  Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we
know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the
translate $x+T$ is contained in $E$. This information is collectively called
the $k$-deck of $E$. One can naturally extend the domain of definition of the
$k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck
of a set in $G$ sufficient to determine the set up to translation? The 2-deck
is not sufficient (even when we allow for reflection of the set, which does not
change the 2-deck) and the first interesting case is $k=3$. We further restrict
$G$ to be cyclic and determine the values of $n$ for which the 3-deck of a
subset of $\ZZ_n$ is sufficient to determine the set up to translation. This
completes the work begun by Gr\"unbaum and Moore as far as the 3-deck is
concerned. We additionally estimate from above the probability that for a
random subset of $\ZZ_n$ there exists another subset, not a translate of the
first, with the same 3-deck. We give an exponentially small upper bound when
the previously known one was $O(1\bigl / \sqrt{n})$.
