Question 93
We consider permutations $f$ on the set $N$ of non-negative integers, i.e. bijective mappings $f$ from $N$ to $N$, with the following properties:
For all $n \in N$, we have $f(f(x)) = x$ and $|f(x) - x| \le 3$.
Furthermore, for all integers $n \gt 42$, we have
$$
M(n) = \frac{1}{n+1} \sum_{j=0}^{n} |f(j) - j| \lt 2.011
$$
Prove that there exist infinitely many integers $K$, such that $f$ maps the set $\{n|0 \le n \le K\}$ onto itself.
For all $n \in N$, we have $f(f(x)) = x$ and $|f(x) - x| \le 3$.
Furthermore, for all integers $n \gt 42$, we have
$$
M(n) = \frac{1}{n+1} \sum_{j=0}^{n} |f(j) - j| \lt 2.011
$$
Prove that there exist infinitely many integers $K$, such that $f$ maps the set $\{n|0 \le n \le K\}$ onto itself.