Contributing expert Stewart Ethier writes:
While it would seem to be a natural alternative to baccarat in the high-roller rooms on the Las Vegas Strip, to our knowledge, trente et quarante has never been dealt in a Las Vegas casino. The three logical places for the game to have surfaced were at the openings of MGM Grand, Monte Carlo, and Paris. The Grand made a big deal of its "Monte Carlo" casino, the Monte Carlo attempted to recreate the atmosphere of its namesake, and Paris featured all things French throughout the property. However, none strayed from traditional baccarat (or blackjack), choosing instead to feature French-style roulette. The game is still popular at Les Casinos de Monte Carlo, however.
Trente et quarante ("thirty and forty" in French) dates back at least as far as the 17th century, having been mentioned in Molière's The Miser (1668). It was studied by the famous 19th-century mathematicians Siméon-Denis Poisson and Augustus De Morgan. Modern studies include those of Thorp and Walden (International Journal of Game Theory, 1973) and Ethier and Levin (Advances in Applied Probability, 2005), available in university libraries.
Known also as rouge et noir ("red and black" in French), trente et quarante is played with six standard 52-card decks mixed together. Suits do not matter but colors do. Aces have a value of one, picture cards have a value of 10, and every other card has a value equal to its nominal value. Two rows of cards are dealt. In the first row, called Black, cards are dealt until the total value is 31 or greater. In the second row, called Red, the process is repeated. Thus, each row has associated with it a total between 31 and 40.
Prior to the dealing of the cards, four even-money bets are available. They're called red, black, color, and inverse. After the cards are dealt, the winning row is the one with a total value closer to 31. A bet on red wins if Red is the winning row. A bet on black wins if Black is the winning row. A bet on color wins if the color of the first card dealt to Black is the same as the color of the winning row. A bet on inverse wins if the color of the first card dealt to Black is different from the color of the winning row. For all four bets, a push occurs if the Red and Black totals are equal and greater than 31.
If the Red and Black totals are both equal to 31, half the amount of the bet is lost. For all four bets, if the Red and Black totals are both equal to 31, there is typically an en prison option similar to that in single-zero roulette. [Note: The first of these two possibilities corresponds to the roulette rule called partager ("to share"), while en prison means that the bet is held until the next coup, when it is either released in full or lost.]
Finally, each of the four even-money bets has an associated insurance bet that can be made for 1% of the original wager. It pays off the loss in the case of a tie at 31 though the bet itself is retained by the casino. The insurance bet is lost if the original bet is won or lost, and it pushes if the original bet pushes.
The house advantage on each of the four even-money bets is 1.095% (pushes included) or 1.200% (pushes excluded). These figures are comparable to those on the banker bet at baccarat: 1.058% (pushes included) or 1.169% (pushes excluded).
Insurance reduces the house advantage to 0.903% (pushes included) and 0.990% (pushes excluded).
As is the case with baccarat, it has been shown that trente et quarante cannot be beaten by card counting.
Stewart Ethier is a Professor of Mathematics at the University of Utah, specializing in probability theory. He is at work on a comprehensive textbook on the mathematics of gambling that will also include historical information about the casino games. The book will be available by 2008.
