  For a family $A\subseteq\left\{ 0,\dots,k\right\} ^{n}$, define the
$\delta$-shadow of $A$ to be the set obtained from $A$ by removing from any of
its vectors one coordinate that equals zero. Given the size of $A$, how should
we choose $A$ to minimise its $\delta$-shadow? Our aim in this paper is to show
that, for any $r$, the family of all sequences with at most $r$ zeros has
minimal $\delta$-shadow. We actually give the exact best $A$ for every size.
