When it comes to research design, the literature largely focuses on prescriptions for strengthening causal inference--identifying confounders, selecting appropriately comparable cases, generating testable hypotheses, etc. This approach to research design features the logic of inquiry heavily, but overlooks the logic of discovery. The logic of discovery is a stage in the research design process that is often bracketed off as an unexplainable moment of inspiation, and is largely missing from the research design literature. We address this gap by advancing a puzzle-based approach to the discovery phase in the life cycle of a research project. We develop a typology of empirical, theoretical, and methodological puzzles, and provide guidance for identifying research questions in each. We argue that not only can the discovery phase of a research design be thought of--and taught--in a systematic way, but that a puzzle-based approach to developing research questions can structure subsequent decisions, such as case selection, identifying the appropriate theoretical framework, and concept formation. We conclude that a puzzle-based approach is a useful way to structure research design, both for practitioners and as a pedagogical tool.