Imagine you are gambling on coin flips with a friend, one of you choosing heads and the other tails. You each stake ten dollars, agreeing that the first player to get five favorable flips wins all the money. But after two heads and four tails, a family member calls you away, breaking off the game. How should you divide the money to reflect the game’s progress?
This scenario is an instance of the problem of points. It was first published in a 1494 Italian textbook by Luca Pacioli (c. 1447-1517), though its origin may be Middle Eastern. Attempted by many over the years, the problem of points lacked a true solution until 1654. When finally found, however, this solution opened up a major field of math—probability theory—with considerable practical application.
The early mathematicians who took up the problem of points generally characterized their approaches as “most correct” rather than definitive, stressing the existence of alternatives. Pacioli simply divided in proportion to the players’ favorable flips at the point of breaking off, treating a 1-2 split and a 30-60 split exactly the same. Niccolò Tartaglia (c. 1500-1557), who considered the problem of points “rather judicial than through computation,” nonetheless found such a method lacking. But looking to the size of the lead, as he did, also produces questionable results; in a game of ten flips, it divides the money the same for a 0-2 split and a 7-9 split. Gerolamo Cardano (1501-1576), the author of a gambling manual tackling the basics of chance, was closest to correct. He realized that the only relevant thing is how far from winning the game each player is. If one player is x points from a win and the other y points, his method divides in the proportion 1+2+…+y, 1+2+…+x. In that 7-9 game to ten points, the first player would get 1 part of the money and the second player 1+2+3=6 parts. But likelihoods simply do not work this way.
Around 1654, Antoine Gombaud (1607-1684), a writer and amateur mathematician commonly known by his pen name Chevalier de Méré, came across the problem of points. It greatly intrigued him. He had, however, more mathematical interest than ability and eventually passed it on to his acquaintance Blaise Pascal (1623-1662).
Pascal’s friend Pierre Fermat (1607-1665) had also heard of the problem, and he soon sent a letter describing his method of division: the first legitimate solution to the problem. It utilizes combinations; that is, it lists all the equally likely ways the game could continue and divides the money in proportion to the number of continuations each player wins.
Pascal recognized Fermat’s approach as correct but potentially time-consuming. In a responding letter, he gave his own method. Like the other, it evaluates likelihoods of winning the game if continued, but is “much shorter and more neat.”
Pascal first examined the alternatives brought about by a single coin flip. For instance, in a 1-2 game to three points, the possible scores after the next flip are 2-2 and 1-3. One of these is a tie and would lead to an equal division of the money; the other is a win for the second player. As these two scenarios are equally likely, Pascal took their average, allotting ¼ of the money to the first player and ¾ to the second. He then continued to go backwards, always averaging the two possible divisions after the next flip. This method works in a game of any length and any score.
Later in the letter, Pascal stated a formula for the fraction of the first player’s wager owed to the second when the score is 0-1 (though the money at stake has various origins in different statements of the problem—in one, for instance, it is the gift of an eccentric older gentleman—Pascal and Fermat assumed each player stakes an equal amount). When the game goes to n points, this fraction is the product of the first n-1 odd integers divided by the product of the first n-1 even integers.
Pascal did not clearly articulate to Fermat how he derived his formula. He tended to skip steps in correspondence, assuming his mathematically talented addressee could fill in the gaps. And he largely sent results as he found them, sometimes making mistakes or admitting confusion. But his Traité du Triangle Arithmétique (also written in 1654, though published posthumously in 1665) presents a full proof for a more general formula, using what is now known as Pascal’s Triangle.1
If the two players are respectively a and b points from winning the game, we shall take the row of the triangle containing a+b elements. The first player–the one a points from the lead–has a fraction of the money proportional to the sum of this row’s first b elements; the other has a fraction of the money proportional to the sum of the other a elements. For instance, in a 7-5 split in a game to 10 points, the respective points from winning are 3 and 5, so we take the row with 8 elements. The first player gets 1+7+21+35+35=99 parts of the money to the second player’s 21+7+1=29 parts.
The “art of chances” soon took off. In 1657, the Dutch astronomer and mathematician Christiaan Huygens (1629-1695) published Libellus de Ratiociniis in Ludo Aleae (On Reasoning in Games of Chance), a probability treatise heavily derivative of his friend Pascal. Huygens presents the problem of points with none of Tartaglia’s fuzziness. He characterizes the division as “exactly [determinable],” suggesting that with this “value of [one’s] expectation” one could easily sell one’s position in such a game.
This newly precise insight into future events demonstrated relevance far beyond gambling. Jacob Bernoulli (1655-1705), whose solutions to five problems at the end of Libellus continued to expand the new field, recognized “application […] in civil, moral, and economic matters.” Particularly in economics, the ability to assign risk a monetary value—as Huygens did—is crucial. The problem of points initially seemed an ill-defined curiosity. But thanks to Fermat and Pascal, it became definite and significant in a way Pacioli and his contemporaries never suspected.
- In Pascal’s triangle, each number not on the edges is equal to the sum of the two above (all edge numbers are equal to 1). The rows can continue indefinitely.
Image Credit:
https://webmuseo.com/ws/musees-narbonne/app/collection/record/295
https://commons.wikimedia.org/wiki/File:Blaise_Pascal_2.jpg
https://commons.wikimedia.org/wiki/File:TrianguloPascal.jpg
https://en.wikipedia.org/wiki/Pascal%27s_triangle
Sources:
https://www.britannica.com/biography/Christiaan-Huygens
Devlin, Keith. Life by the Numbers. New York: John Wilney & Sons, Inc., 1998.
Gullberg, Jan. Mathematics: From the Birth of Numbers. New York: W. W. Norton & Company, Inc., 1996.
Pascal, Blaise. Oeuvres Complètes de Pascal. Bibliothèque de la Pléide, Gallimard edition, 1954.
Pacioli, Luca. Summa de Arithmetica Geometria proportioni et proportionalita.: Continentia de Tutta Lopera. Venice: Paganino Paganini, 1494.
Tartaglia, Nicolo. General Tratatto di Numeri e Misure. Venice, 1556.
Cardano, Georlamo. Practica Arithmetice et Mensurandi Singularis. Milan: J.A. Castellioneus for B. Caluscus, 1539.
Smith, David Eugene. A Source Book in Mathematics. New York: Dover Publications, 1959.
Huygens, Christiaan, and William Browne. Christiani hugenii Libellus de ratiociniis in Ludo aleae. or, the value of all chances in games of fortune ; cards, dice, wagers, Lotteries, &C. Mathematically demonstrated. London: Printed by S. Keimer, for T. Woodward, 1714.
I am not usually a math guy but this was very interesting, good job!
Very concise and precise article, amazingly done Hadley! I love it
Amazing job!! So excited to see a math column coming back to clay!!
interesting article. good job 🙂