  Consider a game played on a simple graph $G = (V, E)$ where each vertex
consists of a clickable light. Clicking any vertex $v$ toggles the on/off state
of $v$ and its neighbors. Starting from an initial configuration of lights, one
wins the game by finding a solution: a sequence of clicks that turns off all
the lights. When $G$ is a $5 \times 5$ grid, this game was commercially
available from Tiger Electronics as Lights Out. Restricting ourselves to
solvable initial configurations, we pose a natural question about this game,
the Most Clicks Problem (MCP): How many clicks does a worst-case initial
configuration on $G$ require to solve? The answer to the MCP is already known
for nullity 0 graphs: those on which every initial configuration is solvable.
Generalizing a technique from Scherphius, we give an upper bound to the MCP for
all grids of size $(6k - 1) \times (6k - 1)$. We show the value given by this
upper bound exactly solves the MCP for all nullity 2 grids of this size. We
conjecture that all nullity 2 grids are of size $(6k - 1) \times (6k - 1)$,
which would mean we solve the MCP for all nullity 2 square grids.
