The Game of Twenty Questions is a classical problem where a questioner must identify a hidden item by asking binary questions. Although simple in form, the game is strategically challenging, especially when the questions are expressed in natural language. It is also a useful abstraction for realistic information-seeking tasks, such as medical diagnosis and troubleshooting. Existing approaches often rely on simplifying assumptions that degrade worst-case performance. This is an issue with serious implications in high-stakes applications. In this work, we introduce Strategic Language Search (SLS), an adversarial counterpart of Twenty Questions, and formalize it and its variants as two-player zero-sum extensive-form games. We then propose Game of Thought (GoT), a framework that applies game-theoretic techniques to approximate a Nash equilibrium (NE) strategy for the restricted variant of the game. Empirical results demonstrate that our approach consistently improves worst-case performance compared to (1) direct prompting-based methods and (2) heuristic-guided search methods across all tested settings.