Just a little observation, this appears to be a baby version of the unsolved 3n+1 problem, whose conjectured answer is known as the Collatz Conjecture.
The only difference is that for odd numbers
n you replace with
3n+1. It's conjectured that you can always get to 1 by following the rules, but to this day it remains unproved.
Fortunately this one is much easier. :)