We propose an approach for showing rationality of an algebraic variety X. We try to cover X by rational curves of certain type and count how many curves pass through a generic point. If the answer is 1, then we can sometimes reduce the question of rationality of X to the question of rationality of a closed subvariety of X. This approach is applied to the case of the so-called Ueno-Campana manifolds. Assuming certain conjectures on curve counting, we show that the previously open cases X4,6 and X5,6 are both rational. Our conjectures are evidenced by computer experiments. In an unexpected twist, existence of lattices D6, E8, and Λ10 turns out to be crucial.