# 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.