Let {xk }k ≧ − r be a simple Bernoulli random walk with x –r = 0. An integer valued threshold ϕ = {ϕk }k≧1 is called a search plan if |ϕk+1−ϕk |≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ 1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.