Symmetric Layer-Rainbow Colorations of Cubes

TL;DR

This paper investigates symmetric colorings of an n×n×n cube with n^2 colors, establishing existence conditions based on modular arithmetic and connecting to combinatorial designs, orthogonal arrays, and Latin squares.

Contribution

It provides a complete characterization of when such symmetric colorings exist, using transportation networks, extending classical combinatorial design theory.

Findings

Colorings exist if and only if n ≡ 0,2 mod 3, with exceptions n=1 and n≠3.

The study links symmetric cube colorings to orthogonal arrays and Latin squares.

Transportation networks are used to prove the existence conditions.

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.