The Fibonacci & Whittaker Roulette System

Leonardo of Pisano was one of the most significant European mathematicians of the last millennium. He's probably better known today under the name Fibonacci, a shortened form of Filius Bonacci - a reference to his being the son of the wealthy merchant, Guglielmo Bonaccio. As a boy, he had accompanied his father on business trips to the Islamic North African sultanate that we now know as Algeria. Here, Pisano became highly skilled at using Hindu-Arabic numerals. Writing down numbers with these was a far more efficient process than when using the long-winded Roman numerals then popular across Europe. Pisano would become an esteemed mathematician, and his widely-read 1202 book, Liber Abaci (Book of Calculation), would be largely responsible for convincing Europe to abandon Roman numerals in favour of their Hindu-Arabic counterparts.

Fibonacci Numbers

In one chapter of Liber Abaci, Pisano tackled the theoretical problem of how quickly rabbits would breed over the course of a year, assuming that no rabbits died, and that each female would produce a pair of baby rabbits every month. Pisano used a sequence of numbers to solve this problem. Calculating the sequence - which he didn't invent himself, but which is today named after him - involved taking the last two numbers, and adding them together in order to create a whole new number. So the sequence would go:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc.

The sequence only needed to go this far in order to answer Pisano's problem, with the result that 233 pairs would have been created by the end of the 12th month - this requires 13 numbers in the sequence.

The problem may be rather theoretical, but this number sequence has been noted numerous times in nature. From calculating the way a snail shell develops, or the number of petals in flowers, or the number of seeds in a seed head, Fibonacci numbers like 34, 55 and 89, come up again and again. And from 89 onwards, if you divide each number in the sequence by the previous one, the answer will be 1.618. This figure, known as the Golden Ratio, is considered very important, and comes up again and again in nature, in building architecture, in musical theory etc.

All in all, then, the Fibonacci sequence is accorded considerable mystical power. Even today, financial market practitioners pay great attention to Fibonacci numbers. If a share price gives up 34% of its recent gains, for instance, this is often seen as an important point at which the price may change direction and start going back up again. The 21% and 55% retracements are also watched with considerable interest. It's probably through its regular use by market traders that the Fibonacci sequence has been applied to betting strategies in Roulette.

How the Fibonacci System Works

As with most systems, it's assumed that you'll be betting repeatedly on a 50/50 selection - such as Red/Black or Odd/Even. We'll also assume you start off with £1 chips.

This system involves betting using the Fibonacci sequence. For reference purposes, the first 20 numbers are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

When you start a new series in this system, you always begin on the second number from the left (that's the second '1'). You then place a £1 bet. If the bet loses, you simply move one place to the right (in this case, to the '2'). If the bet wins, you move two places to the left. (In this case, you would have now successfully completed the series.)

The amount that you bet each time will be the sum of the two numbers to your left in the sequence:

If you're on the 2, you would bet £2 (1 + 1).
If you're on the 21, you would bet £21 (8 + 13)
If you're on 377, you would bet £377 (144 + 233) And so on.

In practice, you don't need to do the addition each time. Just keep a note of which number you're on in the sequence, and bet that amount. As long as you always move one place to the right when you lose, or two places to the left when you win, you'll always be able to keep on top of the betting.

The objective is to keep moving to the left until you land on the very first number - the first '1'. At this point, you will have made a £1 profit on the series, and can start a brand new series - once more starting from the second number on the left.

Writing Out the Sequence by Hand

Some players prefer to write out the sequence by hand, and work out the bets as they go along. That's certainly easier to illustrate here, although the mechanics are slightly different. With this approach, you start off by writing down a 1 as the first number in your sequence. This will also be the amount (£1) that you place as your first bet. If you win, you've made a £1 profit on the series and can start again. But if you lose, you write the bet size (1) down as the second number in your sequence.

