A mobile agent, modeled as a deterministic finite automaton, navigates in the infinite anonymous oriented grid
. It has to explore a given infinite subgraph of the grid by visiting all of its nodes. We focus on the simplest subgraphs, called {\em wedges}, spanned by all nodes of the grid located between two half-lines in the plane, with a common origin. Many wedges
... [Show full abstract] turn out to be impossible to explore by an automaton that cannot mark nodes of the grid. Hence, we study the following question: Given a wedge W, what is the smallest number p of (movable) pebbles for which there exists an automaton that can explore W using p pebbles? Our main contribution is a complete solution of this problem. For each wedge W we determine this minimum number p, show an automaton that explores it using p pebbles and show that fewer pebbles are not enough. We show that this smallest number of pebbles can vary from 0 to 3, depending on the angle between half-lines limiting the wedge and depending on whether the automaton can cross these half-lines or not.