Van der Waerden's theorem

Van der Waerden's Theorem: Conflict Between Necessity and Predictability

With remarkable cyclicality, the idea of the unpredictability of chance is frequently challenged, often using mathematical tools that, unfortunately, cannot alter the intrinsic nature of the phenomenon. This occurs when mathematics is leveraged to gain strategic advantages in gambling games like roulette, giving rise to an illusion that persists over time despite scientific evidence and the laws of probability.

The notion that mathematics can predict the outcomes of random events, such as the spin of a roulette ball, is not only unfounded but also one of the most common fallacies. If this were possible, it would have been discovered centuries ago by mathematicians like Blaise Pascal or Pierre de Fermat, who laid the foundations of probability theory in the 17th century. Yet none of their discoveries ever suggested that the outcomes of random events could be predicted with certainty. This concept clashes with the reality of randomness, a principle that remains unchanged despite advances in computational and theoretical capabilities.

Mathematics and the Illusion of Predictability

Many attempts to apply mathematics to gambling are based on a conceptual error: confusing mathematical necessity with practical predictability. Mathematics, particularly in the context of probability games, describes trends and phenomena on large scales but does not provide tools to predict individual outcomes. The common error lies in believing that random events can be "compensated" or "corrected" based on past patterns, ignoring the fact that each event is independent.

The laws of probability indicate that certain mathematical structures emerge in very large sets. However, these laws do not establish any certainty for individual events. This concept is often misunderstood, leading to a false belief that phenomena like the delay of a particular sequence increase the probability of a specific result in subsequent spins.

Van der Waerden's Theorem and Roulette

Van der Waerden's Theorem, a fundamental result in combinatorial theory, is sometimes invoked to support the idea that there are predictable patterns in roulette. This theorem states that by dividing a set of numbers into a finite number of groups (e.g., two colors, red and black), a uniform arithmetic progression of a predetermined length $k$ will inevitably form, provided the set is sufficiently large.

Applying this theorem to roulette involves representing outcomes as sequences of colors. For example, we can consider $k=3$, meaning an arithmetic progression of three consecutive numbers of the same color, and demonstrate that, after a minimum number of spins $N$, such a progression will inevitably emerge. However, the error lies here: the theorem only guarantees the inevitability of such progressions on sufficiently large scales, not the predictability of individual events.

A Practical Example

Consider $k=3$, $c=2$ (two colors: red and black), and $N=9$. The theorem guarantees that, over nine consecutive spins, at least one arithmetic progression of three consecutive numbers of the same color will form. Here are some possible progressions:

• Step $d=1$: $(1,2,3),(2,3,4),(3,4,5)$, and so on.
• Step $d=2$: $(1,3,5),(2,4,6),(3,5,7)$, and so on.
• Step $d=3$: $(1,4,7),(2,5,8),(3,6,9)$.

The theorem ensures that at least one of these progressions will consist of numbers of the same color. However, it provides no information about which specific progression will emerge or the order in which the colors will appear. This is a mathematical guarantee, not a prediction useful for practical purposes.

Independence of Events in Roulette

A crucial point differentiating roulette from purely mathematical phenomena is the independence of events. Each spin of the roulette wheel is an isolated event: the outcome is not influenced by previous or subsequent spins. This means that, even if a uniform triplet has not occurred in the first eight spins, the probability of it occurring on the ninth spin remains unchanged.

The Gambler's Fallacy

Many players fall victim to the gambler’s fallacy, believing that a particular sequence is "due" after a certain number of spins. This belief has no mathematical basis. The probability of red or black remains constant (approximately 48.65% in European roulette), regardless of prior results.

Conclusion: The Difference Between Inevitability and Predictability

Van der Waerden's Theorem demonstrates the inevitability of certain structures in organized sets. However, confusing this inevitability with the predictability of events is a fundamental error.

Despite the allure of mathematical theories and their potential applicability in other fields, it is crucial to recognize the intrinsic limitations of mathematics in the context of gambling. The randomness of roulette, with its independent and unpredictable nature, remains an insurmountable challenge for any deterministic approach.