Decomposing Cubes into Smaller Cubes

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.