I found this question but I think I don't understand the question properly. The author asks to describe (in coordinates) the faces of the intersection of a cube $C$ of $r$-dimensions, $C=\{0\leq p_{s}\leq 1|s=1,...,r\}$ with a hyperplane $p_{1}+...+p_{r}=\frac{r}{2}$.
If $r=1$ then I have a line segment from 0 to 1 and the hyperplane corresponds to $p_{1}=1/2$ which is mid-point of this line segment. If $r=2$ then $C$ is a unit square with one of its vertex at the origin while the hyperplane is $p_{1}+p_{2}=1$ which is a line segment from the (0,1) to the point (1,0) and it bisects the square. And so on.
What I don't understand is how to "describe the faces of the intersection". Basically, I don't understand what I have to prove. Any kind of hint would be helpful. Thanks.