Match problems to the right solving strategy. Covers working backwards, pattern recognition, simplification, systematic listing, trial and error, means-ends analysis, analogical transfer, and decomposition. Use after comprehension is complete to narrow from "general problem" to a concrete approach before committing time and effort.
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Match problems to the right solving strategy. Covers working backwards, pattern recognition, simplification, systematic listing, trial and error, means-ends analysis, analogical transfer, and decomposition. Use after comprehension is complete to narrow from "general problem" to a concrete approach before committing time and effort.
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.
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 โ 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 โ 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 โ 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 โ 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 โ 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 โ 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 โ 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 โ 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 โ 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:
Is the problem familiar? โ Pattern recognition (Strategy 2)
Is the goal clearer than the path? โ Working backwards (Strategy 1)
Is the full problem too big? โ Simplification or decomposition (Strategy 3 or 8)
Is the solution space finite and small? โ Systematic listing (Strategy 4)
Do you know both endpoints? โ Means-ends analysis (Strategy 6)
Is there a solved problem with the same structure? โ Analogical transfer (Strategy 7)
Does the problem have spatial or relational structure? โ Draw a diagram (Strategy 9)
None of the above? โ Trial and error with tracking (Strategy 5) + forward chaining (Strategy 10)
Combining Strategies
Most real problems need more than one strategy. A typical pattern:
Decompose the problem into sub-problems
Pattern-match each sub-problem to a known type
Apply the strategy specific to each type
Combine results
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 โ 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
