For the least cutoff $c(n)$ after which every $k$ occurs as the number of homothetic cubes in a decomposition of the unit $n$-cube, is $c(n) \gg n^n$? The Lean proof shows $c(n) = o(n^n)$ along odd dimensions.

Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ — in particular, is it true that $c(n)\gg n^n$?

The c(n) \gg n^n conjecture is false: for odd n, one can tile the unit n-cube into k homothetic cubes for every k ≥ U(n) with U(n)/n^n → 0, so c(n)/n^n → 0 along odd dimensions and no absolute constant lower-bounds it.

Determining good bounds for c(n) in general remains open.