There is an algorithm that give us cuboids in $\mathbb{R}^3$, say $Q_1,Q_2,\ldots$, such that $\cup_{i=1}^{\infty} Q_i$ is the simplex with vertices $(0,0,0), (1,0,0) , (0,1,0), (0,0,1)$, and the $Q_i$'s are almost disjoint (i.e $\lambda(Q_i\cap Q_j)=0$ if $i\neq j$)?
In $\mathbb{R}^2$ there are many easy ways to fill a triangle with almost disjoint-rectangles but I had not find the way to generalize this to higher dimensions. Do you have any ideas?