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 Erdos, Hudelson, and Meier.