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

Source 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 $W(B_{2n})$.