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The Trictrac Home Page by David Levy

updated July 9, 2003 Copyright © 1998-2003 by David Levy. All rights reserved.

Welcome to the Trictrac Home Page! This is my slow, ongoing attempt, to explore the game of Trictrac and share my exploration.

Contents Announcing: Soumille, The First Tictrac Program Announcing: The Yahoo Trictrac Group My interest in Trictrac The rules of Trictrac A sample game An illustrated bibliography of works in French on Trictrac Books on Trictrac in English Montmort on Trictrac Disclaimer What's New?

BREAKING NEWS as of July 9 2003!

I have just released version 0.3 of Soumille, The First Trictrac Program. A screen shot appears below. If you would liked to be contacted when there is a general release available, please email me.

Please see a draft of the User Guide

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BREAKING NEWS as of November 14, 2002!

I have just created a Yahoo discussion group for those of us interested in the game of trictrac. If you are interested, click here and click on "Join This Group" in the upper right

hand corner. You will need a Yahoo id.

My Interest in Trictrac Trictrac is a rich, formal game that was immensely popular in French society prior to the revolution. The game was so closely associated with the courts of Louis XIV, Louis XV and Louis XVI that during the revolution, most Trictrac boards were destroyed as symbols of the monarchy. I collect books on card and boards games, but resisted books on Trictrac for a long time. They are in old French and I never expected to be able to read them. One day, I ran across an exquisitely bound but affordable copy of Soumille's Le Grand Trictrac (1756). It contains 270 woodblock prints of Trictrac positions showing the course of play, move by move. With the help of my high school French and a Larousse dictionary, I was able to make some sense out of the game. I have been collecting Trictrac books ever since. I have transcribed the first game in Soumille along with diagrams and descriptions which should give you a good introduction to the game.

Works in English Many modern backgammon books refer to Trictrac only as the French name for backgammon. As these pages show, true Trictrac is a different game and I am aware of only works in English that discuss it: Richard Seymour, The Compleat Gamester. Fiske, Willard. Chess in Iceland and Icelandic Literature with historical notes on other table-games. Florence: The

Florentine Typographical Society. 1905. ix+400pp. Murray, H. J. R. A History of Board Games Other than Chess. Oxford: Clarendon Press. 1952. xii+262pp.

These pages are my attempt to fill that void. If you run across others, please email me.

Disclaimer My understanding of Trictrac is limited by my unpracticed high school French. With the exception of Fiske's Chess in Iceland, I have written more in English about Trictrac than has been written in the previous three hundred years! I am inconsistent about using the (delightful) French vocabulary and translating (awkwardly) into English. I sincerely hope that I am not butchering either the game or the language. Please let me know if you have suggestions.

What's New? November 14, 2002: Added information about the new Yahoo discussion group I created for Trictrac. February 26, 2000: I updated the bibliography with photographs of books from my collection. July 30, 1998: I wrote an essay based on Montmort's Essay D'Analyse sur Les Jeux de Hazard. Montmort was a

pioneer in the study of probability and had some Trictrac examples in his book. June 7, 1998: At long last I have published the rules for Trictrac. I have added a section to the bibliography for

anthologies which have material on Trictrac. May 10, 1998: I have transcribed an annotated game of Trictrac from Soumille's Grand Trictrac. April 26, 1998: I have updated my Trictrac bibliography based on useful feedback from readers.

Soumille Intro

Introduction to the Game of Trictrac by David Levy

updated May 10, 1998 Le Grand Trictrac was published anonymously in 1738. A second edition of 1756 lists the author as "M. l'Abbe S***," correspondent of the Royal Academies of Science in Paris and Toulouse. Secondary sources name the author as Father Bernard Laurent Soumille. The subtitle of Le Grand Trictrac is "an easy way to learn the game, terms and rules without an instructor." Soumille teaches the game by showing several games move by move, explaining the rules, strategy and terms as they arise in the course of the game.

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I have transcribed a complete set of rules but you can learn the basics by playing through the game below. As you will see, the mechanics of trictrac are similar to backgammon. Players alternate rolling the dice and move checkers a number of points corresonding to the pips on the dice. One difference is that the player claims points as a result of the throw before moving checkers. Another is that doubles are played twice, rather than four times as in backgammon. The first game is a simple variant of trictrac (jeu ordinaire) in which each player plays only on his half of the board. The game begins with checkers arranged below. It is between Cloris, a lady and Damon, a gentleman.

Cloris plays with the black checkers and Damon the white. Each player places fifteen checkers on the talon (first point). The first six points make up the petit jan (inner table) and seven through twelve make up the grand jan (outer table). The twelfth point (midpoint) is know as the coin de repos (literally "resting corner"). As you will see, there are rules governing the movement of checkers to and from thecoin de repos.

Trictrac consists of twelve games of twelve points. Points are scored for various plays and positions, known as jans. Most trictrac primers begin with a list of all the jans and the number of points they score. Some of the jans are so rare that starting with them gets in the way of learning the game. It would be like starting to learn bridge by memorizing the scoring table. The most frequent scoring play is to roll a number which allows you to hit an opponent's single checker (demi-case, or blot in backgammon). The play is known as battre sur un demi-case or jan de recompense. I will describe details, examples and other scoring plays as required in the first game. And now, click here to begin.

Illustrated Bibliography of Trictrac by David Levy

updated January 5, 2009 Copyright © 1998-2009 by David Levy. All rights reserved.

Contents Books about trictrac Anthologies with significant material on trictrac

Books about Trictrac Jollivet, Euverte de, sieur de Votilley. 1601-1662. L'excellent jeu du triquetrac, tres-doux esbat és nobles

compagnies. o Paris: Guillemot. 1634. o Paris: Guillemot. 1635. 86pp. o Paris: Preud'homme. 1639. 33pp.

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o Paris: Promé. 1656. 64pp. (Alliey p4) o Paris: Musier. nd. 64pp. o Paris: Raffle. nd [1670c]. 64pp.

Le Jeu du Trictrac comme on le joüe aujourd'huy. Enrichy de Figures. Et d'une méthode tres-aisée, pour apprendre de soy-même à joüer ce Jeu en perfection. o Paris: Chez Henry Charpentier, dans la Grand Salle du Palais, au bon Charpentier. 1698. 87pp. (Alliey p1,

Hargrave, Bodleian) o Le jeu du trictrac, enrichy de figures, avec les jeux du revertier, du toute-table, du tourne-case, des dames

rabattues, du plain et du toc. Deuxième édition, reveuë, corrigée & augmentée. Paris: Chez Henry Charpentier. 1701. 198pp. bound with Suite du trictrac, contenant les règles des jeux du revertier, du toute-table, du tourne-case, des dames rabattues, du plain et du toc, comme on les joüe aujourd'huy. Paris: chez Henry Charpentier. 1699. 111pp.

o Troisième édition, reveuë, corrigée & augmentée. Paris: chez Henry Carpentier. 1715. 198pp. (Alliey p1, Bodleian)

o Troisième édition, reveuë, corrigée & augmentée. Paris: chez Gregoire-Antoine Dupuys. 1730. 198pp. 3. Robbe, Jacq.Trictracus, carminibus elegiacis illustratus.

o Paris: 1710.(Alliey p8) 4. [Soumille, abbé Bernard Laurent 1703-1774]. Le Grand Trictrac. Ou Méthode Facile pour Apprendre san Maître.

