The set $\{0, 1\}^n$ in $\mathbb{R}^n$ can be covered by just two hyperplanes, but as soon as we require that one point of this binary grid should not be covered, the minimum number of hyperplanes required to cover the rest is $n$. This is an important result of Alon and Furedi and all of its known proofs are based on the polynomial method. In this talk, we will discuss several variations of such grid-covering problems and some recent results. These problems have ties with various areas like coding theory, finite geometry, and computer science, and they often require an equally diverse set of techniques to tackle them.