| What kind of complex behavior can arise from simple rules? This question crops up again and again throughout math and science. It also arises in a more mundane context that's near and dear to my heart: board game design.
Board games vary greatly in complexity, from kids' favorites like Clue to strategy games that can last 10 hours or even longer. But many of the most iconic and enduring games strike a delicate balance between the simple and the complex. That's especially true of "combinatorial" games like chess and Go, in which there's no chance or secrecy involved. You can grasp the rules of these games in a few minutes, yet they're so rich that you can play for years and still have more to learn.
For some mathematicians, games aren't just a pastime — they're inextricable from research. As my colleague Jordana Cepelewicz discussed in the April 15, 2024, edition of Fundamentals, the line between silly and serious mathematics is often porous. Sometimes games inspire mathematical discoveries, and sometimes it's the other way around. Computer scientists, too, have long been fascinated by combinatorial games. Theoretical computer scientists have devised ways to quantify the difficulty of these games and relate them to more traditional computational problems. And artificial intelligence researchers have used games as testing grounds for powerful problem-solving systems. What's New and Noteworthy
Mathematicians who study board games often take a playful approach to their other research — indeed, many math problems are like games if you think about them the right way. A classic example is John Conway's Game of Life, in which a pattern of black and white cells on a grid evolves over time according to three simple rules, and different starting patterns produce strikingly different behavior. This unusual game has captured the imagination of generations of professional and amateur mathematicians, and researchers continue to make new discoveries more than 50 years after it was first invented.
Conway called the Game of Life a "no-player never-ending game" — after you specify an initial configuration, there's nothing to do except sit back and see what happens. But many of Conway's other mathematical discoveries were prompted by games you can actually play. In a 2015 biography of Conway, Siobhan Roberts recounts how he spent much of his professional life "playing games, inventing games and reinventing rules to games he found boring." Quanta published an excerpt from Roberts' book, and it's a great read, vividly capturing the creative spirit that animated all of Conway's work.
Games were a persistent theme throughout Conway's career. The computer scientist Shang-Hua Teng came to the subject more recently, after one of his students invented a combinatorial game based on a mathematical theorem and analyzed it using the framework of a discipline called computational complexity, which studies the intrinsic difficulty of different problems. Quanta published a delightful and wide-ranging conversation I had with Teng in 2023: We talked about how computational complexity can shed light on what makes a good board game, the importance of "losing strategically," and how Teng explained his games research to his father, among other topics.
Combinatorial games often involve lots of strategic decision-making, and that's made them popular targets of AI research for decades. AI systems surpassed the world's best human chess players in the 1990s, but it wasn't until 2016 that an AI system finally beat a Go world champion. That system, dubbed AlphaGo by its creators at Google DeepMind, was qualitatively different from its chess-playing precursors, because it hadn't been programmed with any knowledge of Go strategy. Instead, it learned to play by spotting patterns in games between human experts, and mastered the game by playing against itself. Soon after AlphaGo's victory, the computer scientist Michael Nielsen argued in a Quanta essay that this learning-based approach was what made it a major AI milestone. Later versions of AlphaGo went on to master many other games starting from scratch.
Of course, real life is a lot more complicated than even the most intricate game. In 2018, Josh Sokol wrote about the challenges facing researchers trying to solve real-world problems using AI systems modeled after AlphaGo — the technology has advanced a lot since then, but many of the same challenges persist today. AlphaGo and its relatives learn through a process called reinforcement learning, which rewards them when their moves lead to positive outcomes. But these reward signals often aren't available in the real world. Some researchers have explored an alternative approach where AI systems are rewarded for curiosity, and games have proved useful there too. In 2017, John Pavlus wrote a fascinating story about curiosity-driven AI that learned to play the 1985 Nintendo game Super Mario Bros.
From the Game of Life to Go to Mario, games have been a fascinating object of study, providing inspiration for mathematicians and computer scientists alike. Don't underestimate the power of play. | 
