Many if not most of the articles in *Science News* involve some math, whether as an essential tool in conducting research or a way to solve real-world problems, such as how to calculate a safer crowd size during the pandemic, detect gerrymandering in voter districts or cook the perfect steak.

But sometimes we dig into pure mathematics — math that doesn’t address an immediate practical need but is worthy of pursuit for its own sake. That includes last year’s discovery of an “einstein” tile, a long-sought two-dimensional shape that can cover an infinite surface but only with a pattern that never repeats (*SN: 4/22/23, p. 7*).

In this issue, we report on a big advance in combinatorics, which is about as pure mathy as a topic can be (we also revisit the einstein tile). The tale centers on two computer scientists. While trying to solve a seemingly unrelated problem in a distant field, the pair made a breakthrough in a puzzle that mathematicians have been wrestling with for a century.

Combinatorics is a branch of mathematics that involves the counting and arrangement of numbers or other things. An enduring question in combinatorics is whether it’s possible to predict whether an infinitely long list of numbers must include an arithmetic progression: a sequence of equally spaced numbers such as 2, 5, 8, 11, 14, 17.

On first glance, this doesn’t sound like the kind of challenge that brilliant people would devote decades of their lives to figuring out. But as freelance writer Evelyn Lamb, a mathematician herself, explains, people seem hardwired to seek out puzzles and driven to find the answers. “We humans just love going down these rabbit holes, having natural curiosity and building theories about things we see around us,” Lamb told me. “We all have things we’re super-interested in and then start diving deep.”

Arithmetic progressions have fascinated people since antiquity. Today, these sequences and other repeating patterns are part of many areas of math and computer science, providing both challenges and potential solutions. Such patterns can also provide a bit of fun. You don’t have to be a mathematician to get hooked on the plethora of sequence puzzles, whether based on numbers, letters or symbols, that are all over the internet.

I confess that I’m not someone who would put “learn more about combinatorics” at the top of my to-do list. But I was quickly drawn into the tale Lamb tells. Part of the appeal, she says, is that most everyone was introduced to numbers at a very young age, and many of us have played around with arithmetic progressions in school or in games, even if we’ve never heard the term. It’s not hard to become intrigued with something that seems so simple but can quickly become so complicated.

But if numbers can feel almost innate, they can also be intimidating. “If you feel bad at math, you feel like you’re not smart,” Lamb says. Her goal in telling the story of people’s long-held fascination with arithmetic progressions is to help us enjoy math for math’s sake. Read her article, and I think you will.