I have a three-dimensional vector space over integers: $\mathbb{Z}^3$.
I want to split it up into $n$ partitions. For $n=2$ I would simply split it into two sets, one where $x$ just has even numbers,and one where $x$ is odd. But how to do it for $n > 2$? I need some generalization, but lack the math skills. Also just $x$ and $y$ should be used for partitioning, all possibilities for, lets say $(1,1,z)$ should be in one set.