  We establish a criterion for the (failure of) the containment $I^{(m)}\subset
I^r$ for 3-generated ideals $I$ defining reduced sets of points in
$\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutions
of the powers of $I$, specifically the minimal free resolutions for $I^m$ and
$I^r$. We apply this criterion to two point configurations that have recently
arisen as counterexamples to a question of B. Harbourne and C. Huneke: the
Fermat configuration and the Klein configuration.
