Let $N(x) = \big\lceil \frac x9 \big \rceil$ and $T(x) = \big\lceil \frac x2 \big \rceil$.
Consider the operation by which we repeatedly apply $N$ and $T$ (in that order) to a number $x \in \mathbb N$ until the result is less than or equal to $9$ (but larger than $1$). Call this result $r$.
I have a few questions:
(a) What is an easy way (if one exists) to determine what $r$ would be for any $x \in \mathbb N$ ? (If it makes it any easier, assume we can factor $x$.)
(b) How can one know how many compositions of $N$ and $T$ are needed in order to return the greatest $1 < r \leq 9$? For instance, with $x= 2 \cdot 9^3$, we apply $N \rightarrow T \rightarrow N$ to get an $r$ of $9$.
(c) How can one know how many pairs of $N$ and $T$ are needed in order to return the greatest $1 < r \leq 9$? For instance, with $x= 2 \cdot 9^3$, we apply $N \rightarrow T \rightarrow N \rightarrow T$ to get an $r$ of $5.$