The 3n+1 conjecture (a.k.a. the Collatz conjecture) is easy to state, unsolved, probably difficult to prove if true, and can provide us with some pretty pictures. I’m only going to state it and show some pretty pictures.

First, what’s this conjecture? It’s about a simple algorithm: choose a positive integer $ n$. If $ n=1$, don’t do anything. Otherwise follow these instructions:

- If $ n$ is even, divide by $ 2$.
- If $ n$ is odd, multiply by $ 3$ and add $ 1$.

Do this to the result, and keep doing this until the result is 1. For instance, choose $ n=6$. We keep applying the steps above to get the numbers 6,3,10,5,16,8,4,2,1. **The 3n+1 conjecture is that for any positive integer $ n$, this sequence always gets to 1.**

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