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Proof by exhaustion, also known as 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. A proof by exhaustion contains two stages: In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
- 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 way of finding the area or volume of a shape that is not easily defined in terms of traditional shapes. ...
Eudoxus of Cnidus (Greek Εύδοξος) (410 or 408 BC - 355 or 347 BC) was a Greek astronomer, mathematician, physician, scholar and friend of Plato. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ...
Example
To prove that every cube number is either a multiple of 9 or is 1 more or 1 less than a multiple of 9. In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ...
Proof
Each cube number is the cube of some integer n. This integer is either a multiple of 3, or is 1 more or 1 less than a multiple of 3. So the following 3 cases are exhaustive:
- Case 1: If n is a multiple of 3 then the cube of n is a multiple of 27, and so certainly a multiple of 9.
- Case 2: If n is 1 more than a multiple of 3 then the cube of n is 1 more than a multiple of 9.
- Case 3: If n is 1 less than a multiple of 3 then the cube of n is 1 less than a multiple of 9.
[To complete the proof, the claims in cases 2 and 3 can be proved using simple algebra.]
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. ...
Sam Loyd, London Era, 1861 Excelsior by Sam Loyd. ...
Chess is an abstract strategy board game for two players. ...
In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. ...
The first proof of the four colour theorem was a proof by exhaustion with 1,936 cases. 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. Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ...
Mathematicians prefer to avoid proofs with large numbers of cases because these proofs feel inelegant—they leave an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. 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:
Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...
In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ...
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