# Maps to itself... eventually

Algebra Level 3

Let $$f : \mathbb N \to \mathbb N$$ denote a function such that the table below is fulfilled.

 $$x$$ 1 2 3 4 5 6 7 8 9 10 11 12 $$f(x)$$ 8 10 12 5 2 11 9 7 1 4 3 6

Which of the following is true for integer $$1\leq S \leq 12$$?

Bonus: Show that for any such permutation $$\sigma$$, there is an $$n$$ such that $$\sigma ^ n$$ is the identity permutation.

×