Brute force method

Proof by exhaustion, also known as proof by cases, proof by case analysis, 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:

The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion. Computer expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.

In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching.

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:

Mathematicians prefer to avoid proofs by exhaustion with large numbers of cases, which are viewed as inelegant. An illustration of how such proofs might be inelegant is to prove that every year in which the modern Summer Olympic Games is held is divisible by 4.

Proof: the first modern Summer Olympics were held in 1896, and then every 4 years thereafter (neglecting years in which the games were not held due to World War I and World War II). Since 1896 = 474 × 4 is divisible by 4, the next Olympics would be in year 474 × 4 + 4 = (474 + 1) × 4, which is also divisible by four, and so on (this is a proof by mathematical induction). Therefore the statement is proved.

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