Some of the greatest mathematical minds have been dumbfounded for decades over the Hadwiger-Nelson problem, a math 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, an outspoken antiaging researcher, unraveled a problem that has stumped mathematicians for more than 60 years, Quanta Magazine reports.
The problem, now known as the Hadwiger-Nelson problem or the problem of finding the chromatic number of the plane, was suggested by mathematicians in the 1950s and it has piqued the interest of many, including the famously prolific Paul Erdős. Researchers quickly narrowed the possibilities down, finding that the infinite graph can be colored by no fewer than four and no more than seven colors. Other researchers went on to prove a few partial results in the decades that followed, but no one was able to change these bounds.
Aubrey de Grey, a biologist known for his claims that people alive today will live to the age of 1,000, recently uploaded this new proof entitled: “The Chromatic Number of the Plane Is at Least Five”.
In order to offer a simple explanation to non-mathematicians, de Gray broke down his breakthrough result.
“Suppose you have a piece of paper and two pens, red and green ink, and your task is to place dots on the paper in such a way that no pair of dots of the same colour are exactly one inch apart,” he told How Stuff Works.
“But the catch is, it’s a game, and your opponent also has a piece of paper but only one pen, and he puts his dots wherever he likes, and you have to put your dots in the exact same places that he did.
“Is there any way that he can win, i.e. place his dots in such a way that the no-monochromatic-pair rule prevents you from placing your dots in the same places as his?”
“Answer: yes, he can place three dots in an equilateral triangle so that each pair is an inch apart. So now, suppose you have three pens, red blue green, can he still win?
“Answer: it turns out that yes, but it’s harder, and he needs seven dots. So the obvious next question is, what if you have four pens? And I found a way that he can place his dots so that he still wins, but the simplest solution I found needs 1581 dots.”
With four ruled out of the possible answers, mathematicians must now figure out if it’s five, six or seven.
Take a look at the question and see if you can solve the riddle.
Draw a number of equally spaced points — known as vernicles — on a piece of paper.
Now connect all of the points one unit apart from each other with a line.
Now you must figure out the minimum number of colours needed to colour points on a plane so that any two points connected by a line have different colours.
As de Gray highlighted, it doesn’t take much work to come up with a relatively simple unit distance graph that can’t be coloured with just three colours.
When you go to a unit distance graph that can’t be coloured in with four colours, things become a lot more messy as it requires 1581 points.
While de Gray has ruled four out of the mix, five might not be the answer as it’s possible that a graph will come along requiring six or seven colours.
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”.
For now the final answer remains a mystery, but maybe you could be the one to solve it.