# 2.3 Specific heuristics

This list was written by A.H. Schoenfeld:

• draw a diagram (if at all possible)

• examine special cases

• choose special values to exemplify the problem and get a "feel" for it

• examine limiting cases to explore the range of possibilities

• set any integer parameters equal to $1,2,3,...$ in sequences and look for an inductive pattern.

• try to simplify the problem

• exploiting symmetry

• without loss of generality

• consider essentially equivalent problems

• replace the conditions with equivalent ones

• recombine the elements of the problem in different ways

• introduce auxiliary elements (e.g. substitutions)

• reformulate the problem

• change of perspective or notation

• considering argument by contradiction or contrapositive

• assuming you have a solution, and determining its properties (i.e. what would a valid answer look like)

• consider slightly modified problems

• choose sub-goals (obtain partial fulfilment of the conditions)

• relax a condition and then try to re-impose it

• decompose the domain of the problem and work on it case by case

• construct an analogous problem with fewer variables

• hold all but one variable fixed to determine that variable’s impact

• try to exploit any related problems which have similar

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form

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"givens"

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conclusions

• verifying solutions

• Does it pass these simple tests?

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Does it use all the pertinent data?

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Does it conform to reasonable estimates or predictions?

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Does it withstand tests of symmetry, dimension analysis, or scaling?

• Does it pass these general tests?

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Can it be obtained differently?

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Can it be substantiated by special cases?

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Can it be reduced to known results?

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Can it be used to generate something you know?