Prove that the number of hyperplanes such that $$c_1x_1 + c_2 x_2 + ... + c_n x_n = 0, \pm 1, \pm 2, \pm 3, ...$$ which intersect the unit $n$-cube, $0< x_i < 1,$ is at most $$|c_1| + |c_2| + ... + |c_n|.$$
I started out by plotting small values of $c_i$ in the 3D Geogebra grapher and this is true. But I don't know how to prove this. Perhaps we could use induction with decomposing to lower-dimensional hypercubes?
First assume that each $c_i\ge 0$, and set $C:=c_1+\dots+c_n$. Consider the family of hyperplanes $$H_t=\{(x_1,\dots, x_n)\, :\, c_1x_1+\dots+c_nx_n= t\} $$ for all $t\in\Bbb R$. Note that these hyperplanes are all parallel and have the normal vector $(c_1,\dots, c_n)$.
Geometrically it's obvious, but you can also derive it algebraically, that $H_t$ doesn't meet the cube if $t\le 0$ or $t\ge C$ (in the edge cases $t=0$ and $t=C$, if all $c_i>0$, the hyperplane $H_t$ intersects the closed cube in the single corner points $(0,\dots, 0)$ and $(1,\dots, 1)$, respectively; if some $c_i=0$ we get some lower dimensional boundary faces at these edge cases, so they just miss the open cube).
Specifically, the value of $c_1x_1+\dots+c_nx_n$ is always between $0$ and $C$ for the points in the cube, as each $x_i\in (0,1)$.
It follows that among the ones with integer indices, exactly $H_0,\dots, H_{\lfloor C\rfloor}$ will intersect the closed cube, thus $H_1,\dots, H_{\lfloor C\rfloor}$ intersect the open cube if $C\notin\Bbb Z$, and exactly $H_1,\dots, H_{C-1}$ intersect it if $C\in\Bbb Z$.
For the general case, you can apply symmetries of the cube in order to deducing it to the above special case.
More explicitly, if $c_i<0$, consider the reflection through the midplane $x_i=\frac12$, that is $\phi(x_1,\dots, x_n)=(x'_1,\dots, x'_n)$ where $x'_j=x_j$ and $x_i'=(1-x_i)$.
Then $\phi(H_t)=\{(x_1,\dots, x_n) \, :\, c_1x_1+\dots -c_ix_i+\dots+c_nx_n=t-c_i\}$.
This way we swapped the sign of $c_i$ (the normal vector is also reflected), so, e.g. using induction on the number of negative coefficients, we can conclude that $t-c_i$, and thus also $t$, must lie in an open interval of length $C$.
Consequently, $\phi(H_t)$ intersects the open cube at most for $\lfloor C\rfloor$ or $C-1$ integer values of $t$.
Since the cube is invariant under $\phi$, it intersects $H_t$ iff it intersects $\phi(H_t)$.