Leonardo of Pisa wrote about rabbits, not roulette. His Liber Abaci of 1202 used a breeding puzzle to introduce the sequence that now carries his nickname: 1, 1, 2, 3, 5, 8, 13, 21, where each number is the sum of the two before it. Eight centuries later, the same list is one of the most popular staking plans at the roulette table. The Fibonacci roulette system uses it to raise bets after a loss without the violent doubling of the Martingale.
It is a gentler ride. It is also exactly as expensive per dollar as every other system on our roulette strategy hub, because the wheel takes 1/37 of each stake regardless of where that stake sits in a sequence.
The rules in two lines
Choose a unit, for example C$5, and an even-money bet such as black. Start at the first number of the sequence.
- After a loss, move one step forward.
- After a win, move two steps back (or to the start, if you are within two steps of it).
The session cycle is complete when a win brings you back to the beginning of the sequence. A few players use a stricter version that jumps all the way to the start after any win. That version locks in fewer losses, and it reaches the table limit just as easily, because the dangerous part is the climb.
The sequence in Canadian dollars
With a C$5 unit, the first twelve steps look like this:
Swipe the table
| Step | Units | Stake | Total lost if every step so far lost |
|---|---|---|---|
| 1 | 1 | C$5 | C$5 |
| 2 | 1 | C$5 | C$10 |
| 3 | 2 | C$10 | C$20 |
| 4 | 3 | C$15 | C$35 |
| 5 | 5 | C$25 | C$60 |
| 6 | 8 | C$40 | C$100 |
| 7 | 13 | C$65 | C$165 |
| 8 | 21 | C$105 | C$270 |
| 9 | 34 | C$170 | C$440 |
| 10 | 55 | C$275 | C$715 |
| 11 | 89 | C$445 | C$1,160 |
| 12 | 144 | C$720 | C$1,880 |
The early steps crawl. The later ones do not: the ratio between neighbouring numbers settles near 1.618, so after about eight steps the Fibonacci is growing at more than 60% per loss. It starts slower than the Martingale’s 100% per loss, but it is still exponential growth, and exponential growth always meets a ceiling.
A worked sequence: four losses, then the climb back
Here is a realistic stretch at C$5 a unit on black. Nine spins, five losses and four wins.
Swipe the table
| Spin | Result | Stake | Session so far |
|---|---|---|---|
| 1 | Loss | C$5 | minus C$5 |
| 2 | Loss | C$5 | minus C$10 |
| 3 | Loss | C$10 | minus C$20 |
| 4 | Loss | C$15 | minus C$35 |
| 5 | Win | C$25 | minus C$10 |
| 6 | Loss | C$10 | minus C$20 |
| 7 | Win | C$15 | minus C$5 |
| 8 | Win | C$5 | level |
| 9 | Win | C$5 | plus C$5 |
Notice spin five. The C$25 win was the biggest bet of the run, and it still left you C$10 short, because stepping back two places only covers the last two losses. Getting fully clear took three more wins. This is the main difference from the Martingale, where any single win clears the run: the Fibonacci asks for a run of wins to repair a run of losses.
A neat property falls out of the rules. If your results follow the pattern loss, loss, win over and over, you stay exactly level forever. So the Fibonacci treads water if you win one spin in three and edges ahead if you win more. An even-money bet wins 18 times in 37, well above one in three, so many short Fibonacci sessions end slightly up. The losses come from clusters, they are large enough that the average session still loses 2.70% of what it stakes, and the more spins you play, the more room a cluster has to arrive.
Where the C$500 limit stops it
Assume a table running from C$5 to C$500. The eleventh step is C$445, which is allowed. The twelfth is C$720, which is not. So eleven losses in a row end the sequence with C$1,160 gone. On a European wheel, eleven straight losses on an even-money bet have a probability of (19/37)11, about 0.065%, or 1 in 1,527 from a standing start.
