The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes
Abstract
A magic−lined hypercube (or, simply, magic hypercube) of order k and dimension n is an arrangement of the numbers 1,…,kn in a k×⋯× 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≥ 3 and every dimension n, no magic-lined hypercube of order $2$ exists in any dimension n≥ 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,…,2n in a 2×⋯×2 (n-fold) grid \emph{magic-faced} if every 2×2 face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension n≥0, and that it is unique up to a certain natural set of transformations of size 2n(n+1)! when n is even and 2n n· n! when n is odd. As an application, we recover and generalize the classical 4×4 Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ ``most-perfect" 4×4 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(B2n).
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BibTeX
@misc{bhargava2026existence,
title = {{The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes}},
author = {Manjul Bhargava},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26762},
eprint = {2609.26762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26762},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes
AU - Manjul Bhargava
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26762
UR - https://arxiv.org/abs/2609.26762
AB - A magic−lined hypercube (or, simply, magic hypercube) of order k and dimension n is an arrangement of the numbers 1,…,k^n in a k×⋯× 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≥ 3 and every dimension n, no magic-lined hypercube of order $2$ exists in any dimension n≥ 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,…,2^n in a 2×⋯×2 (n-fold) grid \emph{magic-faced} if every 2×2 face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension n≥0, 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· n! when n is odd. As an application, we recover and generalize the classical 4×4 Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ ``most-perfect" 4×4 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).
ER -