  We consider the complexity of problems related to the combinatorial game
Free-Flood-It, in which players aim to make a coloured graph monochromatic with
the minimum possible number of flooding operations. Our main result is that
computing the length of an optimal sequence is fixed parameter tractable (with
the number of colours present as a parameter) when restricted to rectangular
2xn boards. We also show that, when the number of colours is unbounded, the
problem remains NP-hard on such boards. This resolves a question of Clifford,
Jalsenius, Montanaro and Sach (2010).
