Name a positive integer $n$ nice if a square can be divided into $n$ smaller squares. The smaller squares do not need to be of the same size. Since you can always divide a square into $4$ smaller squares it immediately follows, that if $n$ is nice $n+3$ has to be aswell. Since $6, 7$ and $8$ are nice all natural numbers greater than $8$ have to be nice.
This got me thinking about the same problem in higher Dimensions. Let $n_d$ be nice if it divides a Hypercube in $d$ Dimensions into $n_d$ smaller Hypercubes.
Does for all Dimensions $d$ exist a $n_d$ such that all numbers greater than $n_d$ are nice? Is there a simple way to determine wether a number is nice in $d$ Dimensions or not?