Most of the important discoveries in mathematics take place after decades or centuries of effort. If you want to attack the biggest problems, you’ll need to master a lot of highly technical material before you can even begin to say something new.

Such questions don’t interest Richard Schwartz. He likes problems he can read about today and start solving tomorrow — simple problems, fun problems, problems that have the aspect of a carnival game: Step right up and see what you can do with this one! It’s an unusual disposition among research mathematicians. Schwartz embraces it completely. “I don’t think I have a mature attitude towards math,” he said.

Yet none of this is to say that Schwartz is anything but a serious and accomplished mathematician. He is. He received his doctorate at Princeton University under the mentorship of Bill Thurston, one of the most important mathematicians of the last half-century. He is now a tenured professor at Brown University whose most important work has taken place in the field of dynamics, which studies the long-run behavior of iterative processes, like a billiard ball ricocheting on a frictionless table. In 2008 he proved that every triangle with angles all less than 100 degrees contains at least one periodic billiard path — a repeating path that a ball will trace and retrace forever.

Schwartz uses computer experiments in much of his work — he’s on the vanguard in that respect. As he explains, computers complement human mathematical thought in several ways: They draw out patterns that provide hints which lead to proofs that might not have been apparent to the mind alone.

Quanta Magazine spoke with Schwartz about his taste for simple problems, what he called the “miracles” of mathematics, and his upcoming math book for kids about infinity. An edited and condensed version of those conversations follows.

What do you like about mathematics?

I like all kinds of things about it. The first thing is I like the fact that it works, somehow. I like the fact that it’s procedural, and there’s a method to it so you can make progress. I like that you can get to the bottom of questions, unlike, say, politics or religion where you can just talk for years with people and no one will change the other’s opinion.

I also just like shapes and numbers. I’ve always had a love for these kinds of things for some primitive reason I can’t quite explain. Then, I like the intellectual challenge. I like solving problems, trying to solve problems that people can’t solve. There’s kind of a mountain-climbing aspect to it. Finally, I like the beauty of pure mathematics, much in the same way someone might like a work of art.

You said you like simple problems. Why?

I feel if it’s a simple problem that hasn’t been solved, it probably has some kind of hidden depth to it. In other words, there’s something missing in human knowledge that prevents people from solving the problem.

The second thing is that I like doing computer experiments, and so I feel that sometimes I have a chance of making progress. The modern computer is a new tool, and I think of these simple things as excuses for data gathering. Like, I’m just going to program the computer and run some experiments and see if I can turn up some hidden patterns that nobody else had seen just because they hadn’t yet done these experiments.

The third thing, which maybe sounds a little silly, is that the simple problems I like don’t require much background to get into them. I like things where I can just start working. I’m impatient. If I hear about some conjecture in some fancy area of math, I’m lazy about it. I don’t feel like spending six months reading the literature until I get to the point where I’m ready to attack this problem. I like to just get my hands dirty and start right away.

What’s an example of a simple problem?

One problem I got very interested in was the triangular billiards problem. It asks: If you look at billiards in a triangle, is there a periodic billiard path — one that traces the same path over and over again? This was known for acute triangles [where all of the triangle’s angles are less than 90 degrees], but it wasn’t known for obtuse triangles [where one angle is greater than 90 degrees]. The question is: Does every triangle have a periodic billiard path? So I made some progress on this. I proved that as long as all the angles are less than 100 degrees, there is a periodic billiard path.

Could you give me another example?

Another thing I worked on for quite a while and did solve is a problem in outer billiards. Here you have a convex shape in the plane, like an oval, square or pentagon. You start at a point outside the shape and you, well, maybe I should draw a picture of this.