should we divide the stakes if a particular game is incomplete?’ This led Pascal and Fermat to lay the groundwork for probability theory.

I liked the style of the author and the way Devlin dipped into some straightforward mathematics in this book. The history is particularly appealing, with the explanation of how Graunt developed his mortality tables. It also goes on to state that Newton’s first great mathematical discovery, the binomial theorem, is based on Pascal’s triangle.

The book explains with entertaining detail the personalities of Fermat and Pascal. There are also some very fascinating applications of probability mentioned in the book such as how the repeated use of Bayes theorem predicted an attack on the pentagon and also the explanation of why DNA profiling is so reliable.

However the book has the following shortcomings:

  • The dates of birth and death of all the mathematicians mentioned in the book should have been stated.
  • On page 83 the author misses the first Fermat prime 3.
  • The last sentence in the second paragraph on page 102 should say ‘bet 24 to 40, that is 3 to 5, that a sixteen year old will die before the age thirty six’.

This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.