How to solve it

Summary

This book is an attempt at describing basic problem-solving strategies. Based on puzzles and mathematical riddles the author tries to teach the general approach of problem-solving, but I think that assuming that problem-solving skills can be taught by reading a book is a bit of a flawed idea in its own right.

One has to sit down and do problems and figure out how and why things work the way they do, especially in mathematics, if one wants to become good at solving problems.

This book is rather specific to mathematics, even though its title aims at a broader audience. It's about learning how to solve specifically mathematical problems, not general problems. Even though one could argue that there are elements of mathematical problem solving to be found in other problems and that the strategies learned from mathematics can be translated into other realms. But I find the overall title still misleading.

The basic idea of problem solving the George Pólya presents, boils down to this:

• Understand the Problem: What is the unknown? What are we trying to figure out? What data do we have that we can work with? What is the relationship between what we have and what we try to find out?
• Devise a Plan: Look for patterns, solve simpler problems first, then generalize, work backward, invert the problem, draw diagrams, use symmetries or analogies, pattern match to problems previously seen, this reminds me of one of Ray Dalios Principles "ahh it's another one of those"
• Carry out the Plan: If it doesn't work, devise a different plan, adjust. Persistence is key.
• Look back: Review and Learn better ways to solve the same problem with the solution in hand.

Tools of Problem Solving:

• educated guesses and experimentation
• pattern matching => have I solved similar problems before?
• analogies and symmetries
• decomposition (divide and conquer strategy) => dynamic programming is all about this
• specialization => look at a specific case first, then work up from the solved special case
• induction and deduction
• working backwards

Thinking critically and solving problems is about asking the right questions, and recognizing and applying things seen before. Problem solving to a large part is pattern matching and using the tools available to us. When solving a lot of problems, we might even start to develop gut feelings and intuition, because the pattern matching becomes subconsciously internalized.