We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang.

翻译：我们证明，对于固定的 $q$ 和 $m$，随着 $n$ 的增长，将超立方体 ${\bf Z}_q^n$ 划分为 $q^m$ 个维度为 $n-m$ 的子立方体的划分数目渐近等于 $n^{(q^m-1)/(q-1)}$。在证明中，我们引入了星矩阵的爆炸运算，并证明了除分形星矩阵外，任何星矩阵在某种爆炸下都是可扩展的，而分形星矩阵在任何爆炸下仍保持为分形。