La Marche, les Termes, les Règles. Et une grande partie des finesses de ce Jeu, enrichie de 270. Planches ou Figures, avec les décisions des cas particuliers.

Avignon: Chez F.Girard & D. Seguin. 1738. iv+320pp. (NUC: Library of Congress, University of Chicago.)

Avignon: Chez F.Girard & D. Seguin. 1739. iv+320pp.

Paris: Giffart. 1756. vi+323pp. (Alliey pp7-8, NUC: Harvard)

Seconde édition. revuë, corrigée et considérablement augmentée. Par M. L'Abbé S***. Avignon: Chez Alexandre Giroud. 1756. x+438pp. (Alliey pp6-7, NUC: Library of Congress, Yale)

Paris: De Hansy, le jeune. 1766. xvi+357pp. (Alliey p8, NUC: NY Public Library, Cleveland Public Library)

Paris: De Hansy, le jeune. 1790. xvi+357pp. (NUC: Newberry Library, Chicago)

Paris: De Hansy, le jeune. 1801. xvi+357pp. (not in NUC)

o [Falavel, M. J. M.] Principes du Jeu de Tric-Trac mis à la portée des commençans. o Paris: Guyllyn. 1749. 76pp. (Alliey p1)

[Falavel, M. J. M.] Le Jeu du Trictrac, ou les Principes de ce Jeu. Éclaircis par des Exemple en saveur des Commençans; Avec l'Explication des terms par order alphabétique, & une Table des Chapitres servant de Récapitulation générale.

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Paris: Nyon. 1776. xiv+376pp. Alliey pp1-2

Le Jeu de Trictrac, avec figures à l'aide desquelles on démontre tous les coups de ce jeu et l'on pourrait facilement l'apprendre seul sans le secours de personne.

nl. nd. anonymous autograph manuscript on laid paper. 72 unnumbered pages including 11 hand colored drawings of Trictrac positions. Original wraps of blue Holland paper, later covered with marbelled paper. Probably late 18th or early 19th century. Much of the text and most of the illustrations are taken from the second edition of Charpentier.

[Guiton, N.]. Traité Complet du Jeu de Trictrac.

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Avec figures. Contenant les Principes et les Règles de ce Jeu, avec des tables de calculs qui ne se trouvent dans aucun des traités connus. Paris: L. G. Michaud. 1816. xx+310pp. (Alliey pp3-4, NUC: New York Public Library, Free Library of Philadelphia.)

Traité Complet du Jeu de Trictrac, contenant les Principes et les Règles de ce Jeu, et des Tables de Calculs qui ne se trouvent dans aucun des Traités publiés jusqu'à ce Jour; avec Figures. Seconde Édition, dans laquelle on trouvera des Additions essentielles dans les Règles, et des Changemens dans la Distribution des Matières. Suivi d'un Traité du Jeu de Backgammon. Paris: Barrois L'Ainé. 1822. xx+344pp. (Alliey pp3-4, NUC: Yale)

[Lepeintre, Pierre-Marie Michel]. Cours complet de Trictrac, avec un abrégé du Gammon, du Jacquet et du Garanguet, A la portée de tout le monde et à l'usage des amateurs. Par un vieil hermite du Morbihan.

Paris: Guillaume. 1818. vi+170pp+11 leaves with 31 diagrams. (Alliey pp4-5, Guiton 2nd edition, viii)

Paris: Guillaume. 1820. First edition, second issue with cancel title. vi+170pp+11 leaves with 31 diagrams.

[Deshayes]. Lettre en vers sur le jeu de Trictrac, adressée à madame ***, suivie d'un vocabulaire explicatif de tous les terms de ce jeu, par M. Seyahsed, presque amateur.

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o Paris. 1834. (Mery p480) Manouri. Regles du Jeu de Dame, du Jeu de Trictrac, et du Jeu de Loto. par Manouri, membre de l'Académie des jeux de Berlin. o nd [1840c.] Gand. Imp. de Vanderhaeghe-Maya. 31pp. [Trictrac pp18-26] (Bodleian Jessel f857)

12. Alliey, Camille-Théodore Frédéric. Bibliographie complète, analytique, raisonnée et par order alphabétique de tous les ouverages connus en langue française et suivi de quelques autres en diverses languge sur le jeu de tric-trac.

Commerci, Cabasse, 1850. 8pp. (not in NUC)

La Bibliographie Complète du Jeu de Tric-Trac. Commerci, Cabasse, 1855. 8pp. (NUC Cleveland Public Library, Cincinnatti, Providence)

o [Lelasseux-Lafosse, Julien Alexandre Monique]. Le Jeu du Tric Trac.

Le Jeu du Trictrac Rendu facile. Pour toute personne d'un esprit juste et persévérant. Ouvrage contenant des règles et des tables entièrement nouvelles, servant à calculer facilement les chance pour et contre chacun des deux joueurs, et n'exigeant que la connaissance des premieres éléments de l'arithmétique. (La première partie est destinée aux personnes qui ne connaissent absolumnet point ce jeu; la deuxième a celles qui veulleunt y acquerir une grande force.) Par J. L. Ancien élève de l'École Polytechnique, officier d'artillerie, professeur de mathématiques a l'École militaire de la Fleche. Paris: Ledoyen. 1852.

xxx+errata+244pp. xi+errata+296pp. (Bodliean: Jessel d148-9)

Paris: Ledoyen. 1855. xxx+244pp. xi+296pp. Thanks to Thierry Depaulis for finding the author's name in the online records of l'École Polytechnique.

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[Delarue, A. O.]. Traité du Jeu de Trictrac. Novelle edition augmentée du jeu du jacquet par Richard. o Paris: Delarue. [1874]. 100+8pp. (NUC: Lib of Cong)

15. Borentain, H. Le Jacquet. Le Tric-Trac. Le Solitaire. o Paris: Bornemann. 1989. 78pp. o Paris: Bornemann. 1991. 78pp.

