Strategy selection

Strategy Selection

Strategies are general-purpose operators on problems. Polya called them heuristics. Simon and Newell formalized some of them as search operators in a state-space. A good solver has many strategies and picks the right one; a weak solver has few and applies them indiscriminately. This skill is the strategy catalog and the decision rules for matching strategies to problem features.

Agent affinity: polya-ps (overall framing), simon (state-space strategies), newell (means-ends analysis)

Concept IDs: prob-working-backwards, prob-pattern-recognition, prob-simplification, prob-systematic-listing, prob-trial-error-iteration

The Strategy Catalog at a Glance

#	Strategy	When it applies	Key signal
1	Working backwards	Goal is clear, path is obscure	"You need the goal to find the start"
2	Pattern recognition	Problem looks like one you've seen	"This is a ___ problem in disguise"
3	Simplification	Full problem is intractable	Solve for n=2, then generalize
4	Systematic listing	Finite solution space, risk of missing cases	Combinatorics, case analysis
5	Trial and error with tracking	No obvious path, need to explore	Try, evaluate, adjust, track
6	Means-ends analysis	Goal and current state both known	Reduce the difference
7	Analogical transfer	Known solution to structurally similar problem	Map source to target domain
8	Decomposition	Problem is large but separable	Sub-problems with clean interfaces
9	Drawing a diagram	Structure is hidden in prose	Spatial, relational, flow problems
10	Forward chaining	Start is clear, goal is vague	Explore from knowns

Strategy 1 — Working Backwards

Pattern: Start from the goal state and ask "what would have to be true one step before this?" Repeat until you reach the initial state.

When it applies:

• Goal is precisely specified
• Initial state allows many possible first moves (high branching factor)
• Inverse operators exist for most forward operators

Worked example. "Find x such that 3x + 5 = 20." Working backwards: if 3x + 5 = 20, then 3x = 15 (inverse of +5), then x = 5 (inverse of *3). Done.

Strategy 2 — Pattern Recognition

Pattern: Ask "have I seen this problem before?" If yes, transfer the known solution structure.

When it applies:

• The problem has a familiar shape even if the surface details differ
• You have solved similar problems before and remember the method
• The mapping between the new problem and the known solution is clean

Strategy 3 — Simplification

Pattern: Reduce the problem to a smaller or simpler version, solve that, then generalize or scale up.

Worked example. "In how many ways can n students line up?" Try n=1 (1 way), n=2 (2 ways), n=3 (6 ways), n=4 (24 ways). Pattern: n!. The full problem is solved by solving the small cases first.

Strategy 4 — Systematic Listing

Pattern: Enumerate all possibilities in a structured way to guarantee completeness.

When it applies:

• The solution space is finite and small enough to enumerate
• Missing a case would be catastrophic (correctness critical)
• The structure of the enumeration is clear

Strategy 5 — Trial and Error with Tracking

Pattern: Try an approach, evaluate, note what you learned, try a modified approach. Track attempts so you do not repeat failures.

When it applies:

• No obvious strategy applies
• Problem is small enough that multiple attempts are affordable
• Evaluation after each attempt is cheap

This is not random guessing; each trial is informed by the previous one.

Strategy 6 — Means-Ends Analysis

Pattern: Compute the difference between the current state and the goal state. Choose an operator that reduces the difference. Apply. Repeat.

When it applies:

• Both current state and goal are known
• Operators can be ranked by how much they reduce the state difference
• This is the core strategy of Simon and Newell's General Problem Solver

Strategy 7 — Analogical Transfer

Pattern: Find a solved problem with the same structural form, map entities between source and target, transfer the solution.

When it applies:

• A known solved problem shares underlying structure with the target
• The mapping is clean (each entity in the source corresponds to one in the target)
• The solution method in the source has no hidden domain-specific steps

Analogy is powerful but risky: surface similarity without structural similarity produces wrong answers.

Strategy 8 — Decomposition

Pattern: Break the problem into sub-problems with clean interfaces. Solve each sub-problem. Combine.

When it applies:

• Problem is large but separable
• Sub-problems are roughly independent
• The combination step is well-defined

Strategy 9 — Drawing a Diagram

Pattern: Translate the problem into a spatial or relational representation. Geometry, flow diagrams, graphs, state-spaces.

When it applies:

• Problem has spatial, relational, or temporal structure
• The structure is hidden in prose
• The diagram reveals constraints or symmetries not obvious in words

Strategy 10 — Forward Chaining

Pattern: Start from the knowns and generate consequences until you reach (or approach) the goal.

When it applies:

• Initial state is well specified
• Goal is vague or distant
• Operators are well-understood forward rules

The Strategy Decision Tree

A rough procedure for choosing:

1. Is the problem familiar? → Pattern recognition (Strategy 2)
2. Is the goal clearer than the path? → Working backwards (Strategy 1)
3. Is the full problem too big? → Simplification or decomposition (Strategy 3 or 8)
4. Is the solution space finite and small? → Systematic listing (Strategy 4)
5. Do you know both endpoints? → Means-ends analysis (Strategy 6)
6. Is there a solved problem with the same structure? → Analogical transfer (Strategy 7)
7. Does the problem have spatial or relational structure? → Draw a diagram (Strategy 9)
8. None of the above? → Trial and error with tracking (Strategy 5) + forward chaining (Strategy 10)

Combining Strategies

Most real problems need more than one strategy. A typical pattern:

1. Decompose the problem into sub-problems
2. Pattern-match each sub-problem to a known type
3. Apply the strategy specific to each type
4. Combine results
5. Use metacognitive-monitoring to check that combined pieces answer the original question

When Strategy Selection Fails

• Premature commitment. Locking in on the first strategy that comes to mind. Evaluate at least two before picking.
• Strategy without comprehension. Strategies are operators on a problem representation. No representation → no strategy.
• Wrong-level strategy. Applying a structural strategy (decomposition) to a problem that is actually a pattern-recognition problem wastes effort.
• No fallback. If the chosen strategy fails, have a second strategy ready rather than starting over.

Cross-References

• problem-comprehension produces the representation that strategy selection operates on
• mathematical-problem-solving applies many of these strategies to math-specific contexts
• design-thinking-ps adds ideation and prototyping strategies for ill-defined problems
• metacognitive-monitoring evaluates whether a chosen strategy is actually working
• collaborative-problem-solving allows different team members to pursue different strategies in parallel