All too often school teaches us to “guess and check” when a simple exact calculation would lead to the right answer. Guessing the answer to a one variable equation may not further our mathematical knowledge, but is it possible that guessing can lead to deep insights? According to Sanjoy Mahajan, physicist and author of Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving [PDF], the answer is yes. He makes the case that by using certain basic problem solving strategies one can avoid rigorous and complicated calculations while the result will be the same. Moreover, these strategies and the solutions that they yield lead to a deep understanding of the subject matter. The book is full of examples from mathematics, engineering, and physics and although some parts require knowledge of calculus, it should be accessible to motivated high school students. As a bonus, it is freely available from MIT Press. Here is a TEDx talk that the author gave illustrating the street-fighting techniques in his book.