Three system families that flatter losing players, and one Riviera weekend in 1966.
Fibonacci, Labouchere, D'Alembert
Annabel Cavendish
Editor · 14 May 2026
The One Thing All Three Share
Let's establish the ground before we walk onto it.
Total expected loss equals total amount wagered multiplied by the house edge. That's the complete formula. Any system that increases your bet size after losses, or decreases it after wins, or vice versa, changes when and how much you wager, not the mathematical relationship between what you put down and what you get back. The casino collects 2.703% of every chip regardless. our calculation confirms the house edge on any combination of bets on a European wheel is always exactly 1/37, regardless of what order you place them in. The system is irrelevant to the edge. Full stop.
What systems do affect is variance: the shape of the distribution of outcomes across sessions. Some systems increase variance (more dramatic swings), some reduce it. None of them improve the expectation.
Fibonacci: Slower Growth, Same Drain
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, and so on. Applied to roulette, you move one step forward after a loss and two steps back after a win. After six consecutive losses, Fibonacci has you betting 13 units where Martingale would have you at 64. The bust rate at the table limit is lower.
As PlaySmart Ontario's analysis confirms, however, the system is "ultimately doomed to fail" because the wheel has no memory and each spin's probability is 18/37 regardless of history. The recovery mechanism requires two wins to recover from each loss. The long-run drain is still exactly 2.703% of everything put down. The slower buildup is a comfort, not a solution.
Labouchere: The Most Elaborate Failure Mechanism
The Labouchere is the most interesting of the three, and by "interesting" I mean it has the most elaborate failure structure.
You begin by writing a sequence of numbers: say, 1, 2, 3, 4. The sum of your sequence, £10 at £100 per unit, is your profit target. Your first bet is the sum of the first and last number: 1 plus 4 equals 5 units, £500. If you win, you cross off those two numbers and bet the new first and last: 2 plus 3 equals five units again. Eventually you cross off all four numbers and pocket your £1,000 profit. If you lose, you add the lost bet to the end of the sequence and the required next bet grows.
The mechanism: after n consecutive losses on sequence [1,2,3,4], the next required bet is 5+n units. At 44 losses with no wins, the next bet is 49 units, £4,900, approaching a standard £5,000 table limit. The total already lost across those 44 bets, following the mechanics precisely, is the sum of bets 5, 6, 7,..., 48, which equals (5+48) times 44 divided by 2, or 1,166 units: £116,600. Chasing a £1,000 profit target. That is the structure.
The arXiv paper on Labouchere betting systems (1707.00529) confirms the formal result: the probability of losing approaches 1 as rounds approach infinity; with finite bankroll, "the algorithm often halts because it exceeds available funds." The table limit merely determines when that halt arrives.
As Wikipedia's Labouchere article notes, the system's namesake was a Victorian Liberal MP whose magazine "Truth" made its reputation exposing fraud. His own cancellation system was later sold wholesale as an infallible method. There is a kind of biographical symmetry in that.
Norman Leigh and the Nice Exception
In the summer of 1966, a British writer named Norman Leigh took 13 people to the Casino Municipale in Nice and played the Reverse Labouchere: a positive progression that increases bets after wins and decreases them after losses, the exact opposite of the standard version. Two weeks later, the team was banned from every casino in France. Not for cheating. Simply for winning, consistently, in a way the casino found unsettling.
The AbeBooks listing for the 1977 Penguin edition of "Thirteen Against the Bank" confirms the 1966 Casino Municipale dates. The subsequent analysis is less flattering. Casino.org's assessment by R. Paul Wilson is the definitive post-mortem: Leigh "was simply the benefactor of a terrific run of luck." The casino's response was precautionary, not mathematical. The French casino authority's action was an overreaction to an unfamiliar, winning system. No mathematicians studied the specific 1966 sessions and concluded the system was positive-expectation. The EV-invariance of all betting systems is established mathematics that doesn't require a specific study to refute any particular configuration.
Leigh's contribution to gambling literature is a genuinely compelling book about a variance run. Don't mistake it for evidence of a working system.
D'Alembert: Slow, Steady, and Precisely Calibrated to Lose
The D'Alembert is named after the 18th-century French mathematician Jean le Rond d'Alembert, who wrote that "the same event never happens many times in a row." That is the gambler's fallacy stated as a principle of nature. As the LdnMathLab analysis confirms, the method's appeal rests entirely on this fallacy.
The method: add one unit after each loss, subtract one unit after each win. Simple to remember without a notepad.
Our D'Alembert simulation on single-zero roulette with a 10-unit winning goal produces these results: probability of reaching the goal, 89.81%; ratio of losses to total money bet, 2.702%. Exactly the house edge, to four decimal places. The system generates sessions where you win small amounts nearly 90% of the time. The remaining 10% of sessions, you lose steadily. The house doesn't need to cheat; it just needs you to keep coming back.
The non-obvious flaw: D'Alembert assumes wins and losses balance out. They do in frequency, not in value. Because you bet more at higher unit levels during losing streaks, your average bet size during losing periods is larger than during winning periods. The expected loss per unit remains 2.703%, but the system ensures you bet more per spin precisely when you're running badly.
<aside> The net-win formula for a D'Alembert session: W minus D(D+1)/2, where W is wins and D is (losses minus wins). Example: 30 wins, 40 losses. D=10. Net: 30 minus 55 = negative 25 units. </aside>
There are players at Mayfair-level tables who arrive with Labouchere sequences written on their phones, convinced they have optimised the configuration. They have found a more elaborate way to annotate their losses. All configurations share the identical failure structure: sequences grow under losses, compress under wins, and the -2.703% per unit expected value is invariant to whatever is written on the phone. The house edge is indifferent to typography.
