Let $H$ be a finite collection of hyperplanes in $\mathbb{R}^d$ that covers all the points in $\{ 0,1,2 \}^d$ other than the origin $(0,0,\dots, 0)$ which is not covered. Show that $H$ is at least size $2d$.
Alon and Füredi have proven that covering the vertices of an $n$-cube except at one vertex, i.e. covering $\{0,1\}^n$ except the origin, requires at least $n$ hyperplanes here. However, I'm having trouble understanding the proof, and I'm also not sure whether a similar approach applies.