The Power of Pattern Recognition
The human brain has a remarkable ability to recognize patterns. From ancient Polynesians navigating by the stars to modern-day individuals identifying shapes and symbols, humans have proven to be highly skilled at recognizing and utilizing patterns.
However, mathematicians are now exploring the concept of collections without any discernible patterns. This begs the question – how large can collections be before a recognizable pattern must emerge? The answer to this question has significant real-world implications, such as understanding the minimum number of server failures that could cause the internet to crash.
The Quest for Patternless Collections
Recently, University of Wisconsin mathematician Jordan Ellenberg and researchers from Google's Deep Mind have proposed a novel approach to this problem. Their research involves using artificial intelligence to identify large collections without a specified pattern. This can provide valuable insights into worst-case scenarios.
The concept of patternless collections can be illustrated through a popular card game called Set. In this game, players lay out 12 cards with varying colors, numbers, shapes, and shadings. The objective is to identify sets of three cards where each feature is either the same or different on each card. While finding a set is usually possible, it is not always the case and players must continue flipping over cards until a set is found.
But what is the maximum number of cards that can be laid out without forming a set? In 1971, mathematician Giuseppe Pellegrino determined that the largest collection of cards without a set is 20. However, the odds of randomly choosing 20 cards without a set are incredibly low. In fact, this would only occur around one in a trillion times. Additionally, finding these "no set" collections is a highly complex problem.
To identify the smallest collection of cards without a set, researchers could theoretically conduct an exhaustive search of every possible combination using the deck of 81 cards. However, this is an incredibly daunting task, with an estimated 1024 possibilities. Furthermore, the complexity increases if the number of features on the cards is increased.
The Role of AI in Pattern Recognition
Despite these challenges, mathematicians enjoy tackling computationally difficult problems. By approaching these problems in the right manner, they can become more manageable.
While it is easier to identify best-case scenarios, such as the fewest number of cards needed for a set, there are few known strategies for exploring worst-case scenarios – large collections without a set. This is where Ellenberg and his team have utilized large language models (LLMs), a type of AI, to generate and analyze massive collections without a recognizable pattern.
By feeding the LLMs with computer programs that generate set-free collections, the researchers were able to identify larger collections without a set. This method allows for the exploration of disordered collections, providing valuable insights into potential worst-case scenarios.
For example, this approach could be applied to understand the vulnerability of the electrical grid to a malicious attacker. How many substations would need to be destroyed for the grid to fail? By identifying a large set-free collection of substations, researchers can determine the minimum number that could potentially cause a catastrophic failure.
Ellenberg and his team's work exemplifies the power of AI as a problem-solving tool. However, for now, it still requires human ingenuity to guide it in solving complex problems.