P vs NP & Computational Complexity
Stephen Cook / Leonid Levin · 1971
"Cook and Levin formalized the most important open question in computer science: if a solution can be verified quickly (NP), can it also be found quickly (P)? Most experts believe no — and that gap explains why many optimization, routing, and scheduling problems have no efficient perfect solution."
In 1971 Cook isolated a class of problems where checking an answer is easy but finding it seems hard — and proved they are all secretly the same problem. If you solve one fast, you solve them all.
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Turing Machines & the Halting Problem
"Turing proved that no general algorithm can decide in advance whether an arbitrary program will ever halt — some questions are not just hard, they are uncomputable, which sets a hard limit on what automation can ever guarantee."