For each integer n≥ 0, determine whether there exists a bijection A:0,1^n→1,…,2^n such that, for every pair of distinct coordinates i,j∈1,…,n and every choice of the remaining coordinates, the face sum ∑_(varepsilon_i,varepsilon_j)∈0,1^2A(x_1,…,x_i-1,varepsilon_i,x_i+1,…,x_j-1,varepsilon_j,x_j+1,…,x_n) is independent of the chosen 2-dimensional face. Classify all such arrangements up to the natural transformations described in the source. The cited paper claims existence for every n and uniqueness up to transformations of size 2^n(n+1)! when n is even and 2^n n,n! when n is odd.

Coxeter’s question on most-perfect magic squares and magic-faced hypercubes

For each integer $n\ge 0$ , determine whether there exists a bijection $A:\{0,1\}^n\to\{1,\ldots,2^n\}$ such that, for every pair of distinct coordinates $i,j\in\{1,\ldots,n\}$ and every choice of the remaining coordinates, the face sum $\sum_{(\varepsilon_i,\varepsilon_j)\in\{0,1\}^2}A(x_1,\ldots,x_{i-1},\varepsilon_i,x_{i+1},\ldots,x_{j-1},\varepsilon_j,x_{j+1},\ldots,x_n)$ is independent of the chosen $2$ -dimensional face. Classify all such arrangements up to the natural transformations described in the source. The cited paper claims existence for every $n$ and uniqueness up to transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\,n!$ when $n$ is odd.

References

Primary source

arXiv

Additional references

The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes — arXiv — Manjul Bhargava

combinatorics finite-geometry

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new paper claims to classify these magic structures in every dimension, including the previously known four-by-four case, but the claim has not been independently verified.

Coxeter’s question asks for the existence and classification of most-perfect magic squares and their higher-dimensional magic-faced analogues. Manjul Bhargava’s paper claims a complete classification, with uniqueness understood modulo the natural transformation groups it specifies.

Known results

Stewart, 1999: most-perfect squares are pandiagonal, every $2\times2$ subsquare has the magic sum, and entries at the relevant diagonal separation sum to half the magic constant; the reported enumeration includes $48$ inequivalent $4\times4$ squares and $368640$ inequivalent $8\times8$ squares.

September 22, 2026 classification

On September 22, 2026, Manjul Bhargava’s arXiv paper reported a classification of order-two magic-faced hypercubes and their higher-dimensional Weyl-group orbit structure, recovering the $4\times4$ case. This is a claimed resolution, but independent verification is not recorded.

Current status (as of September 2026): Bhargava’s paper claims to settle the $4\times4$ question and classify all higher-dimensional cases, but the resolution remains unverified.

Sources

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