Proof by exhaustion

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This article is about the type of mathematical proof. For the method of calculating limits, see Method of exhaustion.

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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. 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.

[edit] 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.

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.

[edit] 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.

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.

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. 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: