  We show that if $K$ is a knot in $S^3$ and $\Sigma$ is a bridge sphere for
$K$ with high distance and $2n$ punctures, the number of perturbations of $K$
required to interchange the two balls bounded by $\Sigma$ via an isotopy is
$n$. We also construct a knot with two different bridge spheres with $2n$ and
$2n-1$ bridges respectively for which any common perturbation has at least
$3n-1$ bridges. We generalize both of these results to bridge surfaces for
knots in any 3-manifold.
