  For a positive integer $n$, we study the number of steps to reach $n$ by a
{\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the
recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first
suggested by Richard Stanley, is to determine the maximum number $s(n)$ of
steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for
most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property
that $a_{s+1}$ is the integer closest to $\phi n$ where $\phi=(1+\sqrt{5})/2$.
We prove that this is true for only a positive fraction of $n$. We give
explicit formulas for the choice of the starting pairs and the determination of
$s(n)$ by giving a characterization theorem. We also derive a number of density
results concerning the distribution of down and up cases (that is, those $n$
with $a_{s+1}=\lfloor \phi n\rfloor$ or $\lceil \phi n \rceil$, respectively),
as well as for more general `paradoxical' cases.
