An Epistolary Episode
THE UNFINISHED GAME: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern. Keith Devlin. x + 191 pp. Basic Books, 2008. $24.95.
Timothy Gowers, a distinguished mathematician at the University of Cambridge, recently conducted an experiment in collaborative mathematics. He was puzzling over an interesting problem, and rather than go off to work on it in solitude, he posted a note on his blog, inviting others to join him in seeking a solution. There have been hundreds of responses, and the voluminous discussion has spread to other blogs as well as a wiki where participants coordinate efforts and summarize progress.
If Blaise Pascal had had a blog or a wiki, perhaps he would have tried the same strategy when he took up a mathematical challenge in 1654—a problem concerned with figuring the odds in a gambling game. Instead, Pascal wrote a letter to an older colleague, Pierre de Fermat, and the two of them batted the problem back and forth in a correspondence that went on for several weeks, with occasional input from a few others. Most of the letters were later published—after the deaths of both authors—and they became foundation documents in the theory of probability. Keith Devlin now gives us a helpful guidebook to this famous episode of epistolary mathematics.
Here is the essence of the wagering problem that caused all the fuss, as Devlin presents it in a slightly simplified form:
The players, Blaise and Pierre, place equal bets on who will win the best of five tosses of a fair coin. We’ll suppose that on each round, Blaise chooses heads, Pierre tails. Now suppose they have to abandon the game after three tosses, with Blaise ahead 2 to 1. How do they divide the pot?
Since Blaise is leading, it seems he deserves a larger share of the wager. But how much larger? Gamblers and scholars had taken up this question before, at least as far back as Luca Pacioli and Girolamo Cardano a century earlier, but they had failed to settle it. Pascal and Fermat not only got the answer; they also set forth with reasonable clarity how they derived it and why it’s right.
Pascal and Fermat came to the same result through different methods. Fermat proposed a complete enumeration of cases. If the game continued to completion with two more coin tosses, there would be four possible outcomes: HH, HT, TH, TT. Blaise wins in three of these four cases, and so he deserves three-fourths of the stake. The trouble is, another scheme of counting says there are only three cases, because the sequence HT would never be observed: If Blaise wins the fourth toss, the game is over and there’s no point in tossing again. The nub of the whole problem, and the reason it had taken a century to resolve, lies in persuading yourself that Fermat’s count is the correct one and the three-case alternative is wrong. Devlin comments:
So if you are one of those people who finds this alternative argument appealing (or even compelling), take heart; you are in good company (though still wrong).
Pascal’s approach relied on a different style of reasoning, nowadays associated with the terms induction and recursion. Suppose at some stage of the game you need m points to win and your opponent needs n points. After the next round of play, either you will need m–1 points and your opponent will still need n (if you won the toss) or you will need m and your opponent will need n–1 (if you lost). You can continue this analysis to further stages of the game, applying the same logic repeatedly and on each round subtracting 1 from both m and n. Eventually, either m or n or both must be reduced to 0. At that point, the game is over, since someone needs 0 points to win. From this end state, you can work backward through the chain of intermediate steps to assign a numerical value to your probability of winning in the original state.
A disappointment of The Unfinished Game is that Devlin does not give a detailed account of how Pascal’s recursive algorithm works. It is “too technical,” he says, and, besides, Fermat’s approach is better. I am not entirely in agreement. On the other hand, I too have given only a sketch of the algorithm, and readers who find the preceding paragraph of this review hopelessly obscure may well conclude that Devlin made the right choice.
In any case, The Unfinished Game is not primarily a tutorial on the craft of calculating probabilities; the main thing it offers is a chance to look over the shoulders of two eminent mathematicians as they struggle with some of the same elementary confusions that still make probability theory a treacherous terrain. Pascal and Fermat are familiar names in mathematics, but their lives and personalities remain somewhat shadowy. Even with the letters in front of us, and with the social and family background that Devlin fills in, it’s not always easy to make out the characters hiding behind all those elaborate flourishes of 17th-century French courtesy. When Pascal declares to Fermat, “Your first glance is more penetrating than are my prolonged endeavors,” is he sincerely conceding Fermat’s greater genius, or is the phrase mere formulaic deference to a senior colleague?
Pascal and Fermat never met face to face, which seems regrettable for them, but maybe posterity is the beneficiary. If they had been neighbors, meeting at the corner café to talk math every morning, their deliberations over these probability puzzles might never have been written down. Perhaps today’s Internet collaborations will be preserved in a similar way, and teach a future generation how mathematics was done in the 21st century.
Returning to the collaboration I mentioned at the outset, a resolution appears to have been reached: In a blog post dated March 10, 2009, Gowers announced, “Problem solved (probably).”
Brian Hayes is Senior Writer for American Scientist. He is the author most recently of Group Theory in the Bedroom, and Other Mathematical Diversions (Hill and Wang, 2008).