Anthologies For more complete references to the anthologies, please see Depaulis, Les Loix de Jeux, which is the definitive reference to French books about card games. I have attempted to group the various Académie des Jeux to reflect multiple editions of similar works. This is based on my reading of Les Loix de Jeux and the few books which are in my possession. I apologize for the errors I have undoubtedly made. On occasion, you will find portions of the anthologies published separately. For example, I have seen the trictrac sections of the Le Gras Académie des Jeux and the Fournier Almanach des Jeux published separately.

1. La Maison Académique: contenant un recueil general de tous les jeux divertissans pour se rejouyr agreablement dans les bonnes compagnies. par le sieur D.L.M.

Paris: Robert de Nain et Marin Leche. 1654 (Depaulis 11)

Paris. 1659 (Depaulis 13) Paris. 1665 (Depaulis 14) Lyon. 1666 (Depaulis 15) Paris. 1668 (Depaulis 16) Lyon. 1674 (Depaulis 17) Lyon. 1697 (Depaulis 31) The Hague. 1702 (Depaulis

39)

Divertissemens Innocens, contenant les Regles du jeu des echets, du billard, de la paume, du palle-mail, et du trictrac.

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The Hague: Adrien Moetjens, 1696. 496pp (misnumbered 476). (Horr 404)

o [Méré, Antoine Gombaud, Chevalier de, attributed to]. Noveaus Jeu de l'Hombre, comme On le joüe presentement à la Cour & à Paris, Où l'on voit comment se joüe. Espadile forcé. L'Hombre à deux, à trois & à cinq. Avec l'explications des Termes dont on se sert en le joüant.

A La Haye, Chez Jacob van Elinckhysen. 1702. 214pp. [Deapaulis 40]. covers Trictrac pp171-214.

o Ferrand, Pierre. Les Veritez plaisante ou le monde au naturel. o Rouen: chez Maurry. 1702. 32+324pp. (Alliey p2)

Académie Universelle des Jeux: contenant les regles des jeux de cartes permis, du trictrac, des echecs, de la paulme, du mail, du billard, & autres; avec des instructions faciles pour apprendre a les bien jouer.

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Paris: Le Gras. 1718. (Depaulis 54)

Leide: van der Aa. 1721 (Depaulis 56)

Paris: Le Gras. 1725. (Depaulis 61)

Amsterdam: Covens & Mortier. 1728. (Depaulis 64)

Paris: Le Gras. 1730. (Depaulis 67)

Paris: Le Gras. 1732. (Depaulis 68)

Paris: Le Gras. 1737. (Depaulis 71)

Paris: Le Gras. 1739. (Depaulis 73)

Paris: Le Gras. 1743. (Depaulis 77)

Paris: Saugrain fils. 1758. (Depaulis 86)

Amsterdam: Saugrain fils. 1758. (Depaulis 87)

Paris: Knapen. 1765. (Depaulis 101)

Paris: Knapen. 1770. (Depaulis 104)

Académie Universelle des Jeux, avec des instructions faciles pour apprendre a les bien jouer. Nouvelle editions, augmentee & mise en meilleur order.

Amsterdam: aux depens de la Compagnie. 1751. (Depaulis 81)

Amsterdam. 1752. (Depaulis 82)

Amsterdam. 1756. (Depaulis 85)

Amsterdam. 1760. (Depaulis 89)

Amsterdam. 1763. (Depaulis 91)

Amsterdam. 1766. (Depaulis 102)

Amsterdam. 1770. (Depaulis 105)

Amsterdam. 1773. (Depaulis 110)

Amsterdam. 1777. (Depaulis 116)

Amsterdam. 1777. (Depaulis 117)

Amsterdam. 1786. (Depaulis 131)

Amsterdam. 1789. (Depaulis

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134) Amsterdam. 1789. (Depaulis

135)

o Rouen: chez Maurry. 1702. 32+324pp. (Alliey p2) Almanach des jeux, ou Académie portative: contenant les regles du wischt, du reversis, du tre-sette, du piquet, augmente du jeu de trictrac.

Paris: Fournier. 1781. vi+120+(35)+(36)+105+102pp. (Depaulis 122)

Paris: Fournier. 1782. (Depaulis 125)

Paris: Fournier. 1783. (Depaulis 127)

Paris: Fournier. 1783. (Depaulis 128)

Paris: Fournier. 1785. (Depaulis 129)

Paris: Fournier. 1786. (Depaulis 132)

Paris: Fournier. 1787. (Depaulis 133)

Paris: Fournier. 1789. (Depaulis 136)

Paris: Fournier. 1790. (Depaulis 137)

Paris: Fournier. 1802. (Depaulis 139)

Paris: Fournier. 1808. (Depaulis 153)

Académie universelle des jeux: contenant les regles deus jeux de cartes permis; [...]. Nouvelle edition, augmentee du jeu des echecs par Philidor; du jeu de whist, par Edmond Hoyle, traduit de l'anglais; du jeu de tre-sette, du jeu de domino, de l'homme de Broue, etc. etc. Avec figures.

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Lyon: Ballanche. 1802. (Depaulis 140)

Lyon: Ballanche. 1805. (Depaulis 142)

Lyon: Ballanche. 1806. (Depaulis 144)

Lyon: Ballanche. 1810. (Depaulis 157)

Lyon: Ballanche. 1821. (Depaulis 185)

Lyon: Ballanche. 1832. (Depaulis 225)

Petit Académie des jeux

Paris. 1815. (Depaulis 167) Paris. 1817. (Depaulis 175) Metz: Verronnais. 1818.

(Depaulis 177) Paris. 1819. (Depaulis 181) Metz: Verronnais. 1822.

(Depaulis 190) Metz: Verronnais. 1825.

(Depaulis 196) Metz: Verronnais. 1847.

(Depaulis 281)

Grande Académie des jeux: contenant le jeu du trictrac, les echecs, le jeu de dames, de domino, le boston, l'ecarte, le piquet, l'imperiale, la bouillotte, le whist, [...]. o Paris: Caillot [1833]. 252pp. (Depaulis 228)

Nouvelle Académie des jeux o Paris: Masson et Duprey. 1835. (Depaulis 237)

12. Académie des Jeux o Paris: Picard. 1835. (Depaulis 238)

13. Riga, A. Dictionaire generale des jeux de societe, jeux de commerce, jeux de penitence, jeux de jardins, jeux de cartes, jeux prepares et composes, jeux de hasard, etc. o Paris: au Bureau central des dictionnaires. 1837. 180pp. (Depaulis 241)

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14. Delarue, A. O. Académie des Jeux, contentant: la regle de chacun des principaux jeux, soit de cartes, billard, echecs, domino, dames, etc. Nouvelle edition. Mise en order par Richard. o Paris: Delarue. [1846]. (Depaulis 275)

15. Mery, Joseph. Abitre des Jeux.,accompagne de petits poemes historique par Mery. Contenant la Marche, l'histoire, les Regles, la Legislation et la Bibliographie de tous les jeux anciens et modernes, et suivi d'un Apercu sur le Calcul des Probabilites.

