problems

A problem is a situation in which a goal is not currently attained and constraints make the path to the goal nontrivial. Problems can range from simple to complex and from well-defined to ill-defined. They typically involve a discrepancy between the current state and a desired state, requiring analysis and action to reach the goal.

Problems are commonly classified by the nature of the task. Decision problems ask whether a given condition can be satisfied. Optimization problems seek to maximize or minimize a value under constraints. Search problems require finding an object that meets criteria. Constraint satisfaction problems involve configuring variables to satisfy a set of constraints. In mathematics and computer science, problems are further described as well-posed when solutions exist, are unique, and respond to data changes predictably; ill-posed problems lack one or more of these properties. Many problems are computationally hard or intractable, meaning no efficient general solution is known.

Problem solving typically follows a process such as understanding the problem, devising a plan, carrying out the plan, and reviewing results. Methods include algorithms, which guarantee a solution when applicable, and heuristics, which offer practical approximations. Framing and reframing the problem can be crucial, and collaboration often aids in generating effective strategies.

Problems appear across domains, including science, engineering, economics, and everyday life, from designing a product within budget to proving a theorem or organizing a schedule. Puzzles are a subset of problems with a clearly defined path to a unique solution.

Etymology: the word problem derives from Latin problema, from Greek problemma, meaning “a thing put forward” or “a question.”