A Counting Parametric Problem

Algebra Level 3

Define the function \( f(n) : \mathbb{N} \rightarrow \mathbb{N} \) as follows:

\[ f(n) = \begin{cases} n^2 + 1 & \text { if } n \text{ is odd }, \\ \frac{ n } { 2} & \text{ if } n \text { is even } . \\ \end{cases} \]

For how many integral \(n \in [1, 100]\) does \(f(f(...f(n)))=1\) for some number of applications of \(f\)?


This problem is taken from this year's MATHCOUNTS State Competition. I enjoyed solving it, so I'm sharing it with you! Enjoy solving it, and post a creative solution!

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