Tim Roughgarden, a professor at the Institute for Advanced Study, opened a course that asks what computers cannot do, referencing Alan Turing's 1936 work on the halting problem. The course traces the theoretical foundations of computer science, showing that some problems are unsolvable regardless of resources and introducing algorithmic shortcuts that enable efficient solutions for many problems. Examples include Dijkstra's algorithm for finding shortest routes and Karatsuba's multiplication method, which improve performance over naïve approaches. The Traveling Salesperson Problem is presented as an NP‑complete challenge that resists fast algorithms, illustrating the class of problems that are difficult to solve quickly. The lecture discusses the P versus NP question, its historical development involving Hilbert, Gödel, and von Neumann, and its potential impact on cryptography, artificial intelligence, quantum computing, and the nature of computation itself. No prior technical background is required. Materials such as lectures and a chapter index are available online, and the discussion concludes by emphasizing that while many problems admit efficient algorithms, others remain fundamentally hard, leaving the P versus NP question unresolved and shaping future research directions.