Solvable

Solvable is an adjective meaning capable of being solved or resolved. The word comes from the Latin solvāre, meaning to loosen, untie, or pay. In everyday language, something described as solvable is regarded as amenable to a solution or answer.

In mathematics and adjacent fields, solvable carries several precise meanings. An equation or system is solvable if it has a solution, and in many contexts solvable means that the solution can be obtained by a finite sequence of well-defined steps. A polynomial is solvable by radicals if its roots can be expressed using a finite combination of additions, multiplications, and root extractions; this is dictated by Galois theory and explains why some polynomials of degree five or higher lack expressions in radicals. In group theory, a group is called solvable if it has a finite derived series whose factor groups are abelian; this structural property connects to the solvability of polynomials in the Galois framework and has applications in algebra and geometry. In the theory of differential equations, a differential equation is solvable when its solutions can be written in closed form or obtained by quadrature, meaning through integration of known functions.

In computer science and puzzle contexts, solvability refers to the existence of an algorithm or a finite sequence of moves that leads to a desired outcome. A decision problem is solvable if an algorithm terminates with a correct yes/no answer for every input; many problems are decidable, while some are undecidable. Puzzles and games use solvability as an invariant to distinguish reachable or winnable configurations from those that are not.