AbstractA problem posed by Bellman and considered by Isbell is as follows: Suppose one is a mile from a straight shore with no means whatsoever of ascertaining its direction. What is the optimum path to follow so as to (a) minimize the maximum distance travelled in reaching shore, (b) minimize the statistical expectation of the distance travelled, or (c) maximize the probability of reaching shore within a given distance travelled? Isbell found the solution to (a); in the present paper a sequence of approximations to the optimal policy for (b) is considered.