  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.
