In 1936, Alan Turing proved that certain problems lie permanently beyond algorithmic reach — the halting problem demonstrates that no program can universally predict whether another program will stop running. This foundational result establishes a hard boundary for what computation can achieve, decades before physical computers existed.

From impossibility, Tim Roughgarden pivots to efficiency. Dijkstra's algorithm lets mapping applications find shortest routes without checking every path, while Karatsuba's multiplication outperforms grade-school arithmetic. These shortcuts suggest clever algorithms might tame any tractable problem — until the Traveling Salesman Problem reveals a deeper barrier. Despite resembling shortest-path routing, TSP has resisted every attempt at a fast solution.

This resistance birthed NP-completeness: thousands of problems across scheduling, puzzles, and network optimization are computationally equivalent. A fast algorithm for one solves all; proof that any one is inherently hard condemns them all. The P versus NP question, now among mathematics' greatest open problems, emerged from converging traditions in logic and algorithm design.

Roughgarden's course at the Institute for Advanced Study traces this arc without requiring prior background. The stakes extend beyond theory: cryptography assumes certain problems are hard, AI progress depends on tractable optimization, and quantum computing's promise hinges on where the boundary actually falls. Understanding computation's limits is not academic — it shapes what systems we can trust and build.