Playing around with Collatz

Number theory is proving to be the ultimate war of attrition. I’m operating on a sort of mathematical instinct right now, because the actual enjoyment has completely evaporated. The grind has become an understatement; it’s a total drought of inspiration. I know that in mathematics, the breakthrough usually follows the period of greatest frustration, but that doesn't make the current work any less exhausting. I'm just putting in the hours and waiting for the light to break through.

I spoke to Gemini about my dilemma and it suggested the following:

Since you’re looking to inject some life back into the grind, let’s pivot away from dense proofs and toward a problem that feels more like a "numerical magic trick."

One of the most famous and accessible gateways into recreational number theory is the Collatz Conjecture (often called the 3n + 1 problem). It is deceptively simple to understand but has remained unsolved for nearly a century.

The Challenge: The "Hailstone" Numbers

The rules are simple. Pick any positive integer n and follow these two rules:

* If n is even, divide it by 2: n/2

* If n is odd, triple it and add 1: 3n + 1

The Goal: Repeat the process. The "conjecture" is that no matter what number you start with, you will eventually hit the number 1 (and then get stuck in a 4 \to 2 \to 1 loop).

Try This:

Pick a small number like 7 and trace its path. You’ll notice it doesn't just drop straight down; it "hails" upward before crashing:

* Why This is "Fun" (and Frustrating):

* The Chaos: Starting with 26 takes only 10 steps to reach 1. Starting with 27 takes 111 steps, climbing as high as 9,232 before finally plummeting back to 1.

* The Mystery: Mathematicians have tested numbers up to 2^{68} and they all lead to 1. However, no one has been able to prove that it happens for every single number. Paul Erdős once said, "Mathematics may not be ready for such problems."

Your "Recreational" Task

If you want a quick distraction from your formal studies, try to find a number under 100 that takes the most steps to reach 1 (this is called its total stopping time).

I wrote some quick python code to compute this and it’s 118 for number 97.