June Huh’s monochrome chess puzzle paved the way for chromatic geometry.

Most top mathematicians discovered the subject when they were young, often excelling in international competitions.

By contrast, math was a weakness for June Huh, who was born in California and grew up in South Korea. “I was pretty good at most subjects except math,” he said. “Math was notably mediocre, on average, meaning on some tests I did reasonably OK But other tests, I nearly failed.”

As a teenager, Dr. Huh wanted to be a poet, and he spent a couple of years after high school chasing that 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 middle school in the 1990s, he was playing a computer game, “The 11th Hour.” The game included a puzzle of four knights, two black and two white, placed on a small, oddly shaped chess board.

The task was to exchange the positions of the black and white knights. He spent more than a week flailing before he realized the key to the solution was to find which squares the knights could move to. The chess puzzle could be recast as a graph where each knight can move to a neighboring unoccupied space, and a solution could be seen more easily.

Recasting math problems by simplifying them 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 works in only one of them,” Dr. Huh said.

A Puzzle of Mathematical Thinking

A Puzzle of Mathematical Thinking

Here is the puzzle that June Huh beat:

Goal: Exchange the positions of the black and white knights. →

A Puzzle of Mathematical Thinking

Many players stumble through trial and error looking for a pattern. That is what Dr. Huh did, and he almost gave up after hundreds of tries.

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

Recasting a problem into something easier to understand is often key to mathematicians making breakthroughs.

A Puzzle of Mathematical Thinking

Let’s number the squares so that we can keep track of them.

Consider a knight on square 1. It can only move to square 5 while a knight on 5 can move to 1 or 7.

This can be represented as a network diagram — what mathematicians call a graph. The lines indicate that a knight can move between squares 1 and 5 and between squares 5 and 7.

Extending this analysis to the odd-shape chessboard yields this graph:

Now, we can place the knights on this graph, the white knights at spaces 1 and 5, the black knights at 7 and 9.

A Puzzle of Mathematical Thinking

The problem is still to exchange the black and white knights’ positions. For each move, a knight can slide to an adjacent empty node.

The recast version is much easier to figure out. Here is one answer:

A Puzzle of Mathematical Thinking

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

A Puzzle of Mathematical Thinking

“Viewing the same puzzle in this new way, which better reveals the essence of the problem, suddenly the solution was obvious,” Dr. Huh said. “This made me think about what it means to understand something.”

Source: Read Full Article