Proof Strategies

Mathematicians are skeptical people. They use many methods, including experimentation with examples, trial and error, and guesswork, to try to find answers to mathematical questions, but they are generally not convinced that an answer is correct unless they can prove it. You have probably seen some mathematical proofs before, but you may not have any experience writing them yourself. In this chapter you'll learn more about how proofs are put together, so you can start writing your own proofs.

Proofs are a lot like jigsaw puzzles. There are no rules about how jigsaw puzzles must be solved. The only rule concerns the final product: All the pieces must fit together, and the picture must look right. The same holds for proofs.

Although there are no rules about how jigsaw puzzles must be solved, some techniques for solving them work better than others. For example, you'd never do a jigsaw puzzle by filling in every other piece, and then going back and filling in the holes! But you also don't do it by starting at the top and filling in the pieces in order until you reach the bottom. You probably fill in the border first, and then gradually put other chunks of the puzzle together and figure out where they go. Sometimes you try to put pieces in the wrong places, realize that they don't fit, and feel that you're not making any progress.