This is an elegant card game originating in France but played in many parts of Europe. Trente et Quarante means "Thirty and Forty".
The game has many similarities to baccarat. No playing decisions are made at any time.
Two rows of cards are dealt, called Rouge and Noir (red and black). Each card counts its value, face cards count 10 and aces count 1. The cards in the row are dealt until their total sums 31 or greater. The total closest to 31 wins. Players may bet the either on either Rouge or Noir to win. They may also bet on coleur or inverse. Coleur is a bet on whether the first card of the winning row is the same as the colour as its row. Inverse is a bet that the first card of the winning row is not the same colour as its row.
The house derives its edge in the event of a tie at 31, when the house pockets one half of all stakes.
The casino advantage is 1.10%
An insurance bet exists for higher bettors. This must be 1% of the original wager. It cancels out the loss in the event of a tie. This lowers the house edge to 0.9%.
It should be apparent that no card-counting system can beat the red/black bets, since they are perfectly symmetrical.
A similar thought excersize shows that there must be some fluctuation in advantage betwen the coleur and inverse bets. If the first card in the red row is low, then it helps the coleur bet.
I learned from a secondary source that Edward Thorp had investigated the potential of card-counting at Trente et Quarante in his article "The fundamental theorem of card-counting with application to Trente et Quarante" International journal of game theory 2 (1973), 109-119. Thorp apparently discovered that if the game was dealt to the very end and the player could flit between minimum and maximum bets he could gain a 2.5% edge. Neither of these conditions is practical. (I have not been able to obtain the original paper and would appreciate information on where to obtain it)
The best one-level system for counting these two bets counts red A-6 and black 9-K as +1, with black A-6 and red 9-K as -1. When the count is above 23, the coleur bet is favourable. When it is below -23, inverse is favourable.
Even considering we have two wagers to bet on, favourable opportunities do not occur frequently enough to make this worthwhile.
There are ways to improve on a linear system. Firstly, the probability of a tie at 31 could be factored into the system. It can be seen that there are certain end-deck subsets where no tie at 31 is possible and none of the bets have any house advantage. However, simulation revealed the probability of such a tie is fairly constant throughout the shoe. Possibly selective use of the insurance bet may increase expectation slightly.
Secondly, as with baccarat, the games becomes more and more non-linear the greater the number of cards dealt out. For example, seven black tens and one red ace gives rouge a 25% edge. This is worthy of further investigation.