 x Problem Encyclopedia
A problem is an obstacle, impediment, difficulty or challenge, or any situation that invites resolution; the resolution of which is recognized as a solution or contribution toward a known purpose or goal. A problem implies a desired outcome coupled with an apparent deficiency, doubt or inconsistency that prevents the outcome from taking place.

## Problem solving

Generally, an answer is a reply to a question or is a solution, a retaliation, or a response that is relevant to the said question.In law, an answer was originally a solemn assertion in opposition to some one or something, and thus generally any counter-statement or defense, a reply to a question...

or solution. Trying to find a solution to a problem is known as problem solving
Problem solving
Problem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Consideredthe most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of...

. There are many standard techniques for problem solving, such as Proof by Contradiction, or Proof by Exhaustion
Proof by exhaustion
Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds...

, the latter famously being used in the solution to the Thirty-Six Officers Problem
Thirty-six officers problem
The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782.The problem asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 × 6 square so that no rank or regiment will be repeated in any row or column. Such an...

posed by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

. A problem is a gap between an actual and desired situation. The time it takes to solve a problem is a way of measuring complexity
Complexity
In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...

. Many problems have no discovered solution and are therefore classified as an open problem
Open problem
In science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved . Some questions remain unanswered for centuries before solutions are found...

.

From the mid 20th century, the field of theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

has explored the use of computers to solve problems.

## Examples

• Mathematical problem
Mathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...

is a question about mathematical objects and structures that may require a distinct answer or explanation or proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

. Examples include word problems
Word problem (mathematics education)
In mathematics education, the term word problem is often used to refer to any math exercise where significant background information on the problem is presented as text rather than in mathematical notation...

at school level or deeper problems such as shading a map with only four colours
Four color theorem
In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...

.
• In society
Society
A society, or a human society, is a group of people related to each other through persistent relations, or a large social grouping sharing the same geographical or virtual territory, subject to the same political authority and dominant cultural expectations...

, a problem can refer to particular social issues, which if solved would yield social benefits, such as increased harmony or productivity, and conversely diminished hostility and disruption.
A business is an organization engaged in the trade of goods, services, or both to consumers. Businesses are predominant in capitalist economies, where most of them are privately owned and administered to earn profit to increase the wealth of their owners. Businesses may also be not-for-profit...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, a problem is a difference between actual conditions and those that are required or desired. Often, the causes of a problem are not known, in which case root cause analysis
Root cause analysis
Root cause analysis is a class of problem solving methods aimed at identifying the root causes of problems or events.Root Cause Analysis is any structured approach to identifying the factors that resulted in the nature, the magnitude, the location, and the timing of the harmful outcomes of one...

is employed to find the causes and identify corrective actions.
• In chess
Chess problem
A chess problem, also called a chess composition, is a puzzle set by somebody using chess pieces on a chess board, that presents the solver with a particular task to be achieved. For instance, a position might be given with the instruction that White is to move first, and checkmate Black in two...

, a problem is a puzzle set by somebody using chess pieces on a chess board, for others to get instruction or intellectual satisfaction from determining the solution.
• In theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

, there is what is referred to as the Synoptic Problem, regarding the Gospels' relationship to each other.
Discourse
Discourse generally refers to "written or spoken communication". The following are three more specific definitions:...

a problem is a challenge to an assumption, an apparent conflict that requires synthesis and reconciliation. It is a normal part of systematic thinking, the address of which adds to or detracts from the veracity of a conclusion or idea.
• An optimization problem
Optimization problem
In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...

is finding the best solution from all feasible solutions. A good example of this type of problem is the travelling salesperson problem which is based on calculating the most efficient route between many places
• In computability theory
Computability theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

a decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

• In rock climbing
Rock climbing
Rock climbing also lightly called 'The Gravity Game', is a sport in which participants climb up, down or across natural rock formations or artificial rock walls. The goal is to reach the summit of a formation or the endpoint of a pre-defined route without falling...

a problem is a series of rock
Rock (geology)
In geology, rock or stone is a naturally occurring solid aggregate of minerals and/or mineraloids.The Earth's outer solid layer, the lithosphere, is made of rock. In general rocks are of three types, namely, igneous, sedimentary, and metamorphic...

s that forces the climber to climb.
Reading is a complex cognitive process of decoding symbols for the intention of constructing or deriving meaning . It is a means of language acquisition, of communication, and of sharing information and ideas...

, a problem is a combination of a series of words with the overall plotline, which the reader must attempt to decipher.
• In walking
Walking
Walking is one of the main gaits of locomotion among legged animals, and is typically slower than running and other gaits. Walking is defined by an 'inverted pendulum' gait in which the body vaults over the stiff limb or limbs with each step...

, a mobility problem is presented. Motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

is achieved via mechanical interaction of the legs and a surface.
• Popular hip-hop/R&B artist Jay-Z claims to have 99 problems, of which he reports precisely 0 are women. (By comparison, similarly popular hip-hop/R&B artist Kid Cudi reports that, of his own 99 problems, precisely 99 of them are women.)

• Heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

• How to Solve It
How to Solve It
How to Solve It is a small volume by mathematician George Pólya describing methods of problem solving.- Four principles :How to Solve It suggests the following steps when solving a mathematical problem:...

• Question
Question
A question may be either a linguistic expression used to make a request for information, or else the request itself made by such an expression. This information may be provided with an answer....

• Problem-based learning
Problem-based learning
Problem-based learning is a student-centered pedagogy in which students learn about a subject in the context of complex, multifaceted, and realistic problems...

• Problematization
• Worry
Worry
Worry is thoughts, images and emotions of a negative nature in whichmental attempts are made to avoid anticipated potential threats. As an emotion it is experienced as anxiety or concern about a real or imagined issue, usually personal issues such as health or finances or broader ones such as... 