From here on, you always add together the last two numbers of the sequence to give you your bet amount. If you lose, you write down that amount as the next number in your sequence. If you win, you erase the last two numbers (which, of course, will add up to the amount you've just won). You carry on going until you have struck out all of the numbers in the sequence. Remember that all of the numbers (including the first '1') must be eliminated before you have completed the series and made your £1 profit. Here's an example of a game:

Starting sequence: 1

Bet £1. Lose. New sequence: 1 1
Bet £2. Lose. New sequence: 1 1 2
Bet £3. Lose. New sequence: 1 1 2 3
Bet £5. Lose. New sequence: 1 1 2 3 5
Bet £8. Lose. New sequence: 1 1 2 3 5 8
Bet £13. Win. New sequence: 1 1 2 3
Bet £5. Lose. New sequence: 1 1 2 3 5
Bet £8. Win. New sequence: 1 1 2
Bet £3. Win. New sequence: 1
Bet £1. Win. Sequence ended

Whether you prefer to write down the figures as you go, and gradually strike them off, or choose instead to write down the sequence one time only, and then use some sort of weighty token to keep track of where you are, is up to you. We would recommend the latter approach, though. It'll save on paper and unnecessary writing, and since your bet will always be identical to the square you're currently on, it'll be easy to work out how much to place each time.

Natural Miracle or Home-Grown Fiasco?

So does the system work? Our first 100,000 spin session produced a total profit of over £28,000, a feat that was repeated numerous times in subsequent tests. An obvious winner, then, and one that would allow us to mount up massive profits? Well, not entirely. The problem is that this intoxicating success was mixed up with some gargantuan bet sizes. Indeed, at times we were required to chuck in a chip worth over £300,000! Good luck finding a casino that will allow you to start with £1 chips, and end with ones worth over £300,000. As with Martingale, the system works just as long as you do it on paper, with an unlimited bank and unlimited table limits. For practical play, though, this brand simply doesn't work.

What could we do to take the edge off it? The first thing we wanted to try and sort was that maximum bet. If you're starting at £1, the highest many casinos will allow you to go to is £100 per chip. Within the Fibonacci sequence, the closest number to 100 is 89. So what would happen if we got the system to reset itself after a losing bet of £89? That would involve a line that was 11 numbers long. What actually did happen is that we lost. The figures generally weren't five figures over 100,000 spins, but we were still losing £8K or £9K on average. We increased the reset bet to 13 (a losing bet of £233), and the results now became more volatile. Some of the time we lost as little as 5K. Other times, we threw away in excess of £23K. The final result was another big fat minus, though.

What if we decreased the reset figure instead of increasing it? Well, the lower we made the figure, the smaller the losses became. In truth, though, we were never able to beat the basic flat betting odds - which would imply a loss of £2,700. With a reset figure of 2-number lines, we were sometimes beating that £2,700, but this would frequently be offset by a loss of £3,000 or £3,200, leaving us in the red after several big sessions.

Whittacker Variation

The names Fibonacci and Whittacker are often used for similar systems. The essential difference in Whittacker is that you have to lose twice (so two £1 bets in our normal example) before you start adding together the previous two bets. Without any bet limit, the system worked well, regularly earning £17K or more. It didn't have the high bet sizes of the Fibonacci system, but did still get up to some five figure bets - in other words, totally impractical for betting with.

So could it succeed if we tried to limit the maximum bet? Getting the system to reset after a losing bet of £89 (a line 11 numbers long) meant that we polled very similar figures to Fibonacci. Sometimes we were a little better (£7K to £8k), and sometimes a little worse (we had a few £10K-£11K sessions). With larger bet limits, the results again became more volatile, but overall were losing even more money than the 11-number lines. With lower limits, we managed to get very close to the results of flat betting, but we never beat that style of betting.

Let Nature Take the Strain

It seems this system needs the flexibility of the large bet sizes, and that is quite clearly a massive issue. You might be able to find a casino that will allow you to go from £1 to £144 or £233 chips, but chips worth several tens of thousands?? Well, that's not going to happen, even if you have lots of money. Like the Martingale system, Fibonacci works on paper but not in reality. Whittacker makes it marginally less volatile, but you're only delaying the process slightly. You're not compensating for the basic flaw in the system - without unlimited bet sizes, you simply can't make it work all of the time.

Having said that, with the maximum bet limit in place, its losses aren't horrendous. Neither are the logistics of working out the next bet as fiddly as they are for a system like Labouchere. The reason is that the range of possible numbers is fairly limited - you're simply moving up and down the Fibonacci sequence. And there is a certain elegance to working your way up and down the line. Overall, though, Fibonacci and Whittacker offer you little you won't get from a flat betting system. Fantasy players will find it remarkable. The rest of us, though, will see yet another loser.

Pros

Fairly straightforward if you create the sequence in advance and use a token to show you where you are
Losses not excessive once you've placed a ceiling on the bet size

Cons

Needs unlimited bet sizes (and banks!) to be viable.
Simply doesn't work in the long-term