| Proof By Exhaustion |
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In contrast, the Method Of Exhaustion of Eudoxus Of Cnidus was a geometrical and essentially rigorous way of calculating mathematical Limit s. 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:
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. 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 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:
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