Erdős Problem #769

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.