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.

f^{2015}(S) = S

f^{2014}(S) = S

f^{2016}(S) = S

f^{2013}(S) = S

1 solution

A Former Brilliant Member
Jun 14, 2016

Note that the $f^n(S)=S$ if and only if $n$ is a multiple of $4$ .

In the given options, only $2016$ is divisible by $4$ so $f^{2016}(S)=S$ .