npbuilds Polya Method — How to Solve It
George Pólya's How to Solve It (1945) is the spine of problem-statement work. The four-step method tells you when you are; the heuristic dictionary (~67 entries) tells you what to try. Pólya's most important contribution was insisting that Step 1 (Understand the problem) is not trivial — it is where most failures originate.
The Four Steps
- Understand the problem. What is the unknown? What are the data? What is the condition? Restate in your own words. Draw a figure. Introduce suitable notation. Most problems are misstated until Step 1 is taken seriously.
- Devise a plan. Find the connection between the data and the unknown. Search for a related problem with the same unknown. This is the creative step — heuristics live here.
- Carry out the plan. Execute, checking each step.
- Look back. Examine the result. Can you check it? Can you derive it differently? Can you use the result, or the method, on another problem?
The Seven Heuristic Families
The dictionary collapses into seven families. Pick the family by where you're stuck.
| Family | Example heuristics | When to invoke |
|---|---|---|
| Restatement | Have you stated it correctly? Express in different words. Define the terms. | Step 1 — before any solving work |
| Relation-finding | Have you seen the same problem in a different form? Is there a related problem with the same unknown? | Step 2 — when no plan suggests itself |
| Reduction | Solve a simpler / more general / related problem first. Specialize the problem. Generalize the problem. | Step 2 — when the problem is too hard as stated |
| Working backwards | Start from the desired conclusion. Ask what would suffice. Trace backwards to the data. | Search problems, proof problems, design problems |
| Decomposition | Break the problem into sub-problems. Vary the problem. Drop conditions; add them back. | Step 2 — for compound or multi-step problems |
| Analogy | Find an auxiliary problem you've solved before with similar structure. | Cross-domain or transfer cases |
| Verification | Can you check the result? Can you check the argument? Use it for another problem? | Step 4 — never skip |
Diagnostic Cues
| Symptom | Family to try |
|---|---|
| "I don't understand what's being asked" | Restatement |
| "I have no idea where to start" | Relation-finding, then Analogy |
| "This problem is too big" | Decomposition or Reduction (specialize) |
| "I can't find a path forward" | Working backwards |
| "I have an answer but I'm not sure it's right" | Verification |
| "I solved one case but the general result eludes me" | Reduction (generalize) |
How binding-vow uses Polya
Polya's Step 1 maps directly onto six-eyes's Phase 1 (Intake) — the "restate the dump in precise terms" instruction is Polya's "Understand the problem" verbatim. The heuristic families in the table above are the operations frame-rotator (Phase 4) executes:
- Restatement → trivial in Phase 1; deep in
frame-rotator - Reduction (specialize / generalize) →
frame-rotator's level-shifting moves (folded in per v2) - Working backwards → reframing technique when forward-chaining fails
- Decomposition → calls
claim-decomposer(cross-domain — research) - Analogy → suggested by
frame-rotator; can calldomain-translator(neocortex) for cross-domain analogies - Verification → Phase 6 audit (every audit skill is a flavor of Polya verification)
The dictionary itself (the ~67 numbered entries in How to Solve It) is reference material; future skills should cite it by entry rather than re-implementing.
Common Mistakes
- Skipping Step 1 because "I obviously understand the problem" — Reiter-Palmon's automaticity finding shows you usually don't.
- Treating heuristics as solutions — they generate candidate plans, not answers.
- Stopping at Step 3. Step 4 (look back) is where transfer happens; without it, every new problem starts from scratch.
- Confusing Reduction with Decomposition. Reduction simplifies (one easier problem); Decomposition splits (several simpler problems).
- Treating Analogy as proof. An analogous problem suggests an approach; it does not guarantee the approach works.
Connections
six-eyes— Phase 1 maps onto Polya Step 1; Phase 4 implements heuristic familiesframe-rotator(binding-vow) — uses Restatement, Reduction, Working Backwardsclaim-decomposer(research) — implements Decomposition familydomain-translator(neocortex) — implements Analogy family across domains- Vault note:
skill-lab/polya-heuristics-catalog.md
Sources
- Pólya, G. (1945). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press / John Wiley & Sons.
- Pólya, G. (1954). Mathematics and Plausible Reasoning (vol. I & II). Princeton University Press.
- Schoenfeld, A. H. (1985). Mathematical Problem Solving. Academic Press. (Expansion of Pólya's framework with empirical testing.)