Unsolvable

An unsolvable problem is one for which no algorithm can determine the correct answer for every possible input. In computability theory this is described as undecidable, or uncomputable in a broader sense. The notion contrasts with problems that are solvable in principle, even if their solutions are difficult to find. Some problems are solvable only for restricted cases, while others admit no general method at all.

Canonical examples include the halting problem: given an arbitrary computer program and its input, determine whether the program eventually stops. Alan Turing showed that no general algorithm can solve this for all program-input pairs. Related undecidable questions arise in mathematical logic, such as the Entscheidungsproblem, and in number theory, as Hilbert's tenth problem, which asks whether Diophantine equations have integer solutions.

Gödel's incompleteness theorems illustrate a related limitation: in any consistent, sufficiently strong formal system, there are true statements that cannot be proven within the system, which implies certain questions cannot be settled by proof alone. In practice, the label unsolvable is used when no procedure exists to resolve all instances of a problem, whereas open or incomplete problems may simply lack a known solution.

Notes and related concepts include undecidable problems, computability, and uncomputability. In everyday language, "unsolvable" may also describe puzzles or tasks with no possible solution under given constraints.