  Maximal green sequences are important objects in representation theory,
cluster algebras, and string theory. It is an open problem to determine what
lengths are achieved by the maximal green sequences of a quiver. We combine the
combinatorics of surface triangulations and the basics of scattering diagrams
to address this problem. Our main result is a formula for the length of minimal
length maximal green sequences of quivers defined by triangulations of an
annulus or a punctured disk.
