The reverse labouchere roulette system is a supposedly infallible system for winning systematically at roulette. It came to prominence in the 1975 book “Thirteen against the Bank” by Norman Leigh. In this book Leigh describes an attack made by himself and a team of 13 against the (fictionally named) Casino Municipale in Nice which resulted in a large profit and banishment from every casino in France by the government.
Leigh appears to acknowledge that the more famous Labouchere system does not work, blaming its failure on table limits and finite bankrolls. He then reverses the system, an idea he attributes to a book called How To Really Beat the Bank at Monte Carlo by a Lord Beresford, believing that this will turn the system into a profitable one.
In the reverse labouchere bets are increased with wins and decreased with losses. This system seems to rely on the assumption that results occur in 'runs', so that a winning run results in ever increasing profits, and a losing run causes a decrease in stake so that losses are minimized.
Unfortunately the reverse labouchere does not work. Reversing a system does nothing to change its profitability, only the frequency and size of stakes. Put simply, there is no way to beat the house edge at roulette using staking systems.
Suppose the table minimum is 10 units or less. We start by writing the numbers 1-4 in a line:
1 2 3 4
The first stake is calculated by adding the first and last numbers together. In this case we have an initial stake of 1 + 4 = 5 units. This is placed on any of the even money bets (odd/even , high/low, black/red)
If you win the last stake is added to the end of the line:
If you lose the end numbers are deleted:
The next stake is calculated by adding together the end numbers, and the process repeats.
If the entire line is deleted then a new line is begun. Eventually the stake will become so high that it breaks the table maximum bet limit, in which case the line is deleted and the system begins again with a fresh line.
Click here to examine a more in-depth explanation of the results.
We consider only the claims made by Leigh regarding the time spent in Nice. The results of his attack can be summarised as follows:
Day |
Progressions |
Winnings (francs) |
|---|---|---|
1 |
2 |
78,000 |
2 |
3 |
130,450 |
3 |
3 |
66,165 |
4 |
2 |
67,000 |
5 |
2 |
55,385 |
6 |
4 |
123,003 |
7 |
6 |
159,666 |
8 |
5 |
119,595 |
I have constructed a computer simulation which calculates the profit obtained by a team of twelve over the course of a day, using Norman Leigh's own figures as a guide to the number of spins per hour (30). Running this simulation many times allows the distribution of profits to be observed, and although this is not a precise analytical proof it should convince all but the most skeptical that the reverse labouchere roulette system is fundamentally flawed.
The results of the simulation show that the expected result of using the reverse labouchere system is a loss of approximately 1000 francs per day per team. The probability of having a profitable day is around 0.2, and of having a day as good as even the worst of Norman Leigh is 0.000216, approximately 1 in 5000. These results combine to show the failure of the reverse labouchere and contradict the claims of Leigh.
More detailed information regarding the simulation, including a download link, is available on the proof page.
The results of the reverse labouchere simulation cast doubt on the veracity of Leigh's claims. The sheer magnitude of the profits claimed in Thirteen against the Bank when viewed in conjunction with the results of the simulation seem very unlikely indeed.
It is also unlikely that any team using a form of staking system would be banned by the government, as the failure of such systems was well known at the time and system players were routinely welcomed by casino management. Even if the team were lucky enough to make a profit the casino would be desperate for the team to stay, knowing that eventually the team would lose their profits. Forcing Leigh to stop playing only acted to ensure that the team kept their winnings.
There is a distinct lack of evidence surrounding the whole issue. References have been made to French newspaper articles concerning the incident, however these reports appear to be only hearsay. Even if these reports do turn out to be real it proves very little, there are many reasons for which players would be banned from the casino.
Biased wheels have been suggested as an explanation but this is unlikely as it would result in the success of many systems; not only the reverse labouchere. Casinos were also well aware of this problem and would easily have rectified the issue within eight days if this were the case.
There are too many unknowns to explain fully what really happened, although my personal belief is that Normal Leigh created a nice story to sell a few books. I suspect that a large part of the book is indeed true; I believe that he formed a team similar to the one in the book and even attempted the reverse Labouchere in France, however I certainly don't believe that he succeeded.
It should be pointed out that Leigh had an extremely nice deal in France regardless of what happened. The profit distribution agreement ensured that he received 10% of winnings attained from completed progressions as opposed to net profit, while risking none of his own money. Even if the team made a loss overall it would be likely that they would have achieved at least one progression, so Leigh would have still made a profit while carrying no risk himself.