This article is about the type of mathematical proof. For the method of calculating limits, see Method of exhaustion. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the

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 proved separately. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true A proof by exhaustion contains two stages:

• A proof that the cases are exhaustive; i. e. , that each instance of the statement to be proved matches the conditions of (at least) one of the cases.
• A proof of each of the cases.

In contrast, the method of exhaustion of Eudoxus of Cnidus was a geometrical and essentially rigorous way of calculating mathematical limits. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

Example

To prove that every integer that is a perfect cube is either a multiple of 9, or 1 more, or 1 less than a multiple of 9. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times

Proof:
Each cube number is the cube of some integer n. This integer n is either a multiple of 3, or 1 more or 1 less than a multiple of 3. So these 3 cases are exhaustive:

• Case 1: If n = 3p, then n³ = 27p³, which is a multiple of 9.
• Case 2: If n = 3p+1, then n³ = 27p³+27p²+9p+1, which is 1 more than a multiple of 9. For instance, if n = 4 then n³ = 64 = 9x7+1.
• Case 3: If n = 3p−1, then n³ = 27p³−27p²+9p−1, which is 1 less than a multiple of 9. For instance, if n = 5 then n³ = 125 = 9x14−1.

How many cases?

There is no upper limit to the number of cases allowed in a proof by exhaustion. Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving an endgame puzzle in chess might involve considering a very large number of possible positions in the game tree of that problem. In Chess, the endgame (or end game or ending) refers to the stage of the game when there are few pieces left on the board 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 Chess is a recreational and competitive Game played between two players. If you're looking for game tree as it's used in game theory (not combinatorial game theory please see Extensive form game.

The first proof of the four colour theorem was a proof by exhaustion with 1,936 cases. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

Mathematicians prefer to avoid proofs with large numbers of cases. Such proofs feel inelegant to them. A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. Other types of proofs -- such as proof by induction (mathematical induction) -- are considered more elegant. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that However, there are some important theorems for which no other method of proof has been found.

As well as the four colour theorem, other examples of large proofs by exhaustion are:

• The proof that there is no finite projective plane of order 10. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics
• The classification of finite simple groups. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.
• The Kepler conjecture. In Mathematics, the Kepler conjecture is a Conjecture about Sphere packing in three-dimensional Euclidean space.

See also

• Case analysis
• Computer-assisted proof

Case analysis is one of the most general and applicable methods of analytical thinking depending only on the division of a problem decision or situation into a sufficient number of separate A computer-assisted proof is a Mathematical proof that has been at least partially generated by computer

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