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Despite the fact that latin cubes have been studied since in the 1940's,there are only a few results on embedding partial latin cubes, and all theseresults are far from being optimal with respect to the size of the containingcube. For example, the bound of the 1970's result of Cruse that a partial latincube of order $n$ can be embedded into a latin cube of order $16n^4$, was onlyimproved very recently by Potapov to $n^3$. In this note, we prove the firstsuch optimal result by showing that a layer-rainbow latin cube of order $m$ canbe embedded into a layer-rainbow latin cube of order $n$ if and only if $n\geq2m$. A layer-rainbow latin cube $L$ of order $n$ is an $n\times n\times n$array filled with $n^2$ symbols such that each layer parallel to each face(obtained by fixing one coordinate) contains every symbol exactly once.

On Layer-Rainbow Latin Cubes Containing Layer-Rainbow Latin Cubes