The original version of this story appeared in Quanta Magazine.
Algorithms have become ubiquitous. They optimize our commutes, process payments, and coordinate the flow of internet traffic. It seems that for every problem that can be articulated in precise mathematical terms, there’s an algorithm that can solve it, at least in principle.
But that’s not the case—some seemingly simple problems can never be solved algorithmically. The pioneering computer scientist Alan Turing proved the existence of such “uncomputable” problems nearly a century ago, in the same paper where he formulated the mathematical model of computation that launched modern computer science.
Turing proved this groundbreaking result using a counterintuitive strategy: He defined a problem
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