I've gone through several periods of obsession with this problem. One of its addictive qualities is that, even if you're getting no closer to a solution, it's e

I've gone through several periods of obsession with this problem. One of its addictive qualities is that, even if you're getting no closer to a solution, it's easy to find a bunch of related patterns within patterns within patterns, a beautiful balance of order and chaos, a wonderful endless fractal of emerging patterns to play with and explore. For instance: arrange all integers with negatives between positives, as such: 1, -1, 2, -2, 3, -3, 4, -4, etc. Now choose any integer, positive or negative, and follow it to the next integer. For example, if you choose 3, go forward 3 spaces on the list, landing on -4. This is negative, so we go back 4 spaces, landing on -2, which leads to -1. Conjecture: All integers eventually lead back to the 1, -1 loop at the beginning. This can be shown to be a variation of the Collatz; that is, if Collatz is true, this must be true and vice versa. I also attempted to prove Collatz by focusing on the odds only, that's where the real action is... Failed of course as it's not quite rigorous enough to disprove potential looping (it does at least prove that it cannot diverge to infinity), but may be of interest to someone else obsessed with the problem: https://youtu.be/P0F4zbNdbTU https://youtu.be/P0F4zbNdbTU