As for the randomness, I have wondered if Collatz sequences are somehow related to the properties of a common prng with multiplier 3/2, infinite length state vector, and mod 2 on the output with this formula: https://en.m.wikipedia.org/wiki/Linear_congruential_generato.... I assume this could be part of what makes the conjecture both interesting and difficult and beautiful.

Very cool to see there is some patterns hiding in the randomness too!

> The points look quite uniformly distributed to me. If I squint, then maybe I can see some structure, but it's hard to describe and I could be imagining it.

It doesn't, these points look like what happens if you ask someone who doesn't know what a uniform distribution looks like to generate a uniformly distributed set of points though.

Here's what an actual uniform distribution looks like... much less "uniform": https://claude.ai/public/artifacts/00549caf-2ec1-4803-b909-6...

Credit to the book "Struck By Lightning" for making me aware of this fact, many years ago now. Disclaimer that the author is a family friend.

> I've been telling people for years if businesses want employees to have better ideas, they should have more showers in their offices. So far everyone seems to think I'm joking. I'm not.

I have definitely noticed that some of my best ideas or breakthroughs come to me when showering, or sleeping, or eating, or driving, or doing the dishes, or basically any mundane autopilot task where my mind is free to wander. But yeah no, having a shower room in the office is both gross and weird. Maybe offices should.. encourage you to... wash some dishes?

Interesting take. The visualization of the inverse tree highlights just how sparse the “preimage space” is under Collatz iterations. The idea that this sparsity contributes to the apparent randomness is compelling. I’m curious whether modeling the process modulo powers of 2 and 3, or via 2-adic analysis, could formalize some of these heuristic observations. Also, the assumption that most numbers “fall off” rapidly aligns with empirical behavior, but it’s still not clear how to bound exceptional trajectories.