The Wandering Token

A token sits on one square of a 3×3 grid, and that square counts as visited. Players alternate moving the token to an orthogonally adjacent square that has not been visited yet. A player who cannot move loses. The token may be started on any of the 9 squares, and the player to move first is the one who must make the first move from it. For how many of the 9 starting squares does the first mover win with optimal play? Answer with an integer.

• graph theory

Source: Undirected vertex geography; the matching criterion appears in work of Fraenkel, Scheinerman and Ullman on undirected geography games. Statement written for AxiomIQ.