Key numbers
System
Bets after 6 losses (£100 base)
Characteristic failure mode
Mean expected loss per session
Fibonacci
£1,300
Slow ruin; winning streaks must significantly outnumber losses
2.703% of total wagered
Labouchere [1,2,3,4]
Up to £1,200+ depending on sequence
Table-limit crash; £116,600 potential before limit on 44-loss run
Welcome to the lesson on Fibonacci, Labouchere, and D'Alembert.
I'm Annabel, and this is where we address the three roulette betting systems that are not the Martingale, which means they don't get talked about quite as much, which means more people labour under the impression that one of them might actually work.
I'm going to disabuse you of that notion in roughly ten minutes, and I'll try to do it with enough detail that you come away with something genuinely useful rather than merely discouraged.
The three systems are the Fibonacci, the Labouchere, and the D'Alembert.
They are structurally quite different from each other and from the Martingale.
What they share is that none of them alters the expected value of a single spin, and therefore none of them alters your total expected loss across a session.
Total expected loss equals total amount wagered multiplied by the house edge, full stop.
Any system that increases your bet size after losses, or decreases it after wins, or vice versa, changes when and how much you wager, not the mathematical relationship between what you put down and what you get back.
The casino is indifferent to your bet sizing system because it collects two point seven percent of every chip regardless.
Let's start with the Fibonacci.
The sequence, as you may know from nature, art, and a thousand overconfident maths presentations, is one, one, two, three, five, eight, thirteen, twenty-one, and so on, each number the sum of the two preceding it.
Applied to roulette, you move one step forward after a loss and two steps back after a win, which means you recover two losses for every win.
The appeal is that the bet sizes grow more slowly than the Martingale.
After six consecutive losses, Fibonacci has you betting thirteen units where Martingale would have you at sixty-four.
The bust rate at the table limit is lower.
But the wins are proportionally smaller, the recovery mechanism requires winning streaks to meaningfully outnumber losing patterns, and the expected loss per unit wagered is still exactly two point seven percent.
The slower buildup is a comfort, not a solution.
The Labouchere is the most interesting of the three, and by "interesting" I mean it has the most elaborate failure mechanism.
You begin by writing a sequence of numbers: say, one, two, three, four.
The sum of your sequence is your profit target.
In this case, ten units, which at one hundred pounds per unit is a thousand-pound profit target.
Your first bet is the sum of the first and last number in the sequence: one plus four equals five units, five hundred pounds.
If you win, you cross off those two numbers and bet the new first and last: two plus three equals five units again.
Eventually you cross off all four numbers and pocket a thousand pounds.
If you lose, you add the lost bet to the end of the sequence.
The sequence grows.
The next required bet grows.
Here is the number that I think makes this concrete.
Starting with the sequence one, two, three, four at one hundred pounds per unit, if you run up forty-four consecutive losses without a single win, the sequence has grown to forty-eight entries and your next required bet approaches forty-nine units, nearly five thousand pounds.
The total you have already lost across those forty-four bets, following the mechanics precisely, is one thousand one hundred and sixty-six units: one hundred and sixteen thousand, six hundred pounds.
You are chasing a thousand-pound profit target.
That is the structure.
It is a slow, annotated version of the Martingale's catastrophe.
There was a famous attempt to reverse this.
In the summer of 1966, a British writer named Norman Leigh took thirteen people to the Casino Municipale in Nice and played the Reverse Labouchere: a positive progression that increases bets after wins and decreases them after losses, the exact opposite of the standard version.
Two weeks later, the team was banned from every casino in France.
Not for cheating.
Simply for winning, consistently, in a way the casino found unsettling.
The book Leigh wrote about it, "Thirteen Against the Bank," is a gripping read.
The subsequent analysis is less flattering.
Reverse Labouchere has no positive expectation on a negative-EV game.
Leigh was the beneficiary of an extraordinary variance run.
The French casino authority's response was precautionary, not mathematical.
The French government's mathematicians did not need to study the system to confirm this; the EV-invariance of all betting systems is established mathematics.
The system's namesake, Henry Labouchere, was a Victorian Liberal MP and the founder of the magazine "Truth," which made its reputation exposing fraud and swindlers.
His own cancellation system was later sold wholesale as an infallible method.
There is a kind of biographical symmetry in that.
The D'Alembert is named after the eighteenth-century French mathematician Jean le Rond d'Alembert, who wrote in his 1780 "Opuscules mathematiques" that the same event never happens many times in a row.
This is the gambler's fallacy, restated as a principle of nature, by a man who knew enough mathematics to know better.
The system named after him was not actually invented by him, and the gambling strategy postdates his death, but the fallacy is very much his contribution.
The method itself is simple: add one unit after each loss, subtract one unit after each win.
The our analysis ran a simulation of D'Alembert on single-zero roulette with a ten-unit winning goal.
The probability of reaching that goal: eighty-nine point eight one percent.
The ratio of losses to money bet: two point seven zero two percent.
Exactly the house edge, to four decimal places.
The system generates sessions where you win small amounts nearly ninety percent of the time.
It generates sessions where you lose steadily the remaining ten percent.
The maths is unambiguous.
The experience is deceptive.
The house does not need to cheat; it just needs you to keep coming back.
There are players at Mayfair-level tables who arrive with their Labouchere sequences written on their phones, convinced they have optimised the configuration.
They have found a more elaborate way to annotate their losses.
The casino has seen all configurations.
The house edge is indifferent to typography.
Fibonacci, Labouchere, D'Alembert: three different ways to be wrong at the same speed.
