In a series of random walks (random flights) a traveler visits the vertices of a 600-cell (a four-imensional regular polytope). The traveler starts at a given vertex and in each walk, independently of the others, chooses a vertex at random as the destination. In each walk the transition probability depends only on the distance between the starting vertex and the end vertex. In this paper we determine the probability that the traveler returns to the initial position at the end of the nth walk.