  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 or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.. this is a method of direct proof. 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 the curry&ndash;howard isomorphism, proof by exhaustion and case analysis are related to ml-style pattern matching. ==example== to prove that every integer that is a perfect cube is a multiple of 9, or is 1 more than a multiple of 9, or is 1 less than a multiple of 9. proof: each cube number is the cube of some integer n. every 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&nbsp;+&nbsp;1, then n3 = 27p3&nbsp;+&nbsp;27p2&nbsp;+&nbsp;9p&nbsp;+&nbsp;1, which is 1 more than a multiple of 9. for instance, if n&nbsp;=&nbsp;4 then n3 = 64 = 9x7&nbsp;+&nbsp;1. *case 3: if n = 3p&nbsp;â&nbsp;1, then n3 =&nbsp;27p3&nbsp;â&nbsp;27p2&nbsp;+&nbsp;9p&nbsp;â&nbsp;1, which is 1 less than a multiple of&nbsp;9. for instance, if n =&nbsp;5 then n3 = 125 = 9&times;14&nbsp;â&nbsp;1.â ==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, and in general the probability of an erro
Mathematical_proofs Methods_of_proof Problem_solving