The tetrahedron defined by $x\geq 0,y\geq 0,z\geq 0$ and $x+y+z\leq 1$ is going to be cutted on $n$ pieces of equal volume using parallel planes to $x+y+z=1$.
Where do the cuts need to be made?
I think that I need to integrate this region, but ... I don't see how to work with the cuts so that each of the $n$ pieces has the same volume
Start from $(0,0,0)$ and make the $k^{th}$ cut through the points $(\sqrt[3]{k/n}, 0,0)$, $(0, \sqrt[3]{k/n}, 0)$, and $(0,0,\sqrt[3]{k/n})$, for $k = 1, \ldots, n-1$.
The volume of the tetrahedron determined by the $k^{th}$ cut is $\frac{1}{6} (\sqrt[3]{k/n})^3 = \frac{k}{6n} = \frac{k}{n}V$, where $V$ is the volume of the initial tetrahedron. The piece between the $k^{th}$ and $(k+1)^{th}$ cut will have volume $V/n$.