In a joint work with Orson and Park we proved that the number of crossing changes needed to reduce a link to a homotopy trivial link is controlled by the linking number together with the number of components. This is surprising, since it means that the higher order Milnor invariants needed to classify link homotopy have only a bounded impact on the gordian distance up to link homotopy. In a more recent collaboration with Bosman, Martin, Otto, and Vance, we make this control explicit. We prove that the number of crossing changes needed to reduce an n-component link to a homotopy trivial link is roughly the sum of the linking numbers plus n^2. Along the way we will introduce an interesting question in extremal graph theory, and introduce some natural (and accessible) variants on this problem.