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Humans rely on a finely tuned ability to recognize and adapt to socially relevant patterns in their everyday face-to-face interactions. This allows them to anticipate the actions of others, coordinate their behaviors, and create shared meaning-to communicate. Social robots must likewise be able to recognize and perform relevant social patterns, including interactional synchrony, imitation, and particular sequences of behaviors. We use existing empirical work in the social sciences and observations of human interaction to develop nonverbal interactive capabilities for a robot in the context of shadow puppet play, where people interact through shadows of hands cast against a wall. We show how information theoretic quantities can be used to model interaction between humans and to generate interactive controllers for a robot. Finally, we evaluate the resulting model in an embodied human-robot interaction study. We show the benefit of modeling interaction as a joint process rather than modeling individual agents.