  A recent variation of the classical geodetic problem, the strong geodetic
problem, is defined as follows. If $G$ is a graph, then ${\rm sg}(G)$ is the
cardinality of a smallest vertex subset $S$, such that one can assign a fixed
geodesic to each pair $\{x,y\}\subseteq S$ so that these $\binom{|S|}{2}$
geodesics cover all the vertices of $G$. In this paper, the strong geodesic
problem is studied on Cartesian product graphs. A general upper bound is proved
on the Cartesian product of a path with an arbitrary graph and showed that the
bound is tight on flat grids and flat cylinders.
