  We consider a combinatorial question about searching for an unknown ideal
$\mu$ within a known poset $\lambda$. Elements of $\lambda$ may be queried for
membership in $\mu$, but at most $k$ positive query results are permitted. The
goal is to find a search strategy which guarantees a solution in a minimal
total number $m_k(\lambda)$ of queries. We provide tight bounds for
$m_k(\lambda)$, and construct optimal search strategies for the case where
$k=2$ and $\lambda$ is the product poset of totally ordered finite sets, one of
which has cardinality not more than six.
