Consider a particle which is released at some point on a fractal and which moves about the fractal at random. A long standing goal has been to determine a differential equation governing the probability density function which describes this walk. As well as being interesting in its own right, this problem is thought to provide an insight into the problem of anomalous diffusion. Many attempts to derive such an equation have been made, all with limited success, perhaps because of the tension between smoothness required by differential equation tools and the lack of smoothness inherent in fractals. Here we present, for the first time, the equation governing the random walk on a simple fractal—the Koch curve. We show that this equation makes computation of the probability density function for this problem a simple matter.