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We outline the properties of a symmetric random walk in one dimension in which the length of the $nth$ step equals $λn,$ with $λ<1.$ As the number of steps $N→∞,$ the probability that the end point is at $x$ approaches a limiting distribution $Pλ(x)$ that has many beautiful features. For $λ<1/2,$ the support of $Pλ(x)$ is a Cantor set. For $1/2⩽λ<1,$ there is a countably infinite set of λ values for which $Pλ(x)$ is singular, while $Pλ(x)$ is smooth for almost all other λ values. In the most interesting case of $λ=g≡(5−1)/2,$ $Pg(x)$ is riddled with singularities and is strikingly self-similar. This self-similarity is exploited to derive a simple form for the probability measure $M(a,b)≡∫abPg(x) dx.$