It seems like a rather banal thing to be preoccupied with: how best to stack fruit. Your grocer does it effortlessly, probably without thinking twice about it. But for the last 400 years, mathematicians have been toiling away, and making little headway, racking their brains trying to prove something called Kepler's conjecture, a problem which essentially amounts to the fine and delicate art of efficient fruit-stacking.

Yes, it sounds woefully academic, but just because we already intuitively know something doesn't make it worthless to prove. The problem can be extended beyond the produce aisle at your local supermarket. For instance, when 17th-century mathematician and astronomer Johannes Kepler first formulated his conjecture, the primary practical concern that it pertained to was likely the stacking of cannonballs on the decks of ships.

The thing that fruits and cannonballs have in common is that they are spherical, and that's the gist of Kepler's conjecture: What's the most efficient way to pack spheres in three-dimensional Euclidean space?

We know, intuitively, that the best way to do it is to stack them in the shape of a pyramid, with the higher layers of spheres resting in the gaps between the lower layers. The problem comes when trying to offer a mathematical proof for this solution. Kepler couldn't prove his own conjecture, and neither could any mathematician since. That is, it couldn't be proven until now... but it took some help from modern technology to do it.

An international team of mathematicians led by Thomas Callister Hales has finally published a proof of the conjecture, a proof so convoluted that it could only be reliably verified by a computer, reports ZME Science.

Hales and his team opted for a "proof by exhaustion," a brute force method which splits the problem into a possible number of cases and then analyzes all those cases. It's not the elegant solution that mathematicians pine for, but after 400 years of trying to solve the problem, this is the only thing that has worked.

If you want to check it out for yourself, the proof was published in the journal Forum of Mathematics, Pi. A healthy trust in the proof-checking power of computers will save you some time and headaches, however.

So now we can prove what your grocer already intuitively knew, but there's value in that. For instance, it could help researchers understand the atomic distribution of crystals, and it could extend some 2D applications into a 3D space. The algorithms used to solve the problem could even be put to use verifying the safety of driverless cars and similar technologies. So proving Kepler's conjecture is not just an aloof academic victory.

Besides, relying on intuition isn't always wise. Without rigorous proof, we could be led astray. It wouldn't be the first time; human intuition is fallible. It's healthy practice to look for proof, even if the problem really does only pertain to something as menial as fruit-stacking.

Kepler's conjecture: Why it took 400 years to prove what your grocer already knows
The most efficient way of stacking fruit is actually one of the most difficult math problems in history.