  A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$
into subcubes. A subcube partition is irreducible if the only sub-partitions
whose union is a subcube are singletons and the entire partition. A subcube
partition is tight if it "mentions" all coordinates. We study extremal
properties of tight irreducible subcube partitions: minimal size, minimal
weight, maximal number of points, maximal size, and maximal minimum dimension.
We also consider the existence of homogeneous tight irreducible subcube
partitions, in which all subcubes have the same dimensions. We additionally
study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of
$\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal
size. Our constructions and computer experiments lead to several conjectures on
the extremal values of the aforementioned properties.
