Behavioural equivalences can be characterized via bisimulation, modal logics, and spoiler-duplicator games. In this paper we work in the general setting of coalgebra and focus on generic algorithms for computing the winning strategies of both players in a bisimulation game. The winning strategy of the spoiler (if it exists) is then transformed into a modal formula that distinguishes the given non-bisimilar states. The modalities required for the formula are also synthesized on-the-fly, and we present a recipe for re-coding the formula with different modalities, given by a separating set of predicate liftings. Both the game and the generation of the distinguishing formulas have been implemented in a tool called T-Beg.