Possible counts in homothetic cube tilings

Let c(n) be the threshold above which every integer can occur as the number of positive homothetic cubes tiling the unit n-cube. Must c(n) be bounded below by a fixed positive multiple of nⁿ in all dimensions?

Why it matters and what counts as progress

Why it matters: A sub-nⁿ threshold along any dimension sequence would refute a uniform lower-growth prediction without determining all dimensions.

What counts as progress: Give constructions with c(n)=o(nⁿ) along a dimension sequence, or sharpen upper and lower thresholds in other dimensions.

How the problem developed

14 Jul 2026 · Counterexample [1][2]
Star Fleet Reported that for every odd n≥201, all counts k≥n·2ⁿ·ceil(49n/100)ⁿ+2 occur, implying c(n)=o(nⁿ) along odd dimensions and refuting the proposed universal lower bound.

AI contribution

Star Fleet reports a counterexample to Erdős Problem 769’s lower bound

Star Fleet's construction shows that for every odd n at least 201, every integer k at least n·2^n·ceil(49n/100)^n+2 is the count of a tiling of the unit n-cube by positive homothetic cubes. This gives c(n)=o(n^n) along odd dimensions and refutes the proposed universal lower bound c(n)≫n^n; it does not determine the best bounds for c(n) in all dimensions.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationFormal (Lean 4)

Human understandingFew