{
  "abstract": "A $\\textit{magic-lined hypercube}$ (or, simply, $\\textit{magic hypercube}$) of order $k$ and dimension $n$ is an arrangement of the numbers $1,\\dots,k^n$ in a $k\\times\\cdots\\times k$ ($n$-fold) grid such that every line of $k$ numbers parallel to a coordinate axis has the same magic sum. While such hypercubes exist for every order $k\\ge 3$ and every dimension $n$, no magic-lined hypercube of order $2$ exists in any dimension $n\\ge 2$. For hypercubes of order $2$, we thus relax the magic condition from lines to two-dimensional faces or planes. We call an arrangement of the numbers $1,\\dots,2^n$ in a $2\\times\\cdots\\times2$ ($n$-fold) grid \\emph{magic-faced} if every $2\\times2$ face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension $n\\ge0$, and that it is unique up to a certain natural set of transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\\cdot n!$ when $n$ is odd. As an application, we recover and generalize the classical $4\\times4$ Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ ``most-perfect\" $4\\times4$ magic squares, and extend the picture to higher dimensions. In particular, we prove that, in dimension $n$, these most-perfect objects form a single orbit under a certain natural action of the Weyl group $W(B_{2n})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26762",
  "author_total": 1,
  "authors": [
    {
      "name": "Manjul Bhargava"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/49gb2m163xckf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26762",
  "fulltext_url": null,
  "id": "49gb2m163xckf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26762",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214103457",
  "pdf_url": "https://arxiv.org/pdf/2609.26762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/49gb2m163xckf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes",
  "title_text": "The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/49gb2m163xckf",
  "venue": "arXiv",
  "versions": []
}