This German Retiree Solved One of World's Most Complex Maths Problems - and No One Noticed

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It's a problem that the world's most experienced mathematicians have spent decades trying to solve, and the solution had eluded them at every turn - the infamous Gaussian correlation inequality (GCI).

Then, out of nowhere, a retired German statistician figured out the proof while hunched over the sink, cleaning his teeth. But rather than being celebrated by the wider mathematical community, the proof went largely ignored. Because how could such an unlikely figure have outsmarted them all?

"I know of people who worked on it for 40 years," Donald Richards, a statistician from Pennsylvania State University, told Natalie Wolchover at Quanta Magazine. "I myself worked on it for 30 years."
First proposed in the 1950s, but properly formulated in 1972, the GCI principle sounds relatively simple:

If two shapes overlap, such as a rectangle and a circle, the probability of hitting one of those overlapping shapes - say, with a dart - increases the chances of also hitting the other. 

Imagine it like this - you've got a blue rectangle and a yellow circle, and you place one on top of the other, and mark a target in the centre like a dart board.

You throw a bunch of darts at the target, and you'll soon discover that a bell curve - or 'Gaussian distribution' - of positions has formed around the centre, with vast majority of the darts sitting in the overlap.
But it's not just any old vast majority - it's a specific majority that is directly proportional to the number of darts outside the overlapped shapes.

The Gaussian correlation inequality states that the odds of the dart hitting the combined circle and rectangle are always as high as or higher than the probability of it landing inside the rectangle multiplied by the probability of it landing inside the circle.
That might seem like common sense, but try proving it mathematically.

Source: Science Alert