Consider the function $\sigma : \mathbb{Z} \to \mathbb{Z}$ where $\sigma = n - 3$. The orbits are
$\{3n : n \in \mathbb{Z} \}, \{3n + 1 : n \in \mathbb{Z} \}, \{ 3n + 2: n \mathbb{Z} \}$
What exactly are they really doing? I tried listing out a few numbers $\sigma (1), \sigma(2), \sigma(3)$ etc... but they just keep going on. How are they finding these? What happens if the function is more complicated like $\rho = n^2 + 1$? What strategy are they using? What is their thinking towards this problem?