# Montmort

Pierre de Montmort (1678-1719) was one of three sons born into a French family. His father wanted him to study law but Montmort decided to rebel against his father as he had no desire or passion for the area of study and instead pursued mathematics (Hacking, 1970). He left college after a number of weeks and eventually left home at the age of eighteen, at which point he went abroad. When visiting his cousin in Germany, Montmort gained access to Nicolas Malebranche’s La Recherché de la Verite, a book in his cousin’s library. According to Bernard de Fontenelle his discovery of this work made him into the philosopher and mathematician that he became (Dupont, 1978: 117-125), and at the age of twenty one he began his studies in France under Nicolas Malebranche (1638-1715), focusing on the newest algebra and geometry (O’Connor and Robertson, 2015).

In 1708 Montmort published his first work on chance called Essay d’Analyse sur les Jeux de Hazard. He was the first to solve the mathematical problem concerning the chances (probability) of drawing a certain value of a card from a pack, and looked at how to calculate the mathematical expectation of several dice games (Epstein, 2009). It is also believed that Montmort is responsible for naming Pascal’s triangle (Miller).

Pierre Rémond de Montmort, Essay d’analyse sur les jeux de hazard (Paris, 1713), p. 53.

Here we can see how Montmort analyses the number of outcomes for games involving dice (‘dés’ in French). Case a shows the outcomes for two dice, one numbered 1-6 and the other numbered 1-4 which means the minimum score when both dice are rolled is 2 and the maximum score is 10. All 24 outcomes are listed so that the columns show the number of ways of scoring a two, three, four … ten.

In case b Montmort adds another dice numbered 1-3. He uses the known outcomes from the dice in case a to create the first row and uses three copies of this since the third dice must be one, two or three. The new minimum score is 3 (first column) and the maximum score is 13 (last column). Since there are 11 possible scores there are 11 columns listed in order. In case c he adds a fourth dice numbered 1-2 and in case d he adds a fifth dice again numbered 1-2.

It can be suggested that the work of Montmort was cutting edge by the standards of his time since he analysed the chances involved in game playing, especially card games, which was not a popular topic of study (Hacking, 1970). One card game in particular that he wrote about was the game Pharaoh.

Pierre Rémond de Montmort, Essay d’analyse sur les jeux de hazard (Paris, 1713), p. 80.

He supposes the banker has four cards and lists all 24 possibilities denoting the cards by a, b, c and d. If the player and banker bet A each then there are twelve hands which pay 2A to the banker and six hands which pay 2A to the player while the rest result in a draw and the bet is returned. The banker’s advantage in this example is s=A + ¼A   . In fact, Montmort discusses this game in great detail and gives the following table as a complete explanation of the game. Notice the last row which contains the example discussed above.

Pierre Rémond de Montmort, Essay d’analyse sur les jeux de hazard (Paris, 1713), Pharoan table.

Montmort collaborated with John and Nicolas Bernoulli, two men who also contributed greatly to the world of mathematics especially in probability and game playing. As a result the second edition of the work (which is the edition collected by Edward Worth), was published in 1713 including copies of their correspondence. Essay d’Analyse sur les Jeux de Hazard was regarded as a primary source for mathematical problems concerning games of chance during this period (Seneta, 2016) and it is notable that the mathematics that was used is very similar to the notation still used today.

Sources

Epstein, Richard A. The Theory of Gambling and Statistical Logic (Academic Press: Cambridge, 2009), p 4.

Hacking, I., Montmort biography, Dictionary of Scientific Biography (New York 1970-1990).

Dupont, P., Nicolas Bernoulli’s proof of de Montmort’s formula concerning the ‘jeu du treize’ (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Nature. 112 (1-2) (1978), 117-125

O’Connor, J.J. and E.F. Robertson, Montmort, MacTutor History of Mathematics (University of St Andrews, 2015).

Seneta, E., Pierre Rémond de Montmort. Encyclopedia of Mathematics, 2016.

Miller, J., Pascal’s Triangle, Earliest Known Uses of Some of the Words of Mathematics.

Text: April Kiernan and Fionnán Howard