Paris: Gabriel de Gonet. 1847. 506pp. (Depaulis 282)

o Van Tenac, Charles. Album des Jeux de hasard et des combinaisons en usage dans les salons et dans les cercles. o Paris: Gustave Havard. 1847. 544pp. (Depaulis 289)

Passard, Francois-Lubin. Almanach des Jeux: Académie nouvelle comprenant les regles des principaux jeux de cartes, de combinaison et d'exercice, avec un traite du jeu de whist entierement noveau. par Hilaire Le Gai. o Paris: Passard. 1849. 212pp. (Depaulis 299) o Paris: Passard. 1850. 189pp. (Depaulis 304)

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Rules for the Game of Trictrac by David Levy

updated July 22, 2007 Copyright © 1998-2007 by David Levy. All rights reserved.

Contents Introduction

o Object of the Game o Equipment o Notation o Vocabulary

Conduct of the Game o Setup o A Turn at Trictrac o Checker Movement o Limits on Checker Movement

Scoring o Rare Jans o Filling a Table o Other Scoring Plays o An oddity o Écoles o Obsolete Scoring Plays o Recording Points

Ending the game

Introduction Object of the Game Trictrac is a game for two players. The players move checkers on a board, scoring points for various plays and positions. The winner of the match (partie) is the first to win twelve games (trous) of twelve points each. Trictrac is not fundamentally a racing game.

Equipment

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Trictrac is played on a board which generally resembles a backgammon board. It consists of twenty four points (fleches), twelve on a side, with a bar separating each side into two tables (jans) of six points. Each player has fifteen checkers (dames) two dice (des) and a dice cup (cornet). Points are scored with three disks (jetons de bredouille) and games are scored by placing a peg (fiche) in one of twelve holes (trous) opposite the points alongside the board.

Notation It seems that each trictrac book has developed its own notation for naming or numbering the points on the board and for recording positions. I will adopt a notation that should be familiar to backgammon players rather than trying to stay true to some of the older trictrac literature.

The figure on the left represents the trictrac board. I have not found a way to illustrate the trous for recording games. On a real trictrac board, there would be small holes along each side of the board where the numerals one through twelve appear in the diagram. I will refer to the points as B1 through B12 (B for Black) on the top half of the board and W1 through

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W12 on the bottom. When I refer to a point by number alone, it should be taken to mean the point from the perspective of each player. For example the one point means B1 from Black's perspective and W1 from White's.

Vocabulary The board is divided into four tables or jans which have two names, one from each player's perspective.

Table Black White

B1 to B6 petit jan jan de retour

B7 to B12 grand jan

W7 to W12 grand jan

W1 to W6 jan de retour petit jan

Some of the points have colorful French names: The one point is the talon or stock. The game begins with all checkers on the talon. The twelve point is the coin de repos. The coin is the most difficult point in the grand jan to make (that is, to

occupy with two checkers) because special rules limit the movement of checkers to the coin de repos. The eight point is the case de diable (the devil's point) because experience has shown that, after the coin de repos,

it is the most difficult point to make in the grand jan. The eleven point is the case d'écolier (schoolboy's point) because novice players tend to make it too early in the

game to their disadvantage. Some books refer to the six and seven points, together known as the coin bourgeois.

Conduct of the Game Setup

Each player begins with fifteen checkers on the talon (stock), B1 for Black and W1 for White.

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A Turn at Trictrac A player begins the turn by throwing two dice and announcing the pips, high die first. The player then examines the position and determines whether the throw permits him to score any points. The player announces the score and move the markers to record the score. The player then moves checkers corresponding to the numbers thrown. The opponent then announces any points earned by the player's throw and rolls the dice.

Checker Movement Black moves clockwise from B1 to B12, then W12 to W1. White moves counter- clockwise from W1 to W12, then B12 to B1. As long as the player keeps his checkers on his half of the table, he is said to play an ordinary game (jeu ordinaire). The player is said to head for home (passer au retour) when he moves the first checker on the opponent's half of the board. As we will see, there are rules which limit when a player may passer au retour. Checker movement is similar to backgammon. A player moves checkers a number of points matching the numbers thrown on the dice.

For example, if Black opens the game with a 3-2, Black may move one checker from the one point to the four point and another to the three point. In clearing two checkers off the talon, Black is said to play tout à bas.

Black's alternative play is to move one checker to the six point, known as playing tout d'une. Black is said to pass over (faire étape or reposer) the intermediate point (here, either the three or four point).

Doubles are treated as two identical numbers, not four, as in backgammon. If possible a player must move checkers corresponding to the number on each die. If a player can move either but not both of the two numbers, the player must move the higher number. Like backgammon, once a player has all checkers in the jan de retour, the player begins to bear checkers off the board.

Limits on Checker Movement

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Checker movement is limited by a number of rules. A checker may not move to or pass over a point occupied by one or more opponent's checkers. This differs from backgammon where you can hit a single checker (demi-case or blot) and send it to the bar.

Lois du Coin You may not move any checkers to the opponent's coin de repos, although you may pass over it. You may not move a single checker to your own coin de repos, if it is empty. The coin de repos must be occupied with two checkers on a single throw.

With an opening 6-5, black must clear two checkers from the talon. Black cannot play a single checker to the twelve point - the coin de repos must be occupied with two checkers.

If you throw dice which allow you to land two checkers on the opponent's empty coin de repos, you may use them to make your own coin de repos. You are said to capture your own coin par puissance.

Six-six brings the two checkers on black's seven point to white's empty coin de repos. Black may use this roll to move the two checkers to his coin de repos, making it par puissance.

Finally, if you can make your coin de repos both directly and par puissance, and you choose to make it, you must do so directly.

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It would appear that five-five would let black make the coin de repos with either the two checkers on the seven point or the eight point. The rule states that if black is to take the coin, it must be directly (from the seven point) rather than par puissance.

If a player has exactly two checkers on his own coin de repos, the player cannot move one without moving the other.

Lois du Plein If a player has at least two checkers on each of the six points in the petit jan (points one through six), the grand jan (seven through twelve) or the jan de retour (the opponent's one through six points), the player is said to fill a table (faire un plein or remplir). A player cannot fill the fourth table (the opponent's grand jan) because he cannot occupy the opponent's coin de repos. A player can fill a table in more than one way.

