Symmetric Layer-Rainbow Colorations of Cubes

Abstract

Can we color the $n^3$ cells of an $n\times n\times n$ cube $L$ with $n^2$ colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that $L_{ij\ell}=L_{j\ell i}=L_{\ell ij}$ for distinct $i,j,\ell \in \{1,\dots,n\}$, and $L_{iij}=L_{jj i}, L_{iji}=L_{jij}, L_{ij j}=L_{jii}$ for $i,j\in \{1,\dots,n\}$? Using transportation networks, we show that such a coloring is possible if and only if $n\equiv 0,2 \mod 3$ (with two exceptions, $n=1$ and $n\neq 3$). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940's. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.