Smart Math guide · Problem Solving

Look for Invariants and Patterns

Repeated structure can suggest a general rule, but the rule still needs justification.

All levelsWorked example includedSelf-check included
The method

How to use it

  1. Compute several organized cases.
  2. Identify what changes and what stays fixed.
  3. State a conjecture and test more cases.
Worked example

See it in action

Odd numbers 1+3+5+7=16 suggests the sum of first n odd numbers is n² for n=4.

Check your thinking

How to verify it

Test n=5: add 9 to get 25.

Watch out

Common mistake

Examples suggest a rule; they are not by themselves a proof for all cases.