  The localization game is played by two players: a Cop with a team of $k$
cops, and a Robber. The game is initialised by the Robber choosing a vertex $r
\in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the
start of each turn, the Cop probes $k$ vertices and in return receives a
distance vector. If the Cop can determine the exact location of $r$ from the
vector, the Robber is located and the Cop wins. Otherwise, the Robber is
allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The
Cop then again probes $k$ vertices. The game continues in this fashion, where
the Cop wins if the Robber can be located in a finite number of turns. The
localization number $\zeta(G)$, is defined as the least positive integer $k$
for which the Cop has a winning strategy irrespective of the moves of the
Robber. In this paper, we focus on the game played on Cartesian products. We
prove that $\zeta( G \square H) \geq \max\{\zeta(G), \zeta(H)\}$ as well as
$\zeta(G \square H) \leq \zeta(G) + \psi(H) - 1$ where $\psi(H)$ is a doubly
resolving set of $H$. We also show that $\zeta(C_m \square C_n)$ is mostly
equal to two.
