## Variation on Thomas Donald – The Donald-Natanson System

Several years ago one of the authors of this article, a professional mathematician, made a critical change to the Thomas Donald System. He argued the following:

Assume one always bet on red and the initial bet is €1. After black turns up he increases his bet by a unit, and after red turns up he reduces his bet by a unit. But what should he do, if he have bet €1 on red and have won? According to T. Donald, the bet should remain invariable since there is no such thing as a zero-sum bet or a negative bet. “But why?” asked the mathematician. And when he analyzed the problem, he came to an interesting conclusion.

The literal application of his roulette system, of course, is impossible. If a player bets €1, the next bet should be equal to zero. According to Natanson, the zero bet is simple: the player passes during the next spin of the wheel. Nonetheless, the player plays as if he had bet on red. And the player must be alert to the results of the round, so as to know how to bet next time. Assume that the ball falls on red again. The player has won and now should reduce the bet again. The following bet (according to the roulette system) should be equal to -1.

And what is the negative bet on red? It is the bet on black! Therefore, whatever happens, there is only one rule:

- when black is fallen on, the bet increases, and when red is, the bet decreases.

Imagine, for example, that at the first three starts of roulette red turns up every time. After the first round we have won €1, we pass on the second round, and bet -€1 the third turn by placing a euro on black.

Before the fourth spin, we decrease the bet to -€2. We put €2 on black.

Therefore, it can be proven that from 2N starts of roulette red and black turn up N times the winnings will equal N initial units. Irrespective of number of times red is landed upon (and accordingly, black), the “property of invariance” holds true: the sequence in which red and black alternate does not influence the size of the win.

Let us assume that the roulette is started 36 times. Your income (positive or negative) is shown in the table.

Numbers of Times Red Wins |
Income |

14 | -22 |

15 | -6 |

16 | +6 |

17 | +14 |

18 | +18 |

19 | +18 |

20 | +14 |

21 | +6 |

22 | -6 |

23 | -22 |

For example, if red is landed upon 20 times then with an initial bet of €1, player wins €14. If red is landed upon 17 times, the player also wins €14. Interestingly, the income distribution is symmetrical in the middle of the table.

The above table demonstrates that what happens when the frequencies of red and black differ is insignificant (with other types of roulette system, players would lose). Donald counted on the fact that over time the frequencies would be roughly similar. Natanson followed in his footsteps, but intensified the system.

Finally, we can’t forget zero.

According to Donald, the next bet should be risen when zero is landed on. In Natanson’s modification, it should be raised modularly. In other words, if the bet is positive, it should be raised by one and if it is negative it should be lowered by one. Unfortunately, the appearance of zero breaks the beautiful “property of invariance,” and makes it impossible to determine one’s income. However, consider what happens when zero turns up only once in 36 spins.

First, assume zero is fallen upon when the bet was positive. In this case, the zero is the equivalent to black, and therefore the income is defined according to the same table as above. For example, when red is fallen upon 20 times, black 15 times, and zero once, the winnings are €14. However, this doesn’t mean that zero does not have any influence: it reduces the expected number red wins.

Now, assume zero is fallen upon when the bet is negative. Now it is equivalent to red. If red is fallen upon 20 times, then because of zero the number of its occurrences equals 21. Instead of €14 (according to the table) we only win €6. But if red was fallen upon less than 18 times, our income would increase.

Finally, assume zero is fallen upon when there’s no bet. We can do whatever we like: with rise of the bet zero will be equivalent to black, with a reduction, it’ll be equivalent to red. However don’t forget the background: if red is fallen upon more often than black, it is necessary to raise the bet, and visa versa. The more frequently both colors turn up, the better. Mr. Donald would be pleased.

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