Polya's Problem Solving Heuristics

How To Solve It: George Polya's 1945 classic lays out four steps (understand the problem, devise a plan, carry out the plan, look back) plus a catalog of named mental moves, and it has sold well over a million copies to readers far beyond mathematics.

George Polya: Born in Budapest in 1887, he drifted through law, languages and philosophy before landing in mathematics, then fled Europe in 1940 and spent decades at Stanford teaching teachers how to think.

Polya's Inventor's Paradox: The more ambitious plan may have a better chance of success, because a broader problem can be easier to crack than the narrow one you started with.

Polya's Auxiliary Problem: When stuck, Polya advises introducing a helper problem or element, such as a new line in a diagram, that you solve first as a stepping stone to the real question.

Polya's Working Backwards: Start from what you want to prove and ask what it would follow from, then keep stepping back until you hit something you already know.

Pappus Of Alexandria: Polya traced working backwards to this late ancient Greek geometer, who described the method of analysis as treating the thing sought as already found.

Polya's Specialization: Try the problem with small numbers or a simple special case, because patterns that show up there often point toward the general solution.

Polya's Look Back Step: After solving, Polya urges you to check the result, ask if you could get it a different way, and see where else the method might work, the step most solvers skip.

Polya's Analogy Heuristic: Ask whether you know a similar problem with a known answer, then borrow its structure, a move that powered many discoveries by mapping a new puzzle onto an old one.

Polya's Draw A Figure: Sketching the situation is listed as a basic move because a picture forces you to notice what is given and what is missing.

Polya's Decomposing And Recombining: Break the problem into parts, solve them separately, then put them back together, which Polya treats as a core move for large or tangled problems.

Polya's What Is The Unknown Question: His opening prompt, which asks what you are trying to find, what is given and what condition links them, quietly fixes the most common cause of failure, which is misreading the problem.

Mathematics And Plausible Reasoning: Polya's 1954 two volume work argues that discovery runs on guessing and weighing evidence, with proof coming afterward, an unusual stance for a rigorous mathematician.

Mathematical Discovery: This two volume sequel from the early 1960s applies the heuristics to dozens of worked problems and was written with teachers in mind.

Basel Problem: Polya used Euler's discovery that the sum of reciprocal squares equals pi squared over six as a model of plausible reasoning, because Euler checked his guess numerically long before he could prove it.

Archimedes: The word heuristic comes from the Greek verb that gives us Eureka, the cry attributed to Archimedes, which Polya cites to show how old the study of discovery is.

Rene Descartes: His unfinished Rules for the Direction of the Mind sketched a universal method of problem solving, and Polya treated it as an ancestor of his own project.

Jacques Hadamard: His book The Psychology of Invention in the Mathematical Field appeared the same year as How To Solve It, and it explored the same mystery of how ideas arrive.

Henri Poincare: His story of a key idea arriving as he stepped onto a bus after days of fruitless work is the classic case of incubation, the unconscious phase that Polya's methods try to feed.

Allen Newell: A future founder of artificial intelligence took Polya's course at Stanford, and Polya's idea of rules of thumb that guide search fed straight into early AI heuristics.

General Problem Solver: Newell and Herbert Simon built this 1950s program around means ends analysis, a move in the same family as Polya's working backwards and auxiliary problems.

Imre Lakatos: The Hungarian philosopher of mathematics grew up with Polya's heuristic approach in the air, and his Proofs and Refutations made the messy path of discovery central to the story of mathematics.

Alan Schoenfeld: After years of classroom studies, this Berkeley researcher found that Polya's heuristics were too vague for students to use on their own, and showed that each one needs detailed teaching and practice.

Polya Random Walk Theorem: Polya proved in 1921 that a drunkard wandering on a street grid will always return home, but in three dimensions the chance of ever coming back drops to about 34 percent.

Polya Enumeration Theorem: Polya's counting method for objects under symmetry can tally how many distinct arrangements, such as chemical isomers, exist when rotations and flips count as the same.