What Would It Mean to Solve the Conjecture?
How would one go about trying to solve the Collatz Conjecture? Well, there are two main approaches. The first approach is that you must show that every positive integer n, when placed into the Collatz System, eventually falls to 1.
Researchers sometimes refer to these input_n values as "hailstones". In actual weather phenomenon, hail occurs when conditions are warmer up above, freezing down below, and there's a rare upwards wind draft. Rain tries to fall down to earth, but then freezes and is blown back up again into the atmosphere, where it melts, gets larger, and tries to fall back down again. Watching a Collatz input_n value's orbit is a lot like watching a hailstone. They go up then down, up then down, up then down chaotically... yet inevitably eventually fall down to the ground.
But what if the conjecture is false? What if they don't all fall to 1? What would that mean? Well, then one of two things must have happened. Either some particular input_n value (a hailstone, if you will) must have gotten stuck in a different loop than the familiar 4 >> 2 >> 1 loop; some other loop way out there that humanity has not yet discovered. We've tested the conjecture for all n values up to 10^18 and so far 4 >> 2 >> 1 is the only loop we've ever spotted. The other way the conjecture could fail is if some particular hailstone rises up and up... and never falls back down at all. Researchers call this divergence. Thus the conjecture really has two halves. To prove the Collatz Conjecture, one must establish that the x3+1 Collatz system has only one loop and that it does not have divergence. This is the classical approach to the problem.
The alternate technique is to start from 1 and work your way back up the tree. The tree begins with 1 then 2, 4, and 8, but then it branches at 16, offering two paths. Those two paths then divide again at 64 and at 10, splitting into four. The full structure, all the branches included, is called The Collatz Tree. It's a totally different way of thinking about the problem. To use this technique to prove the conjecture, you would have to show that every positive integer is found somewhere in this great branching tree.
So far this has never been done.
Could you find your way through this maze?