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 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, as they seem inelegant. 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, such as
- The proof that there is no finite projective plane of order 10.
- The classification of finite simple groups.
- The Kepler conjecture.
Read more about this topic: Proof By Exhaustion
Famous quotes containing the words number of, number and/or cases:
“To finish the moment, to find the journey’s end in every step of the road, to live the greatest number of good hours, is wisdom. It is not the part of men, but of fanatics, or of mathematicians, if you will, to say, that, the shortness of life considered, it is not worth caring whether for so short a duration we were sprawling in want, or sitting high. Since our office is with moments, let us husband them.”
—Ralph Waldo Emerson (1803–1882)
“One may confidently assert that when thirty thousand men fight a pitched battle against an equal number of troops, there are about twenty thousand on each side with the pox.”
—Voltaire [François Marie Arouet] (1694–1778)
“... and the next summer she died in childbirth.
That’s all. Of course, there may be some sort of sequel but it is not known to me. In such cases instead of getting bogged down in guesswork, I repeat the words of the merry king in my favorite fairy tale: Which arrow flies for ever? The arrow that has hit its mark.”
—Vladimir Nabokov (1899–1977)