Maps to itself... eventually

Algebra Level 3

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

\(f(x) \)810125211971436

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.


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