  In this paper, we consider the following problem: what is the minimum number
of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0,
1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is
uncovered? The $k=1$ case is the well-known Alon-F\"uredi theorem which says a
minimum of $n$ affine hyperplanes is required, proved by the Combinatorial
Nullstellensatz.
  We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset
sums, and completely solve the fractional version of this problem, which also
provides an asymptotic answer to the integral version for fixed $n$ and $k
\rightarrow \infty$. We also use a Punctured Combinatorial Nullstellensatz
developed by Ball and Serra, to show that a minimum of $n+3$ affine hyperplanes
is needed for $k=3$, and pose a conjecture for arbitrary $k$ and large $n$.

Combinatorics