That looks remote, but straight losing runs are not the only way up the ladder. Because a win only moves you back two steps while a loss moves you one forward, a choppy stretch with losses outnumbering wins two to one climbs steadily. Following the step number spin by spin gives the chance of being pushed to the C$720 step at some point:
Swipe the table
| Even-money spins played | Chance of reaching the C$720 step |
|---|---|
| 100 | 9.4% |
| 200 | 19.3% |
| 500 | 42.9% |
| 1,000 | 67.9% |
Compare that with the Martingale on the same table, where the chance of hitting the wall within 100 spins is about 36%. The Fibonacci postpones the problem. Over enough spins, it still arrives, and the bill is bigger when it does: C$1,160 against the Martingale’s C$635.
A bigger unit, the same ceiling
The unit you pick moves the wall closer. At C$10 a unit on the same C$5 to C$500 table, the sequence runs C$10, C$10, C$20, C$30, C$50, C$80, C$130, C$210, C$340, and the tenth step would be C$550. Nine straight losses now end it, with C$880 gone, and nine losses in a row have a probability of about 0.25%, or 1 in 403. Counting the choppy routes up the ladder as well, the chance of being pushed to the C$550 step is about 23% over 100 spins, 43% over 200 and 76% over 500. Doubling the unit roughly doubles the chance of hitting the limit in an ordinary session, while each completed cycle only earns C$10 instead of C$5.
Why the edge stays at 2.70%
Every bet in the sequence is one even-money wager, and on average it gives up 1/37 of its size. The step number decides how big the stake is, and nothing else. So the expected result of any Fibonacci session is simply the total staked multiplied by minus 2.70%.
The worked sequence above put C$95 through the wheel over nine spins (5 + 5 + 10 + 15 + 25 + 10 + 15 + 5 + 5). Its expected cost was C$95 ÷ 37, about C$2.57. That particular sequence happened to finish C$5 up; a sequence with one fewer win would have finished behind. Across thousands of sessions, the small wins and the occasional C$1,160 hit average out to that same 2.70% of everything staked. Type any of the stakes into the roulette odds calculator and you will see the per-bet figure directly.
Fibonacci on other bets
Some players run the sequence on dozens or columns, which pay 2 to 1. A single win high up the ladder then covers more of the run, since it returns twice the stake. The cost is frequency: a dozen loses on 25 of the 37 pockets, so losing runs are longer and the ladder climbs faster. The trade is roughly even, and the price is unchanged at 2.70%.
Inside bets make no sense with this system. A straight-up number loses 36 times in 37, so a Fibonacci on a single number reaches C$720 stakes with depressing speed.
Choosing a unit and a stopping point
If you want to play the Fibonacci, start from the budget, never from a profit target. Suppose you set aside C$200 for an evening. With a C$5 unit, six losses in a row already cost C$100 and the seventh stake is C$65, so the budget is spent within one ordinary bad patch. With a C$2 unit, the same six losses cost C$40, and you get far more spins for the same money.
Set a time limit too, and stop at whichever limit comes first. Do not reset the sequence with a bigger unit after a bad run; that turns a contained loss into a larger one. The bankroll and session limits guide works through budgets in detail, and our responsible gambling page lists the limit tools and support services in each province.
One more practical point. Single-zero tables are easy to find at the online casinos we compare on the roulette casinos homepage, and they cost half as much per spin as American tables. That does more for a Fibonacci player than any choice of unit.
What the sequence is good for
A short aside before you go. Fibonacci numbers turn up in some unexpected places: the spiral of seeds in a sunflower head, the branching of some plants, the ratio 1.618 that architects have argued about for centuries. Plenty of roulette players seem to feel that a sequence with that pedigree must carry some hidden order into the game. It does not. The wheel is the one place where the sequence is purely a bookkeeping device, a tidy way of writing down how much to bet next. Used that way, as a pacing tool with a firm ceiling and money you have already decided to spend, it is a perfectly pleasant way to structure a session.