This Biologist Cracked a Problem That’s Stumped Mathematicians

October 25, 2018

An amateur mathematician has just partially solved a problem that has plagued mathematicians since 1950.

Aubrey de Grey – a biologist better known for trying to radically extend the human lifespan and for predicting that the first person to live to be 1,000 years old has been born – has published a paper on the preprint server arXiv that narrows down the answer to the 68-year-old Hardwig-Nelson problem. Mathematicians have known for years that the answer to this problem (which we’ll get to in a moment) is 4, 5, 6, or 7. de Grey shows in his paper that it’s definitely not 4. that leaves only 5, 6, or 7.

Now that you have De Grey’s answer, here’s a question.

Take a piece of canvas and draw a bunch of points (called vertices) on it. If the distance between any points is 1 unit, draw a line between them. The mathematician doesn’t care whether the “unit” is an inch or a mile. It doesn’t matter as long as all connected vertices are the same distance from each other. (The line connecting the points is called an “edge.”) Mathematicians call this a unit distance graph. What you end up with will look like this.

Now it’s time to go to the store and buy paint to color all the dots.

Now ask yourself. What’s the minimum number of colors of paint you’d need to color any graph to ensure that no two dots have the same color edge?

It’s easy to come up with a unit distance diagram that can’t be colored with just three colors. Here’s a good example.

But it’s much harder to come up with a unit distance graph that can’t be colored with four colors. computers can’t do it on their own. no full-time mathematician has been able to do it for 68 years, until De Grey came up with this monstrosity.

De Grey’s graph has 1,581 vertices on it. and they’re arranged in such a way that you can’t get it just right with four colors of paint. There has to be at least five colors to get it right.

But that doesn’t mean that five is the absolute minimum. Mathematicians know that it’s possible for a graph to need six colors of pigment, or even seven. Back in 1950, mathematician John Isbell proposed a strategy involving seven colors to solve any graph.

The absolute minimum required remains a mystery. But thanks to de Grey, we know it’s more than four.