A roll of 2-1 fills the table in only one way, from the nine point. The one would be played from the four, five or seven points. A roll of 5-2 fills the table in two ways, playing the two from the nine point, or playing five, then two from the four point. A roll of 4-2 fills the table in three ways: two from the nine point, four from the seven point or four then two from the five point. Doubles can allow you to fill in at most two ways. A roll of 1-1 fills from the nine point. A roll of 2-2 fills from either the seven or nine points.

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As we will see in the section on scoring, a player earns points for filling a table or for preserving a table already filled. Filling a table in two or three ways scores double or triple. There are two rules governing checker movement that relating to filling tables. You must fill a table, if possible. You must preserve a table, if possible.

Lois du Retour A checker may not stop on a point in the opponent's grand or petit jan if the opponent is still able to fill it. A checker may pass over an empty point in the opponent's grand jan to an empty point in the petit jan even if the opponent is still able to fill the grand jan.

Lois de la Sortie When a player gets all checkers to the jan de retour, the player may begin removing checkers from the board. The player can remove a checker from the six point with a six, and so on. Unlike backgammon, a player cannot move a checker with a number if the number can be played inside the jan de retour. So you cannot remove a checker from the five point with a five if there is a checker on the six point which can be played to the one point. Like backgammon, you can remove a checker from a lower point with a high number if there are no checkers on the point corresponding to the higher number. For example with no checkers on the six point and checkers on the five point, a player can remove a checker from the five point by rolling a six.

Scoring In trictrac, there are a number of jans, plays which are worth points to one of the players. It is said that the term jan refers to the god Janus who had two faces, since many of the scoring plays in trictrac have two faces. There has been no change in scoring plays since the early 18th century. Most sources say that there are eight jans, but list other scoring plays which they do not call jans. Other sources list eleven jans. Although this is a bit confusing, the good news is that the sources are completely consistent as to how to score the game.

Rare Jans The first five of the jans can occur only at the beginning of a game, and even then quite infrequently.

Jan de trois coups or Jan de six dames The jan de trois coups is scored on the third throw of a game. If a player can move to leave one checker each on the two through seven points, it is worth four points. The player does not have to play the third throw to leave all the blots in order to claim the score.

Jan de deux tables The jan de deux tables requires: a player has only two checkers cleared from the talon the opponent has not made the coin de repos the player rolls a number which brings a checker to each of the two coins.

The player scores four points for the jan de deux tables if the roll was not doubles, or six points for doubles. The laws of the coin de repos prevent you from moving to the coins. The player clears two checkers from his talon.

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Black scores six for the jan de deux tables and then moves two checkers from the talon to the seven point.

Contre jan de deux tables The contre jan de deux tables requires: a player has only two checkers cleared from the talon the opponent has made the coin de repos the player rolls a number which brings a checker to each of the two coins.

The opponent scores four points for the contre jan de deux tables if the roll was not doubles, or six points for doubles.

Jan de mézéas A player scores for the jan de mézéas when: a player has cleared only two checkers from the talon the player has made the coin de repos the opponent has not made the coin de repos the player rolls a one (4 points) or two ones (6 points)

Contre Jan de mézéas An opponent scores for the contre jan de mézéas when: a player has cleared only two checkers from the talon the player has made the coin de repos the opponent has made the coin de repos the player rolls a one (4 points) or two ones (6 points)

This is quite rare.

Filling a Table Petit Jan

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For filling the petit jan with non-doubles, a player scores: four points for filling in one way eight points for two ways twelve points for three ways

For filling the petit jan with doubles, a player scores: six points for filling in one way twelve points for two ways

For each turn in which a player preserves the petit jan, the player scores: four points with non-doubles six points with doubles

Grand Jan Scoring is identical to the petit jan.

Jan de Retour Scoring is identical to the petit jan.

Other Scoring Plays Battre sur une demi-case A player scores for hitting an opponent's single checker, that is, rolling a number which would allow the player to move a checker to a point occupied by a single opposing checker. As we have seen, however, the rules prohibit actually moving to the point.

Black rolls five-two which permits him to hit the checker on W6 from the coin de repos. Black scores four points (see the table below) and makes the case de diable (B8) with checkers from the three and six point.

Note that you can hit a blot from the coin de repos even though there are only two checkers on the coin. This is true despite the rule that the last two checkers on the coin must leave together. Here, the checkers are not moving, so the law does not apply (but see battre le coin for the opposite result). As with filling a table, it is possible to hit a checker up to three ways with non-doubles and up to two ways with doubles. It is also possible to hit more than one checker with the same roll of the dice. For hitting a checker in either petit jan with non-doubles, a player scores: four points for hitting one way eight points for two ways twelve points for three ways

For hitting a checker in either petit jan with doubles, a player scores: six points for hitting one way twelve points for two ways

For hitting a checker in either grand jan with non-doubles, a player scores: two points for hitting one way four points for two ways six points for three ways

For hitting a checker in either grand jan with doubles, a player scores:

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four points for hitting one way eight points for hitting two ways

Battre a faux When you hit a checker with the sum of two dice (tout d'une), one of the intermediate points must be free, or you cannot score for hitting a checker. The play is said to be a false hit, and in fact points go to the opponent.

Black rolls a five-two, which appears to let him hit the checker on W6 from the coin de repos. However the five is blocked by the checkers on W8 and the two is blocked by the checkers on W11. Black has made a false hit and White scores four points.

The false hit scores the same number of points as the actual hit would have scored. A false hit in either petit jan by non-doubles scores four points. A false hit in either petit jan by doubles scores six points. A false hit in either grand jan by non-doubles scores two points. A false hit in either grand jan by doubles scores four points.

Battre le coin A player scores for hitting the opponent's coin de repos if: The player has made his own coin de repos. The opponent's coin is empty The player rolls numbers which would bring two checkers to the opponent's coin de repos.

Black has made his coin de repos; White has not. Black scores for hitting the coin de repos since six-six brings two checkers from B7 to W12. Black then moves two checkers from the talon to B7.

To hit the coin from one's own coin requires extra checkers on the coin. For example six-one would hit only if there were three (or more) checkers on B12. One-one would hit only if there were four (or more) checkers on B12. This is a consequence of the rule that the last two checkers on the coin must leave together. This differs from hitting a checker, where the last two checkers on the coin de repos can hit.

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It is also important to contrast battre le coin with the jan de mézéas. The jan de mézéas occurs only at the beginning of the game when there are only two checkers cleared from the talon. Hitting the coin is worth four points by non-doubles and six points by doubles.

Nombre non joue If a player cannot move either or both of the numbers thrown, the opponent gets two points for each number not played.

Sortir The first player to clear all of his checkers off the board wins four points by non-doubles or six points by doubles.

An oddity Now that you have seen all of the scoring plays, what is the greatest number of points that can be scored on a single roll? Can you produce the position and roll?

