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Claim. M. Hudelson, Dissecting $d$-cubes into smaller $d$-cubes, J. Combin. Theory Ser. A 81 (1998), no. 2, 190--200, proves that $c(d)=O((2d)^{d-1})$, where $c(d)$ is the least integer such that the unit $d$-cube splits into $k$ homothetic cubes for every $k\ge c(d)$, and that $c(d)<6^d$ whenever $\gcd(2^d-1,3^d-1)=1$; more generally $c(d)=O((2k)^d)$ whenever $\gcd(2^d-1,k^d-1)=1$, and the paper derives specific bounds for $d\le5$. The paper's abstract states these results, and the zbMATH review of the paper (Zbl 0891.05018) states the general bound and reports the paper's bound $c(4)\le809$. These are upper bounds on the quantity that Problem 769 asks to bound; the general bound improves by a factor of order $n$ the bound $(2^n-2)((n+1)^n-2)-1$ that Burgess and Erdős proved, which has order $2^nn^n$.

Covers. The upper bound $c(n)\ll(2n)^{n-1}$ for every $n$, the bound $c(n)<6^n$ under the stated gcd condition, and the bound $c(4)\le809$, as part of the request for good bounds. Not covered: the order of $c(n)$ and the question whether $c(n)\gg n^n$, which these upper bounds do not decide: the bound $6^n$ would answer it no if infinitely many $n$ met the gcd condition, which Erdős conjectured and which is not known.

Depends on. No page of this wiki.

Acceptance. refereed: Journal of Combinatorial Theory, Series A 81 (1998), no. 2, 190--200, February 1998. The site credits the result in its remarks on the problem, which it labels OPEN, so that remark is not acceptance and no reviewed is listed.