  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 these
results are far from being optimal with respect to the size of the containing
cube. For example, the bound of the 1970's result of Cruse that a partial latin
cube of order $n$ can be embedded into a latin cube of order $16n^4$, was only
improved very recently by Potapov to $n^3$. In this note, we prove the first
such optimal result by showing that a layer-rainbow latin cube of order $m$ can
be embedded into a layer-rainbow latin cube of order $n$ if and only if $n\geq
2m$. 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.
