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We study Fibonacci walks, which are sequences of positive integers satisfying the recurrence w k+ 2= w k+ 1+ w k. Richard Stanley suggested studying n-slow Fibonacci walks, which are Fibonacci walks with w s= n and s as large as possible. Stanley conjectured that for most n, there is a slow Fibonacci walk reaching n= w s with the property that w s+ 1 is the integer closest to ϕn where ϕ=(1+ 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 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 w s+ 1=⌊ ϕ n⌋ or⌈ ϕ n⌉, respectively), as well as for more general “paradoxical” cases.