Problems

Many years ago, I stumbled upon a book that has been surprisingly helpful to me: How to Solve It by the mathematician George Pólya. While the book is about math, the principles are useful more generally.

The book is a simple set of instructions on how to solve mathematical problems. Pólya offers four steps:

  1. First, you have to understand the problem.

  2. After understanding, make a plan.

  3. Carry out the plan.

  4. Look back on your work. How could it be better?

Each step includes a set of questions and suggestions. The questions and suggestions Pólya offers for the first two steps are particularly helpful for problems in general.

Step one: Understand the problem

  • What are you asked to find or show?

  • Can you restate the problem in your own words?

  • Can you think of a picture or a diagram that might help you understand the problem?

  • Is there enough information to enable you to find a solution?

  • Do you understand all the words used in stating the problem?

  • Do you need to ask a question to get the answer?

Step two: Devise a plan

  • Guess and check

  • Make an orderly list

  • Eliminate possibilities

  • Use symmetry

  • Consider special cases

  • Use direct reasoning

  • Solve an equation

  • Look for a pattern

  • Draw a picture

  • Solve a simpler problem

  • Use a model

  • Work backward

  • Use a formula

  • Be creative

Pólya also offers a set of heuristics to help in problem-solving:

  • Analogy: Can you find a problem analogous to your problem and solve that?

  • Generalization: Can you find a problem more general than your problem?

  • Induction: Can you solve your problem by deriving a generalization from some examples?

  • Variation of the problem: Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?

  • Auxiliary problem: Can you find a subproblem or side problem whose solution will help you solve your problem?

  • Related problem: Can you find a problem related to yours that has already been solved and use that to solve your problem?

  • Specialization: Can you find a problem more specialized?

  • Decomposing and recombining: Can you decompose the problem and recombine its elements in some new manner?

  • Working backward: Can you start with the goal and work backwards to something you already know?

  • Draw a figure: Can you draw a picture of the problem?

  • Auxiliary elements: Can you add some new element to your problem to get closer to a solution?

Surprisingly, often just the simple step of asking what is the problem that needs to be solved and then taking the time to carefully define the problem—actually writing it out—makes all the difference. It changes the issue from a vaguely-defined one where everyone is talking past each other to a clear problem to be solved. I’ve found it to be a remarkable focusing tool.