Écoles If a player fails to claim all of the points to which he is entitled, the opponent scores the points, sending the player to school (école). This is similar to muggins in cribbage. The rules around écoles are fairly complicated, as they must take into account when to claim the penalty, false claims, etc. The penalties also apply for certain illegal checker movements (failing to fill a board). As of this time, I have not studied these in detail.

Obsolete Scoring Plays Jan de Rencontre The jan de rencontre occurs at the beginning of the game where the second player's roll is the same as the first. This scores four for non-doubles and six for doubles. This was abandoned as scoring play sometime in the seventeenth century.

Pile de misere The pile de misere occurs when a player is unable to bring his checkers onto the opponent's half of the table and ends up with all fifteen checkers on his own coin. This scored four points for non-doubles and six for doubles. Preserving the pile de misere scored four points for non-doubles and six for doubles. Some late seventeenth century books say that this had already been abandoned, yet Soumille (1756) states that some play it and some do not.

Margot la Fendue Margot la fendue punished a player for a near miss. When a player rolls a number which would land on an empty point between two blots, the opponent scores two points for non-doubles and four points for doubles. This was obsolete by the late seventeenth century.

White's six-five allows a checker to hit from W11 to B3, splitting the two blots on B2 and B4. White loses two points.

Recording Points A trictrac match consists of twelve games of twelve points.

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Partie Bredouille and Partie Simple If a player wins a game by scoring twelve points in a row, he wins a double game, known as a partie bredouille, or a lurch in many other games. Bredouille means empty-handed. A single game is known as a partie simple. Either the first or second to score can win a partie bredouille. If Black scores first, he wins a partie bredouille if he scores the first twelve points. If Black scores first, White can win a partie bredouille by scoring the next twelve points. If Black scores, then White scores, then Black scores again, neither player can win a double game. Black is said to remove the possibility of a double game (débredouiller).

Jetons de Bredouille Three markers known as the jetons de bredouille are used to score points and to keep track of whether a partie bredouille is possible. The game begins with the jetons on the playing surface against the side of the board nearest the two talons, halfway between the players. The first to score takes a single jeton and moves it along the middle of the board, slightly closer to his side than the opponent's to mark points. The jeton is moved: between the two and three points to mark two between the four and five points to mark four against the bar on the side of the talon to mark six against the bar on the other side to mark eight against the far side to mark ten

Either player can score a partie bredouille. The second player to score moves the remaining two jetons to mark points. Now only the second player can win a partie bredouille. If the first player scores again, the first player continues to move the one checker along, but removes the second of the two jetons marking the opponent's score (débredouille). This indicates that neither player can win a partie bredouille.

Marking Trous A player winning a game marks the score by placing a marker (fiche) in a hole (trou) alongside the points on his side of the board. The hole by the talon indicates one point and the hole by the coin de repos indicates twelve points. A player marks one trou for a partie simple and two for a partie bredouille. It is possible to win more than two trous on a single roll. For example, with a game score of six points to none in favor of Black, Black scores twenty points for a total of twenty six. This is worth two parties bredouilles (the first twenty four points make twelve in a row, twice) with two points left over. Black scores four trous and can leave a single jeton marking the remaining two points (but see ending the game, below). For another example, consider a score of six-six, with no partie bredouille possible. Black scores twenty points. Black wins a partie simple with the first six points, a partie bredouille with the next twelve and has two points left. Black scores three trous and can leave a single jeton marking the remaining two points (but see ending the game, below).

Ending the game A player who wins one or more trous on his own throw of the dice has the choice of continuing the game (tenir) or starting a fresh game (s'en aller). If a player wins a trou on the opponent's throw, the game continues. In either case, the partie winning the trou rolls again to begin the next game.

Tenir If a player chooses to continue, the opponent loses any points earned toward the next trou. The winner, however, keeps any points in excess of those required to win the trou. The opponent, before his turn, can still mark any points earned by the winner's throw (for example, for battre à faux).

S'en Aller If a player chooses to restart, both players return the checkers and the jetons de bredouille to the starting position.

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Richard Seymour's The Compleat Gamester. 1734.

Grand Trick-Track by Richard Seymour The earliest English material I know on the game of trictrac is Richard Seymour's The Compleat Gamester. This went through a number of editions. I have included a photograph of the Sixth Edition of 1739, but quote from the Fifth edition of 1734.

The Compleat Gamester: In Three Parts. viz. I. Full and easy Instructions for playing the Games chiefly used at Court and in the Assemblees, viz. Ombre, Quadrille, Quinitille, Piquet, Basset, Faro, and the Royal Game of Chess. II. The true Manner of playing the most usual Games at Cards, viz. Whist, All-Fours, Cribbidge, Put, Lue, Brag &c. With several diverting Tricks upon the Cards. III. Rules for playing at all the Games both Within and Without the Tables: likewise at English and French Billiards. Also the Laws of each Game annexed to prevent Disputes. Written for the Use of the Young Princesses, by Richard Seymour, Esq.; The Fifth edition. London: E Curll. 1734. viii+132+94pp. [ Horr 1161, Jessel 1494]

II. The Noble and Courtly Game called GRAND TRICK-TRACK, is a French Diversion; and most commonly used by Persons of the first Quality. It is thus played: The Table-Men are to be placed on the Side of the Tables. Next, it is also to be observed, that besides the Table-Men which you play, there are 3 other Pieces to be used, call Markers, whether Half Crowns, or Halfpence, or any other Coin: These are to mark the Throw of your Dice, on the Points of the Tables which are advantageous to you: For Example, If in you first Tables you make single Toots, in 3 Casts, or Throws, you mark with one Marker 4. There are 12 Holes on the Sides of your Tables, with pegs in them, for the Use of this Game. Note then, that 12 Marks gained on the Points of your Tables, make an Hole, and 12 Holes make up the Game, if you agree to it; else less, or more: If you fill up your Points, for every single Throw on the Dice, you make 4, and for Doublets 6; and may hold your Game as long as you think convenient, that is, play on without breaking up your own, and your Adversary's if you believe you shall get no Advantage by beginning again. If you hold with your double Men in your Tables, before you can make a Point, and your Adversary cannot fill his Tables, you are obliged with your Man to pass over into his Tables, tho' it be commonly a Disadvantage; but if he throws so well, as to fill up, then it alters the Matter, and you cannot pass. Note, that when you have marked 12 with your Marker, which, as I said before, makes up one Hole, you may go off, break up your Tables, and begin again, provided you have the Dice; or else you cannot. If in playing this Game you touch a Man rashly, as intending to play it, and think to change it for another, you are obliged to play it as you before intended. Note, As to those Men that are obliged to pass over into the Adversary's Tables, if he hits them, he marks thus: For every single Throw 4; for Doublets 6; and if at any Time, by your good Fortune in throwing, you can mark over and above 12, you must then mark a Hole, or else 2, if you go double, and the Overplus remaining is called to the Good,

