# A calculus of the absurd

#### 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

• ∗ form

• ∗ "givens"

• ∗ conclusions

• verifying solutions

• – Does it pass these simple tests?

• ∗ Does it use all the pertinent data?

• ∗ Does it conform to reasonable estimates or predictions?

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

• – Does it pass these general tests?

• ∗ Can it be obtained differently?

• ∗ Can it be substantiated by special cases?

• ∗ Can it be reduced to known results?

• ∗ Can it be used to generate something you know?