  Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all
but one point at least $k$ times? Problems of this nature have been studied for
decades, with a general lower bound having been established by Ball and Serra.
We solve this problem for various types of grids, in particular showing the
tightness of the Ball--Serra bound when one side is much larger than the other.
In other cases, we prove new lower bounds that improve upon Ball--Serra and
provide an asymptotic answer for almost all grids. For the standard grid
$\{0,\ldots,n-1\} \times \{0,\ldots,n-1\}$, we prove nontrivial upper and lower
bounds on the number of lines needed. To prove our results, we combine linear
programming duality with some combinatorial arguments.
