We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph possessing a given target property. We show that if $p\geq \frac{\ln n+\ln\ln n+ω(1)}{n}$, then one can find a Hamilton cycle with high probability after exposing $(1+o(1))n$ edges. Our result is tight in both $p$ and the number of exposed edges.