### Fibonacci and the Whittaker Progression

#### What the Fibonacci is about

The Fibonacci sequence is named after Italian mathematician Leonardo Bonacci (1170 - 1250) also known as Fibonacci. His 1202 book Liber Abaci (Book of Calculation) introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. The Fibonacci numbers are or Fibonacci sequence is shown in the following integer sequence: 1, 1, 2, 3, 5, 8, 21, 34, 55, 89, 144, …

By definition, the first two numbers in the Fibonacci sequence are 1 and 1. Each subsequent number is the sum of the previous two.

#### How the Whittaker Progression works

The Whittaker (Whittacker) Progression is a betting strategy developed for the use with even money chances as the High/Low, Odd/Even and Red/Black bets in roulette. It uses the Fibonacci method to determine the amount of units wagered and is related to the Labouchère method in terms of how sequence is played. When you lose, the sum of your last two unsuccessful wagers equals the amount of the next bet, and you write off this and the previous number when you win.

The pattern of bets differs in some descriptions found on the internet from the Fibonacci according the first two numbers in the sequence.
Fibonacci: 1, 1, 2, 3, 5, 8, …
Whittaker: 1, 2, 3, 5, 8, …

We recommend to use the original Fibonacci sequence for the Whittaker progression. While a pure Fibonacci betting progression can stretch to table limits, the Whittaker usually requires stopping at a predetermined point in the sequence like the Labouchère. You decide before starting your game about the exact sequence you will use – so if you only want a short and low-risk set, you might want to play 1, 2, 3, 5, 8.

#### How you play the Whittaker

First you'll need to set a goal for how much money you are willing to lose or drop down in the worst case. It's usually simplest to express this in a certain number of 'units' – where the unit value can be any amount you like. For instance, you could limit your losses to 19 units with a maximum bet amount of 8 units. You can set the unit value to whatever the table limit permits (8 x unit value).

Once you know how many units you want to risk at maximum, you can start your game wagering 1 unit. If you win your 1 unit bet, you just stay with 1 unit for the next bet. If you lose a 1 unit bet you write down the lost unit and continue wagering 1 unit. If you lose this 1 unit too you write down that loss too. Your line would now look like this:

1 | 1

For your next bet sum up your last two lost wagers which would be 2 units (1 + 1 = 2 units). For the sake of this example, let's say you lose again. Now, the lost bet amount goes to the end of your line, making it look like this:

1 | 1 | 2

Your next bet would now be for 3 units (1 + 2 = 3 units). Let's say you lose again, write down the lost amount and your new line looks like this:

1 | 1 | 2 | 3

Your next bet would now be for 5 units (2 + 3 = 5 units) not yet reaching your set loss limit. Let's say you win write down your bet amount and cross off the two most right numbers in your line. Your line would now look like this:

1 | 1 | 2 | X | X

Your next bet would now be for 2 units as it is the most right number in your sequence. Let's say you win again and cross off two bet amounts again. Your line would now look like this:

1 | X | X | X | X

Your next bet would be 1 unit. You win again, cross off the last number in the line and complete a game. You lost four bets and won three bets and gained a net profit of 1 unit. You continue wagering 1 unit and if you lose you have to step up the progression ladder again.

Compared to the Martingale system, Whittacker betting strategy is less risky in what concerns the player's bankroll and requires a smaller bankroll. The benefit in using the Whittacker strategy over the standard Fibonacci is that limiting the length of the progression means you can only lose a certain amount – especially if you decide to use the highest number as a stop loss in case of a bad streak. The Whittaker progression should only be used with a good betting selection.

The disadvantage of the Whittaker Progression is the fact that the bet amounts can grow rapidly with several consecutive losses. A method to slow down the rapid rise of the bet size is to divide the sum of the last 2 lost bets by 2 and continue accordingly.

#### RNG and Live Dealer Tests of the Whittaker Progression for 32000 bets

For demonstration purpose we have tested a simple roulette system. The rules are just to bet on the previously hit chance of High/Low using the Whittaker strategy with a stop after step 10 of the progression which determines the highest bet amount with 89 units. The first test run shows the results of the progression on a No Zero RNG roulette wheel to demonstrate how it work for real 50:50 chance games. The almost equal distribution of Won/Lost bets with 15709/16291 produces a loss of 972 units seeing a maximum drawdown to -1621 units. The average game length is 3.3 bets and the longest game took 76 bets before either completing the sequence or reaching the stop loss at step 10 of the progression.

The results for single zero roulette wheels are shown in the other two tests. As you can see in the charts causes the zero higher losses of 4058 units in the RNG and 5496 units in the Live Dealer test run. The longest observed losing streaks are 12 spins long and 16 spins long respectively. The RNG test showed 15353 won and 16647 lost bets. The live wheel showed 15323 won and 16677 lost bets. The average game length is 3.5 bets for both and the longest games took 65 (RNG) and 71 (Live Dealer) bets before either completing the sequence or reaching the stop loss at step 10 of the progression.

#### RNG and Live Dealer Tests of the Fibonacci Progression for 32000 bets

In this test we have used the same simple roulette system as before. The first test run shows the results of the progression on a No Zero RNG roulette wheel. The almost equal distribution of Won/Lost bets with 15624/16376 produces a profit of 5547 units seeing a maximum drowdown to -45335 units. The highest bet amount is 28657 units.

The results for single zero roulette wheels are shown in the other two tests. As you can see in the charts most of the time the bet amounts stay relatively moderate but at times they increase up to a maximum of 28657 units in the RNG and 17711 units in the Live Dealer test run. The longest observed losing streaks are 16 spins long and 12 spins long respectively. Due to zero the RNG test showed 15520 won and 16480 lost bets. The live wheel showed 15364 won and 16636 lost bets.