Credit…Ruth Fremson/The New York Times

Most top mathematicians discovered the subject at a young age and have often excelled in international competitions.

Math, on the other hand, was a weakness for California-born and South Korean-raised June Huh. “I was pretty good at most subjects except math,” he said. “Math was remarkably mediocre on average, which means I did reasonably well on some tests, but I almost failed other tests.”

As a teenager, Dr. Huh became a poet, and he spent a few years after high school pursuing this creative pursuit. But none of his writings were ever published. Upon entering Seoul National University, he studied physics and astronomy and considered a career as a science journalist.

Looking back, he sees flashes of mathematical insight. He played a computer game, The 11th Hour, in middle school in the 1990s. The game involved a puzzle of four knights, two black and two white, placed on a small, oddly shaped chess board.

The task was to swap 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 rearranged into a diagram where each knight can move to an adjacent unoccupied square, and a solution could be more easily seen.

Reformulating mathematical problems by simplifying and translating them in a way that makes a solution more obvious has been the key to many breakthroughs. “The two formulations are logically indistinguishable, but our intuition only works in one of them,” said Dr. huh

## A riddle of mathematical thought

## A riddle of mathematical thought

Here is **the riddle that posed June Huh**:

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**Goal: **Swap the positions of the black and white knights. →

## A riddle of mathematical thought

Many players stumble through trial and error in search of a pattern. Dr. Huh done, and **he almost gave up** after hundreds of tries.

Then he realized that the odd board and the L-shaped movements of the knights are irrelevant. What matters is the relationships between the squares.

Rephrasing a problem in **a little easier to understand** is often the key for mathematicians making breakthroughs.

## A riddle of mathematical thought

let us **Number the squares** so we can track them.

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## A riddle of mathematical thought

consider **a knight on square 1**. It can only move to square 5, while a knight on 5 can move to 1 or 7.

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## A riddle of mathematical thought

It can be **shown as a network plan** — what mathematicians call a graph. The lines indicate that a knight can move between squares 1 and 5 and between squares 5 and 7.

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## A riddle of mathematical thought

Extending this analysis to the odd chessboard **gives this graph**:

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Now we can place the knights on this diagram, the white knights on squares 1 and 5, the black knights on squares 7 and 9.

## A riddle of mathematical thought

The problem remains with swapping the positions of the black and white knights. On each move, a knight can slide to an adjacent empty node.

The rewrite is **much easier to find out**. Here is an answer:

## A riddle of mathematical thought

Using the chart with **the numbered nodes as a decoder ring**we then find the moves on the original board.

## A riddle of mathematical thought

Ruth Fremson/The New York Times

“Looking at the same puzzle in this new way that better reveals the essence of the problem, **Suddenly the solution was obvious**said Dr. huh “It made me think about what it means to understand something.”

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He only rediscovered mathematics in his final year at 23. That year, Heisuke Hironaka, a Japanese mathematician who won a Fields Medal in 1970, was a visiting professor at Seoul National.

dr Hironaka taught a course in algebraic geometry, and Dr. Huh took part long before he got his PhD and thought he could do an article on Dr. Write Hironaka. “He’s like a superstar in almost all of East Asia,” said Dr. Huh about Dr. Hironaka.

Initially, the course attracted more than 100 students, said Dr. huh But most students quickly found the material incomprehensible and dropped out of class. dr Hu continued.

“After about three lectures, there were five of us,” he said.

dr Huh started talking to Dr. Hironaka having lunch to discuss math.

“Mostly he talked to me,” said Dr. Huh, “and my goal was to pretend I understood something and responded correctly so the conversation would continue. It was a challenging task because I really didn’t know what was going on.”

dr Huh graduated and started with Dr. Hironaka to work on a master’s degree. In 2009, Dr. Huh to about a dozen graduate schools in the United States to do a Ph.D.

“I was pretty confident that despite all my math class failures, I had an enthusiastic letter on my undergraduate degree from a Fields Medalist that I would be accepted into 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 exciting few weeks,” said Dr. huh

In Illinois, he began work that brought him exposure to combinatorics, a branch of mathematics that works out how many things can be mixed. At first glance it looks like playing with Tinker Toys.

Imagine a triangle, a simple geometric object – what mathematicians call a graph – with three edges and three vertices where the edges meet.

One can then begin to ask questions such as: B. 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 chromatic polynomial.

More complex chromatic polynomials can be written for more complex geometric objects.

Using tools from his work with Dr. Hironaka proved Dr. Huh Read’s conjecture, which described the mathematical properties of these chromatic polynomials.

In 2015, Dr. Huh, along with Ohio State University’s Eric Katz and Hebrew University of Jerusalem’s Karim Adiprasito, developed the Rota conjecture, which involved more abstract combinatorial objects known as matroids instead of triangles and other graphs.

For the matroids, there is another set of polynomials that have behavior similar to chromatic polynomials.

Their proof drew on an esoteric piece of algebraic geometry known as Hodge’s 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,” said Dr. huh “The truth is that even the top experts in this field don’t know what it really is.”