  We often rely on censuses of triangulations to guide our intuition in
$3$-manifold topology. However, this can lead to misplaced faith in conjectures
if the smallest counterexamples are too large to appear in our census. Since
the number of triangulations increases super-exponentially with size, there is
no way to expand a census beyond relatively small triangulations; the current
census only goes up to $10$ tetrahedra. Here, we show that it is feasible to
search for large and hard-to-find counterexamples by using heuristics to
selectively (rather than exhaustively) enumerate triangulations. We use this
idea to find counterexamples to three conjectures which ask, for certain
$3$-manifolds, whether one-vertex triangulations always have a "distinctive"
edge that would allow us to recognise the $3$-manifold.
