  In this paper, we use the notion of twisted subgroups (i.e., subsets of group
elements closed under the binary operation $(a,b) \mapsto aba$) to provide the
first structural characterization of optimal play in the Explorer-Director
game, introduced as the Magnus-Derek game by Nedev and Muthukrishnan and
generalized to finite groups by Gerbner. In particular, we reduce the game to
the problem of finding the largest proper twisted subgroup, and as a corollary
we resolve the Explorer-Director game completely for all nilpotent groups.
