The d’Alembert betting strategy is a technique for playing roulette. The technique was discovered by the French mathematician Jules D’Alembert in the 18th century. It makes use of two properties of roulette, namely that the total number of spins is finite and that there are relatively few winning numbers. Three conditions must be satisfied in order for the strategy to be effective:

- All payouts have to be larger than zero.
- The number of spins must be an even number, which implies that the number of red and black slots is odd.
- The amount staked has to be at least 3 times the payout with a total outlay up to 7 times the payout in order for it not to have a negative expected value.

In essence, the strategy is based on two winning numbers; these winning numbers are attained by distribution of all the spins amongst ‘red’ and ‘black’ outcomes. By distributing the spins in this way, there is a good chance that at least one of the two adjacent reels will land on either ‘red’ or ‘black’.

The strategy can be implemented by placing 3 coins on each of the 3 tables/reels. Imagine that red is to win 1 coin for every head on the first table/stroke, 2 on the second and 3 for every head on the third table/stroke. Black is to win 1 coin for every head on the first table/stroke, 2 on the second and 3 on the third table. In this way, there will be an even number of red and black outcomes as required by condition 3 above. The value of a single spin can then be calculated as follows:

1 if red comes up

3 if red comes up and another table/reel has a head

5 if same as before but now another table/reel has a red outcome

8 if both tables have a head.

If the payout is “p” for heads, this means the expected value of the game is (1 + 3 + 5 + 8)/3 “p”. Thus, if the payout is several times larger than the wager (e.g., “p” = 2.5, stake “x”), the game has positive expectation; otherwise it has negative expectation (“e.g. p” = -5, stake “x”).

By the same reasoning, if a player has a negative expectation with respect to a single spin in a given game in this way, the player can just double up all of their 3 coins and keep playing in this way. Although it is mathematically possible to double up all three coins in one spin at any point where the strategy is working, it is not feasible from a practical point of view because it would cost an average of approximately 5 times the payout for each additional coin.

This strategy can be used to make a profit by exploiting the properties of the game.

If play is stopped and restarted, and three coins are put on each of the tables/reels then one can calculate that if red comes up in one table, then another table will have a black outcome in two spins. If it does not, then in three spins another table will have a black outcome.

The strategy may not be viable after a single black result on a table/reel, but this situation is very rare. This means that the player can safely double up their three coins in any game that they are not losing money to at the time; in other words, in any game where the payout is approximately equal to or greater than 3 times the stake. This will ensure that an overall profit is made on the long term.

If the payout is exactly three times the stake (2.5 if one coin is placed), then such an amount of profit cannot be guaranteed; however, there is still a good chance of profit because any loss of one coin on a table/reel will be compensated by two wins on adjacent tables/reels in “two” turns.

Based on the above, the strategy will continue to work until a period of two losses on adjacent tables/reels.

If a player’s stake is greater than 3 times the payout, then the player should stop playing after one net loss on a table/reel and wait until another table has a net net profit before continuing to play.

If the player’s stakes are smaller than 3 times the payout, then they can safely increase their stake by 1 coin or stay for another spin without any risk of losing money overall.

There is one condition for this strategy to work: all payouts have to be larger than zero. This implies that the result of the last spin determines the payout; i.e., if red or black wins, then 1 coin is given. If a green zero (or green 00 if 3 coins are placed) wins then nothing happens and therefore no payout can be made.

The strategy is particularly good on the table/reel that has just lost, although it is also of use on any of the three tables at any time. The best timing to use the strategy is just after a zero has won and just before a zero wins. This ensures that there is always a chance of making a profit after each spin.

There is only one disadvantage to this strategy: that it does not work on an American-style roulette table with two zeros or more on the wheel. The reason for this is that in American roulette, the number of total spins is infinite. Thus, any single strategy designed to win eventually will probably fail after a very long period of time.