Early Attempts at the Problem
While I had first heard of the Collatz Conjecture in high school, it was during the summer of 2017 that I first really fell in love with the problem. Like so many others, I became totally mesmerized by the Stopping Times Graph and Max Value Graph and spent endless hours trying to decipher their mysteries.
Look back, I'm very proud of my early work on the problem. Years later, watching Marc Grinnell's lecture, I learned that I had all by myself discovered and travelled fairly far down each of the main major paths taken by early researchers to the problem in the 1970's. From trying to predict the future movements of the hailstones, to constructing the Collatz tree from the ground up, even to more theoretical approaches like recognizing that any convergent loop must, in a certain sense, be a kind of approximation of ln(2)/ln(3) to balance out the x3 steps with ÷2 steps... The trouble is — these are all dead ends.
When you look for patterns out in the Collatz System, to help you predict the "flight paths" of the hailstones, you actually do find them. The Collatz System is filled to the brim with patterns! The issue is, these tend to be local patterns, helping you predict the hailstones' movement 1 or 2 steps ahead. No matter how clever a pattern you come up with, it will only ever help you predict a certain finite number of steps ahead. When you take the full Collatz System and "press go", your clever patterns always dissolve in your hands. You can only ever peer so far into the "fog" of the Collatz world, before all is lost to the chaos and mayhem of the dynamical system.
I needed to develop a new approach.