Most top mathematicians discovered the subject when they were young, often excelling in international competitions. In contrast, math was a weakness for Jun Hu, who was born in California and raised in South Korea. “I was very good in most subjects except math,” he said. “Maths was particularly mediocre, on average, meaning that on some tests I did quite well, but on other tests, I almost failed.” As a teenager, Dr. Hu wanted to be a poet and spent a few years after high school pursuing this creative pursuit. But none of his writings were ever published. When he entered Seoul National University, he studied physics and astronomy and considered a career as a science journalist. Looking back, he recognizes flashes of mathematical insight. In high school in the 1990s, he played a computer game, “The 11th Hour.” The game consisted of a puzzle with four knights, two black and two white, placed on a small, oddly shaped chessboard. The task was to exchange the positions of the black and white knights. He spent more than a week struggling before realizing that the key to the solution was figuring out which squares the knights could move to. The chess puzzle could be reformulated as a graph where each knight can be moved to an adjacent unoccupied space and a solution could be more easily seen. Reformulating mathematical problems by simplifying them and translating them in a way that makes a solution more obvious has been the key to many discoveries. “The two formulations are logically indistinguishable, but our intuition only works on one of them,” Dr Hu said.
A puzzle of mathematical thinking
A puzzle of mathematical thinking
Here is the puzzle June Huh beat: The New York Times Objective: Swap the positions of the black and white knights. →
A puzzle of mathematical thinking
Many players stumble through trial and error looking for a pattern. This is what Dr. Hu did and he almost gave up after hundreds of attempts. Then he realized that the strange tableau and the L-shaped movements of the knights are irrelevant. What matters are the relationships between the squares. Reformulating a problem into something more understandable is often the key for mathematicians to make discoveries.
A puzzle of mathematical thinking
Let’s number the squares so we can keep track of them. The New York Times
A puzzle of mathematical thinking
Consider a knight on square 1. It can only move to square 5 while a knight on 5 can move to 1 or 7. The New York Times
A puzzle of mathematical thinking
This can be represented as a network diagram – what mathematicians call a graph. The lines show that a knight can move between squares 1 and 5 and between squares 5 and 7. The New York Times
A puzzle of mathematical thinking
Extending this analysis to the odd-shaped checkerboard yields this graph: The New York Times Now, we can place the knights on this chart, the white knights in spaces 1 and 5, the black knights in 7 and 9.
A puzzle of mathematical thinking
The problem is still to exchange the positions of the black and white knights. For each move, a knight can slide into an adjacent empty node. The reworded version is much easier to understand. Here is an answer:
A puzzle of mathematical thinking
Using the graph with the numbered nodes as a decoder ring, we then find the moves in the original matrix.
A puzzle of mathematical thinking
Ruth Fremson/The New York Times “Seeing the same puzzle in this new way, which better reveals the essence of the problem, suddenly the solution was obvious,” said Dr. Hu. “That made me think about what it means to understand something.” Item 1 of 9 1 of 91 of 9 It wasn’t until his senior year of college, when he was 23, that he rediscovered math. That year, Heisuke Hironaka, a Japanese mathematician who had won a Fields Medal in 1970, was a visiting professor at Seoul National. Dr. Hironaka was teaching a course on algebraic geometry, and Dr. Hu, long before he got his Ph.D., thinking he could write an article about Dr. Hironaka, watch. “He’s like a superstar in most of East Asia,” Dr. Hu said of Dr. Hironaka. Initially, the course attracted more than 100 students, Dr Hu said. But most students quickly found the material incomprehensible and left the class. Dr. Hu continued. “After three lectures, there were about five of us,” he said. Dr. Hu began to have lunch with Dr. Hironaka to discuss mathematics. “Mostly he was talking to me,” Dr. Hu said, “and my goal was to pretend I understood something and react in the right way so the conversation would continue. It was a challenge because I really didn’t know what was going on.” Dr. Hu graduated and began working toward a master’s degree with Dr. Hironaka. In 2009, when Dr. Hu applied to about a dozen graduate schools in the United States to pursue his Ph.D. “I was pretty sure that despite failing my undergraduate math classes, I had an enthusiastic letter from a Fields Medalist, so I would be accepted by many, many graduate schools.” All but one turned him down — the University of Illinois Urbana-Champaign put him on a waiting list before finally accepting him. “It’s been a very anxious couple of weeks,” Dr Hu said. In Illinois, he began the work that brought him to the fore in the field of combinatorics, an area of mathematics that calculates the number of ways things can be mixed up. At first glance, it looks like you’re playing with Tinker Toys. Consider a triangle, a simple geometric object—what mathematicians call a graph—with three edges and three vertices where the edges meet. One can then start asking questions like, given a certain number of colors, how many ways are there to color the vertices where none can be the same color? The mathematical expression that gives the answer is called a color polynomial. More complex color polynomials can be written for more complex geometric objects. Using tools from his work with Dr. Hironaka, Dr. Hu proved Read’s conjecture, which described the mathematical properties of these color polynomials. In 2015, Dr. Huh, along with Eric Katz of Ohio State University and Karim Adiprasito of the Hebrew University of Jerusalem, proved the Rota conjecture, which involved more abstract combinatorial objects known as matroids instead of triangles and other graphs. For matroids, there is another set of polynomials, which behave similarly to color polynomials. Their proof gave rise to an inner piece of algebraic geometry known as Hodge theory, named after William Vallance Douglas Hodge, a British mathematician. But what Hodge had developed, “was just one example of this mysterious, ubiquitous occurrence of the same pattern in all mathematical disciplines,” Dr. Hu said. “The truth is, we, even the leading experts in the field, don’t know what it really is.”