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provided you do not break your Game: You cannot go off, nor break your Tables by your Adversary's Throws. And note, that if you chance to make more or less than is right, it is in his Power to take the Advantake, (sic) put you back, or oblige you to mark full. This is what is most considerable in your first Tables. Now, as to your Adversary's second Tables: For every Man you hit of his with a single Die, you mark but 2; and for Doublets 4; though in his first Tables, 4 for each single Die, and 6 for Doublets. If you chance to hit a Blot or two in your Adversary's Tables, and cannot pass by reason of his Man standing in your Way, and hindering you, it is allowable for him to take the Advantage of marking by your own Throws in both Tables, as before mentioned. The Ace-Point of both Corners in the second Tables, cannot be divided here, nor fill the Corners, as at other Games; though in Lieu of that Convenience, if the Dice favour, for each single Cast, you mark 4; and for Doublets 6. Then as that Part of the Game, called Gens des Retour, or the Back-Game, which is the latter Part: Next, bearing off your Men, as it is used at Back-Gammon, you play your Men as fast as you can, into his Tables, endeavouring to fill up the Points, as at the Fore-Game; which being done, you bear off your Men; only there is on Distinction between this and Back-Gammon (noted before) that as Doublets thrown at the last Cast gives considerable Advantage to the Gamester there, it is here of no Value, nor gives any Addition to the Throw. Note, that if at any Time you break up your Tables, and disorder your Men, except by gaining 12 Points, you can mark a Hole, it is in your Adversary's Power to oblige you to hold your Game on still and to play all the Table-Men you have so touched and disordered to his own Advantage. Note also, that we distinguish the Single from the Double, in this Manner: If your Adversary, by his ill Fortune in throwing, has no Points on his Tables marked, altho' your Throw is single, yet still you may mark a double Point; but if otherwise, he has such good Luck by the Dice, to have any Point to mark, then he comes double; which you are to take off again, if you can hit him. These are all the Passages that are considerable in this Game; which tho' easily to be comprehended, by those who divert themselves with playing often at Tables, and especially such who have any Skill in Tick-Tack, of which, this Game is observed to make a Compleatment, by adding more Parts and Embellishments; yet the most ready Way for a young Gamester, who is desirous to learn it, is to see it performed by two Gamesters; and then taking notice of these Instructions, he will presently be let into the Secret.

Montmort and the Grand Jan by David Levy

updated August 10, 1998

Contents Introduction Montmort's Jeux de Hazard Trictrac Problems

o Problem 1 o Problem 2

The First Roll Montmort's Calculation But is this Trictrac?

o Problem 3

Introduction The search for literature about historical games leads to some fascinating side trips. The theory of mathematical probability arose almost entirely as a result of early interest in games of chance. So the early works of Pascal, Fermat, De Moivre and others have much interesting material on early games. One early work on probability has a small section on trictrac, consisting of three problems. The first and third problem are trivial; the second caused me to consider some fascinating issues about preserving a grand jan.

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Montmort's Jeux de Hazard Pierre Remond de Montmort lived from 1678 to 1719. His work on probability was originally published in 1708 and a second greatly expanded edition appeared five years later:

[Montmort] Essay D'Analyse sur Les Jeux de Hazard. Seconde Edition. Revue & augmentée de plusieurs Lettres. Paris: Jacque Quillau. 1713. xlii+414pp.

The book poses problems in probability and combinations about the games of the day and attempts to develop mathematical solutions. In solving the problems, Montmort extended the knowledge of probability beyond what had been published earlier by Pascal and Fermat in the mid-17th century. For a full account of the historical importance of the work, see Todhunter's 1865 History of the Theory of Probability (or the 1949 reprint). While I will focus on the trictrac problems, you will get a flavor of the style of the early works on probabilities.

Trictrac Problems In Section 148 (page 189), Montmort notes that to play trictrac well, it would be useful to determine quickly the chances of hitting a checker or filling a jan. He gives an example where Black is trying to fill the petit jan. Recall that filling the petit jan is worth four points each way for non-doublets and six points each way for doublets

Montmort comments that there are few players who can determine at a glance that there is one roll which wins twelve points, ten which win eight, three which win six, sixteen which win four and six which fail to fill the petit jan. But these details which are overlooked by ordinary players are nonetheless important for good play.

The table below supplies the details (which, thanks to Witold Szczeponik, have been reviewed and corrected).

Number of Throws

Number of Points

Throws

1 12 2-2

10 8 5-4 5-2 4-2 4-1 3-2

3 6 5-5 4-4 1-1

16 4 6-5 6-4 6-2 5-3 5-1 4-3 3-1 2-1

6 0 6-6 6-3 6-1 3-3

The First Problem What are the chances of making the coin de repos in the first two throws of the game? Recall that you can only make the coin de repos with two checkers together and that you can make the coin par puissance by having two checkers land on the opponent's empty coin. Montmort notes that there are three sequences which make the coin.

1. 6-5 followed by 6-5

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2. 5-5 followed by 6-6 3. 6-6 followed by 5-5 or 6-6 (the latter, par puissance)

Thus the chances are 2/36 * 2/36 + 1/36 * 1/36 + 1/36 * 2/36 = 7/1296.

The Second Problem

Montmort gives the position at left and asks what are the chances that White can play two throws without breaking the grand jan? Assume that White cannot play to the other side of the board (See the Lois de Retour). So if White rolls 6-6, the only play is W6 to W12. The other six cannot be played.

The First Roll Before reviewing the calculations, let us look at some of the subtleties of checker play. How would you play each of the following rolls as white in this position? 4-1 3-2 3-1 2-2 5-2 4-3 5-4 4-1: Move one checker from W6 to W11. (Return to Diagram)

Before you move, you have only one way to play a six -- from W6 to W12. By using 4-1 to move the checker from W6, if your next roll contains any six you can move only the lower die and will preserve your board. The alternative W7 to W12 is better only if you roll 4-2 or 3-3 (three chances in 36) on the next roll, but is worse if you roll 6-5, 6-4, 6-3, 6-2 or 6-1 (10 chances in 36). Killing sixes is a common technique to preserve a board in backgammon and applies equally here. Kill the six! The alternative W6 to W10 and W7 to W8 maintains the jan with 4-2 (two chances), but breaks on 6-5 and 5-1 (four chances). Keeping the spare on the highest possible point will help maintain the board. Preserve a five!

3-2: The same. Move one checker from W6 to W11. (Return to Diagram) Moving both checkers to W9 loses on 6-5 and 5-1 (four chances), gaining only on 3-3 and 3-2 (three chances). Preserve the five!

