Editing Proof by exhaustion (section)

==Number of 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 a [[chess endgame]] [[chess problem|puzzle]] 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 1834 cases.<ref>{{Citation |last=Appel |first=Kenneth |last2=Haken |first2=Wolfgang |last3=Koch |first3=John |year=1977 |title=Every Planar Map is Four Colorable. II. Reducibility |journal=Illinois Journal of Mathematics| volume=21 |page=504| mr=0543793 | issue=3 |doi=10.1215/ijm/1256049012 |doi-access=free |quote=Of the 1834 configurations in 𝓤}}</ref> 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.

In general the probability of an error in the whole proof increases with the number of cases. 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 [[mathematical beauty|elegant]]. However, there are some important theorems for which no other method of proof has been found, such as
* The proof that there is no finite [[projective plane]] of order 10.
* The [[classification of finite simple groups]].
* The [[Kepler conjecture]].
* The [[Boolean Pythagorean triples problem]].