Dissectingd-Cubes into Smallerd-Cubes

In this paper, we explore the following question: Given integersdandk, is it possible to subdivide ad-dimensional cube intoksmallerd-dimensional cubes? In particular, we investigate bounds on the integerc(d) which is the smallest integer for which it is possible to subdivide thed-cube into any numberk⩾c(d) smallerd-cubes. We derive specific bounds ford⩽5, and furthermore, we investigate, for givenk, the asymptotic behavior ofc(d) for thosedsuch that gcd(2d−1, kd−1)=1. Specifically, we show that if gcd (2d−1, 3d−1) thenc(d)<6dand that if gcd(2d−1, kd−1) thenc(d)=O((2k)d). Finally, we derive the general asymptotic boundc(d)=O((2d)d−1) which improves the currently known bound ofc(d)=O((2d)d).