  We consider the problem of constructing Latin cubes subject to the condition
that some symbols may not appear in certain cells. We prove that there is a
constant $\gamma > 0$ such that if $n=2^k$ and $A$ is $3$-dimensional $n\times
n\times n$ array where every cell contains at most $\gamma n$ symbols, and
every symbol occurs at most $\gamma n$ times in every line of $A$, then $A$ is
{\em avoidable}; that is, there is a Latin cube $L$ of order $n$ such that for
every $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not
appear in the corresponding cell of $A$.
