This amateur mathematician just made an enormous “lucky” breakthrough in solving the 60-year-old Hadwiger-Nelson problem

Some of the greatest mathematical minds have been dumbfounded for decades over the Hadwiger-Nelson problem, a maths puzzle relating to untouchable colours. So they must have their proverbial tails between their legs, then, with the news an amateur has taken a significant leap towards solving the problem.

Aubrey de Grey, a Cambridge-educated anti-ageing researcher from London, has reportedly been able to narrow down what the possible solution could be.

The Hadwiger-Nelson problem pertains to the minimum number of colours you’d need to colour points on a plane (that’s “plane: a flat surface on which a straight line joining any two points on it would wholly lie,” not a plane of the Boeing 737 persuasion), whilst ensuring that points one unit away from one another aren’t represented in the same colour.

It helps, I’m told, to picture a graph comprising any number of dispersed points on a plane.
If each of these dots were imbued with a colour, and you were to connect the dots, how many different colours would be required so that no two connecting dots shared the same colour?

If this sounds like something a child could master, then you’re probably one of those people who barks “I could have done that,” at literally any piece of modern or conceptual art. Just kidding. The trick is to remember that the number of connected points on the hypothetical graph is theoretically infinite.

The problem, pioneered by Princeton mathematician Edward Nelson, originates back to 1950. Soon after, a breakthrough was made, with mathematicians deciphering that the number had to be somewhere between four and seven. But since then, mathematicians have continued to toil away at the problem.

Now comes another breakthrough, from de Grey, who recently uploaded this new proof to the pre-print research site arXiv.org entitled: “The Chromatic Number of the Plane Is at Least Five”.

De Grey’s solution demonstrates that a graph with 1,581 vertices requires at least five different colours, disproving the minimum of four that was previously proffered. The amateur played around with the Moser spindle, a gadget comprising an undirected graph of seven vertices and 11 edges. He realised that the problem, when it has amassed 20,425 points, requires more than four colours – the first time the problem’s answer has been narrowed down since last century.

As for de Grey, he remains remarkable humble, in the way that incredibly capable human beings are want to do. Speaking to Quanta about his discovery, he offered plainly: “I got extraordinarily lucky”.

Image: jeanbaptisteparis, used under Creative Commons