Ideas
Invariants
A quantity that does not change under the allowed moves. Showing an invariant separates reachable states from impossible ones.
Pigeonhole principle
If n + 1 objects sit in n boxes, some box holds at least two. Generalizations bound the most-loaded box.
Telescoping
Rewrite a sum or product as differences (or ratios) of consecutive terms so almost everything cancels.
Vieta jumping
Given a Diophantine equation, treat one variable as fixed and use Vieta's formulas to 'jump' to a smaller solution, reaching a contradiction or base case.
Extremal principle
Pick the largest, smallest, or otherwise extreme object in a configuration and study its forced properties.
Strong induction
Prove P(n) by assuming P(k) for all k < n. Useful when the inductive step needs more than the immediate predecessor.
Generating functions
Encode a sequence as the coefficients of a formal power series and manipulate the series instead of the sequence.
Coloring arguments
Color cells or vertices to expose a parity / counting obstruction (classic for tiling problems).