# Retired Man Solves Decades-Old Maths Problems While Brushing His Teeth

The bell curve, known as the normal distribution or the Gaussian distribution. marekuliasz/Shutterstock

During what was an unremarkable summer’s morning, a 67-year-old man had a modern-day eureka moment, allowing him to suddenly crack a probability problem that has stumped statisticians since the 1950s.

Quanta Magazine recently reported the story of an unassuming retired German statistician called Thomas Royen, who came up with a mathematical proof for the Gaussian correlation inequality (GCI) during a brief flash of inspiration while brushing his teeth in July 2014.

The GCI principle is all about geometry and probability. It was first proposed in the 1950s and more eloquently formulated in 1972. Although remarkably simple to understand, it has eluded mathematical explanation.

Here's an example of how it works: Imagine you are throwing darts at a target made up of any two convex symmetrical shapes that are centered on the same point. More simply put, a target made up a rectangle with a circular cut-out laid over the top of it. The number of darts that hit the circle, rectangle, or both will appear in a normal bell-curve distribution, known as a Gaussian position, with the majority of the darts hitting in the overlapping section.

GCI says that the probability of a dart hitting the overlap is always equal to or higher than the individual probability of it landing inside the rectangle multiplied by the individual probability of landing in the circle. This makes common sense, but it has proved to be a pain to explain mathematically.

“I know of people who worked on it for 40 years,” Donald Richards, a statistician at Pennsylvania State University, told Quanta Magazine. “I myself worked on it for 30 years.”

Royen, however, found a remarkably sleek solution to it. The same day he had his eureka moment, he typed up the calculations on Microsoft Word (as he didn’t know how to use the fancy word processor used by professional mathematicians) and sent the paper to the pre-print website arXiv. The paper is called “A simple proof of the Gaussian correlation conjecture." He sent the solution to a few statisticians in the US who he thought might be interested.

“I got this article by email from him,” Richards added. “And when I looked at it I knew instantly that it was solved.”

His work has gone more or less unrecognized in the field. However, that’s largely due to Royen’s relaxed attitude towards the whole event, choosing to avoid the long and arduous peer-review system. With no career to try and promote, he’s simply happy that the proof is now found.