To Kai Ang (12), being “good” at math doesn’t necessarily mean finishing problems first. His level of expertise lies not in rote calculations or speed competitions, but in the most complex problems that challenge him even without a time limit.

“I’m very bad at being super fast and being a human calculator,” he said. “I’m much better at thinking about problems and making connections between things and explaining how I got to where I got. That’s why I prefer the more open-ended [competitions].”

Beyond the various competitions, Kai does college-level math research. At Stanford University Mathematics Camp (SUMAC) the summer of his sophomore year, he compiled a proof of Polya’s Recurrence theorem using parts of papers and his own logic.

“We’re going to do some random walks in several dimensions,” he said in his description of the problem. “The first dimension is just a number line, and we start at the place zero. We’re going to take a number of steps, infinitely many, and for each step you either go right one or left one, and that’s random. Polya’s recurrence theorem says that we’re going to return to where we started eventually. Then you do this for two dimensions, like on a coordinate grid…  and you’re still going to return to where you started.”

But at three dimensions and higher, these one-step walks will not necessarily return to the starting point. While the logic may seem intuitive—more dimensions equals more ways to stray from the origin—proving it requires combinatorics and probability, generating functions, inequalities, calculus and asymptotic approximations.

“In the math that’s beyond [the high-school level], there are a whole different bunch of areas of math, and they’re all interconnected in weird ways,” Kai said. “It’s endless.”

After he solves a problem, Kai doesn’t keep it to himself. As president of Math Club, he found an outlet for sharing his passion for math with students of all skill levels.

“[Kai’s] set up a really elaborate system of classes where anyone who’s interested in contest-style math problems can just drop in and learn new things every week,” Math Club officer Jimmy Lin (11) said. “He’s certainly opened it up to people who are less experienced in math competitions but are willing to learn. Whenever he’s teaching someone, he genuinely cares about their understanding and he wants to make sure that they see a topic at the same level of depth that he does.”

Kai describes himself as a truth-seeker, constantly looking to build connections between the truths he’s already established.

“Solving problems is a struggle, but it’s really rewarding when you put the stuff together. It seems like magic sometimes,” Kai said. “Once I learn something really cool, I’ll write it in a handout. I guess the point of writing a paper and teaching would be to just share what I know and share my excitement.”