Proof by Exhaustion

Example

Prove that the difference of two consecutive squares between 30 and 100 is odd

1. The squares between 30 and 100 are 36, 49, 64 and 81 1.1. Case 1: $49-36=13$ which is odd 1.2. Case 2: $64-49=15$ which is odd 1.3. Case 3: $81-64=17$ which is odd
2. Therefore, the difference of two consecutive squares between 30 and 100 is odd

Note

• Also known as proof by cases, or proof by brute force
• This is suitable when the number of cases is finite