---
id: wiki-2026-0508-problem-solving-process
title: Problem Solving Process
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Problem Solving, Polya Method, Engineering Problem Solving]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [cognition, engineering, methodology, debugging]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: multi
  framework: structured-problem-solving
---

# Problem Solving Process

## 매 한 줄
> **"매 a problem well-stated is half-solved."**. 매 Polya 1945 *How to Solve It* 의 4단계 (understand → plan → carry out → look back) 가 매 modern engineering, debugging, AI agent design 의 backbone. 매 2026 LLM agent (Claude, OpenAI Operator) 의 ReAct/CoT 도 매 본질적으로 Polya 의 자동화.

## 매 핵심

### 매 Polya 4 stages
- **Understand**: 매 restate, 매 identify knowns/unknowns/constraints.
- **Plan**: 매 decompose, 매 analogous problems, 매 work backwards.
- **Carry Out**: 매 execute, 매 verify each step.
- **Look Back**: 매 check, 매 generalize, 매 alternate methods.

### 매 strategies
- **Decomposition**: 매 break into sub-problems (divide & conquer).
- **Analogy**: 매 find similar solved problem (case-based reasoning).
- **Inversion**: 매 work backwards from goal.
- **Specialization**: 매 try simpler instance (n=1, n=2 first).
- **Generalization**: 매 solve more general version sometimes easier.
- **Symmetry / Invariants**: 매 find quantity that doesn't change.

### 매 응용
1. Debugging: 매 reproduce → bisect → fix → verify → write test.
2. System design: 매 understand requirements → decompose → component design.
3. LLM agent: 매 ReAct loop = Polya in code.
4. Math/algorithms: 매 examples → conjecture → prove → optimize.

## 💻 패턴

### Pattern 1: Debugging as Polya
```python
def debug(bug_report):
    # 1. UNDERSTAND
    repro = build_minimal_repro(bug_report)
    # 2. PLAN
    suspected_modules = trace_stack(repro)
    plan = git_bisect_plan(repro, suspected_modules)
    # 3. CARRY OUT
    bad_commit = git_bisect(plan)
    fix = author_fix(bad_commit)
    # 4. LOOK BACK
    add_regression_test(repro)
    update_runbook(bug_report.symptom)
    return fix
```

### Pattern 2: ReAct LLM agent (Polya 자동화)
```python
SYSTEM = """For each task, follow:
1. THOUGHT: restate the problem and constraints (UNDERSTAND).
2. PLAN: decompose into 1-3 next actions.
3. ACTION: call exactly one tool.
4. OBSERVE: read result.
5. REFLECT: did this advance? if not, revise plan (LOOK BACK).
Repeat until done.
"""
```

### Pattern 3: Decomposition tree
```python
@dataclass
class Problem:
    statement: str
    children: list["Problem"] = field(default_factory=list)
    solved: bool = False
    solution: Optional[str] = None

def solve(p: Problem):
    if can_solve_directly(p):
        p.solution = direct(p); p.solved = True; return
    p.children = decompose(p)
    for c in p.children: solve(c)
    p.solution = combine([c.solution for c in p.children])
    p.solved = all(c.solved for c in p.children)
```

### Pattern 4: Five-Whys root cause
```text
Symptom: 매 dashboard p99 latency 5x baseline.
Why? — 매 DB queries slow.
Why? — 매 missing index.
Why? — 매 migration didn't add it.
Why? — 매 PR template doesn't require index check.
Why? — 매 no automated linter.
→ 매 Root: missing tooling, not "lazy engineer".
```

### Pattern 5: Pre-mortem (inverted planning)
```python
def pre_mortem(plan):
    # 매 imagine the plan failed in 6 months
    # 매 ask team: what went wrong?
    failure_modes = team_brainstorm("It's 6mo from now. Project failed. Why?")
    return prioritize_mitigations(failure_modes)
```

### Pattern 6: Working-memory checkpoint
```markdown
<!-- 매 problem_log.md while solving -->
## 매 KNOWN
- API returns 500 on POST /orders > 1000 items.
## 매 UNKNOWN
- Is it timeout, memory, or DB lock?
## 매 TRIED
- [x] Reproduce locally → reproduces at 1500.
- [x] Add timing logs → DB INSERT is slow.
- [ ] Check lock contention.
## 매 NEXT
- pg_stat_activity during repro.
```

### Pattern 7: Solution generalization (look back)
```python
# 매 after fixing one instance — 매 ask: where else does this pattern occur?
def generalize(fix):
    pattern = abstract(fix)  # e.g., "missing pagination in list endpoints"
    similar = scan_codebase_for(pattern)
    return apply_fix_to_all(similar)
```

## 매 결정 기준
| 상황 | Strategy |
|---|---|
| 매 stuck at "understand" | Restate problem to rubber duck / LLM |
| 매 plan unclear | Try simpler case (n=1) first |
| 매 carry-out fails | Bisect; isolate variable |
| 매 done — but is it right? | Look back; alt method; edge cases |
| 매 recurring class of bugs | Generalize the fix; tooling |
| 매 LLM agent loop stuck | Force REFLECT step; reduce action set |

**기본값**: 매 always do Look Back — 매 most engineers skip it; 매 90% of compounding leverage lives there.

## 🔗 Graph
- 부모: [[Cognitive Psychology]]
- 변형: [[Polya Method]] · [[Debugging]] · [[Root Cause Analysis]]
- Adjacent: [[ReAct]] · [[Chain of Thought]]

## 🤖 LLM 활용
**언제**: 매 system-prompt scaffolds, 매 incident runbooks, 매 onboarding docs, 매 interview prep.
**언제 X**: 매 trivial 1-line tasks — 매 over-formalization slows.

## ❌ 안티패턴
- **Skip understanding**: 매 jump to coding → 매 wrong problem solved.
- **Skip looking back**: 매 ship fix, never abstract → 매 same bug class returns.
- **No working-memory log**: 매 forget what you tried.
- **One strategy only**: 매 if decomposition fails, try inversion / analogy.
- **LLM as oracle**: 매 use as plan-critic, not plan-author.

## 🧪 검증 / 중복
- Verified (Polya 1945, Newell & Simon 1972, Yao et al. 2022 ReAct).
- 신뢰도 A (foundational).

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Polya 4단계 + ReAct + 7 패턴 |