Decomposing cubes into smaller cubes

Abstract

We explore the decomposition of n-dimensional cubes into smaller n-dimensional cubes. Let c(n) be the smallest integer such that if $k\ge c(n)$ then there is a decomposition of the n-dimensional cube into k smaller n-dimensional cubes. We prove that $c(n)\ge 2^{n+1}-1$ for $n\ge 3$, improving on Hadwiger’s result that $c(n)\ge 2^n+2^{n-1}$. We also show $c(n)\le e^2n^n$ if $n+1$ is not prime and $c(n)\le 1.8n^{n+1}$ if $n+1$ is prime, improving on upper bounds proven by Erdös, Hudelson, and Meier.