3-1: Move one checker from W6 to W10, killing the six but preserving a five. (Return to Diagram) W7 to W11 gives up 6-5, 6-4, 6-3 and 6-2 and gains only on 5-2 and 4-2. W6 to W9 and W7 to W8 gives up 6-5 and 5-2 and 5-1 and gains on 4-3 and 3-3.

2-2: The same. Move one checker from W6 to W10, killing the six but preserving a five. (Return to Diagram) 5-2: Play W6 to W8 and W7 to W12. (Return to Diagram)

The alternative W6 to W11 and W7 to W9 gives up 6-4 and 2-2 and does not gain on any rolls. Preserve the four! 4-3: W6 to W9 and W7 to W11. (Return to Diagram)

The alternative, two checkers to W10 gains on 2-2, but loses on 6-3 and 3-1. Preserve the three! 5-4: W6 to W10 and W7 to W12. (Return to Diagram)

Preserve the two! The alternative, two checkers to W11 would force you to break with 6-2.

Montmort's Calculation

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So, now that we know how to play the first roll, we can look at Montmort's calculations. I have added some clarifying detail.

Case Chances in 36 first roll

First Roll Location of Spares after first Roll

Chances in 36 second roll

Second Roll Product

1 2 6-5 W12 W12 1 6-6 2

2 3 6-4 5-5 W11 W12 3 6-6 6-1 9

3 4 6-3 5-4 W10 W12 6 6-6 6-2 6-1 1-1

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4 4 6-2 5-3 W9 W12 10 6-6 6-3 6-2 6-1 2-1 1-1

40

9 1 4-4 W10 W11 8 6-6 6-2 6-1 2-1 1-1

8

5 2 4-3 W9 W11 12 6-6 6-3 6-2 6-1 3-1 2-1 1-1

24

6 4 6-1 5-2 W8 W12 15 6-6 6-4 6-3 6-2 6-1 3-1 2-2 2-1 1-1

60

7 6 6-6 5-1 4-2 3-3

W7 W12 12 6-6 6-5 6-4 6-3 6-2 6-1 4-1 3-2 3-1 2-2 2-1 1-1

126

8 4 4-1 3-2 W7 W11 23 6-6 6-5 6-4 6-3 6-2 6-1 5-1 4-1 3-2 3-1 2-2 2-1 1-1

92

10 3 3-1 2-2 W7 W10 27 6-6 6-5 6-4 6-3 6-2 6-1 5-2 5-1 4-2 4-1 3-2 3-1 2-2 2-1 1-1

81

11 2 2-1 W7 W9 32 breaks on 5-5 5-4 4-4

64

12 1 1-1 W7 W8 35 breaks on 5-5

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total 36 565

So with careful play, White can preserve the grand jan in 565 of 1296 cases.

But is this Trictrac? The backgammon writer Danny Kleinman has noted on many occasions, "It's not how many, but how good or bad." With that thought, let's take another look at the position:

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Assume the score stands at 11 trous to 11 and that Black has ten points and White has two toward the final trou. Also assume that Black cannot score on his own throw of the dice. You roll 4-1 and score another four points, making six. Do you still play W6-W11, killing sixes and preserving fives? Check the scoring rules carefully!

If you do, you will win the partie next term if you roll 6-6, 2-2 or 1-1, scoring six for preserving the board by doublets. However, with any other six, you will score only four points but Black will win two points and the partie for the six you are unable to play (nombre non jouer)! The play we rejected earlier, W7 to W12 has four match winning doublets -- the three above plus 3-3. It does not immediately lose the match on 6-5, 6-4, 6-3, 6-2 or 6-1 since those rolls can all be played. Thus considering only the immediate match-ending rolls, preserving sixes is better after all.

Problem 3 Montmort discusses the impossibility of examining a situation in Trictrac and determine which player has the best position. The only solvable problems are those in which both of the players have already borne off many of their checkers. He gives the following example:

Pierre, White, is on roll, playing against Paul, Black. What is the value of the Pierre?

I have to say that Montmort has a convoluted way to solve this, which may explain why he ended up with the wrong answer! Let the stake be A and S be the value of the game to Pierre. Pierre wins A if he does not roll 2-1 or 1-1. Let x be the value to Pierre after rolling 2-1 and y be the value to Pierre after rolling 1-1. Then:

1 S = (33A + 2x + y)/36

After rolling 2-1, let u be the value to Pierre if Paul does not roll 2-1 or 1-1. Let h be the value if Paul rolls 2-1 and t be the value if Paul rolls 1-1. Then:

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2 x = (33u + 2h + t)/36

To find the value of u, note Pierre wins if he does not roll 1-1, so

3 u = 35/36*A

To find the value of h, note that Pierre wins if he does not roll 1-1, or if he rolls 1-1 and so does Paul.

4 h = (35*36 + (1*1)/(36*36) ) * A

To find the value of t, note that Pierre wins if does not roll 1-1, or if he rolls 1-1 and Paul rolls 2-2, 2-1 or 1-1. So:

5 t = (35*36 + (1*4)/(36*36) ) * A

Substituting equations 3, 4 and 5 into 2, both Montmort and I get

6 x = 45466 / 46656 * A

Recall that y is the value after he rolls 1-1. Again there are three cases, depending on whether Paul rolls 2-1, 1-1 or another throw. Let q be the value after another throw, p after 2-1 and n after 1-1. Then:

7 y = (33q + 2p + n)/36

since Pierre wins unless he rolls 2-2, 2-1 or 11.

8 q = 32/36 A

After 1-1, 2-1, Pierre wins if he does not roll 2-2 2-1 or 1-1 or, if he does, if Paul rolls 1-1. Thus

9 p = (32/36 + (4*1)/(36*36) )* A

After 1-1, 1-1, Pierre wins if he does not roll 2-2, 2-1 or 1-1, or, if he does, if Paul rolls, 2-2, 2-1 or 1-1. Thus:

10 p = (32/36 + (4*4)/(36*36) ) * A

Substituting 8,9 and 10 into 7, both Montmort and I get

11 y = 41496 / 46656 * A

Substituting 6 and 11 into 1, Montmort (or his printer!) gets

12 S = 46641 / 46656 * A

which fails the reasonableness test. Clearly Paul wins more than 15 times in 46656. My math yields

13 S + 46441 / 46656 * A

which I also get by taking a more rational (meaning verifiable) approach to the problem! I also note that carrying the stake throughout the problem is confusing and misleading. The chances of Pierre winning are 46441 / 46656. The value of the position to him is (46441 - 215 ) / 46656 times the stake. Inserted from <http://web.archive.org/web/20090627104906/http://pages.sbcglobal.net/david.levy/trictrac/essays/montmort.htm>

Monday, August 01, 2011