Humans are very good at spotting patterns, or repeating features people can recognize. For instance, ancient Polynesians navigated across the Pacific by recognizing many patterns, from the stars’ constellations to more subtle ones such as the directions and sizes of ocean swells.
Very recently, mathematicians like me have started to study large collections of objects that have no patterns of a particular sort. How large can collections be before a specified pattern has to appear somewhere in the collection? Understanding such scenarios can have significant real-world implications: For example, what’s the smallest number of server failures that would lead to the severing of the internet?
Research from mathematician Jordan Ellenberg at the University of Wisconsin and researchers at Google’s Deep Mind have proposed a novel approach to this problem. Their work uses artificial intelligence to find large collections that don’t contain a specified pattern, which can help us understand some worst-case scenarios.
Patterns in the card game Set
The idea of patternless collections can be illustrated by a popular card game called Set. In this game, players lay out 12 cards, face up. Each card has a different simple picture on it. They vary in terms of number, color, shape and shading. Each of these four features can have one of three values.