Proof by exhaustion

Proof by exhaustion 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.

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.

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 a few dozen - for example, rigorously solving an end-game puzzle in chess might involve considering a dozen or more possible lines of moves. Sometimes there may be hundreds of cases.

The first proof of the four colour theorem was a proof by exhaustion with almost 2,000 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 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 :-

• The proof that there is no finite projective plane of order 10
• The classification of finite simple groups.
• The Kepler conjecture.