The Collatz conjecture is also known as the "3n + 1" problem. It's an easy problem to explain and check, and has been tested up into the nineteen figure range. But it's only now that anyone has come forward claiming a proof that it always works.
The Collatz conjecture was first posed in 1937. It's simple enough that an elementary school student can make their way through it. Start by picking a number. Designate that number 'n'. If n is an even number, divide it by two. If it's an odd number, multiply it by three and add one. This last maneuver is what earns the Collatz conjecture its nickname, the "3n + 1" problem. Take the product of this division or multiplication and repeat the process. If it's even, divide it by two and if it's odd multiply it by three and then add one.
Do this enough and you will always eventually get the number one. This has been tested on numbers up to a billion billion and it has always worked. However, there was no proof that it would always work with any theoretical number. Pick a high enough number and it could, possibly, break the entire process down. Because the math is so tantalizingly simple, many people have had a go at proving it at one point or another since its publication. The simple math hid a thorny problem, however, and no one has ever proved the Collatz conjecture.
Recently, a mathematician at the University of Hamburg has published a proof of the conjecture. It has not been conclusively confirmed by others, but it could represent the conclusion of a problem that has stumped people for almost a century. Of course, it it doesn't work out, perhaps an io9er can take a shot at it. How hard could it be? The toughest math in it is "3n + 1."
Update: It looks like the proof has been withdrawn. Io9ers are free to come up with their own for the next three quarters of a century.