Invariants

Find what the chaos cannot touch.

A process runs wild: numbers merge, chips jump, cards flip. An invariant is a quantity that every legal move preserves. If the start and the target disagree on an invariant, the target is unreachable, and you have proven impossibility without examining a single sequence of moves.

First candidates to test: the sum, the product, the parity, a coloring count. If the move replaces $a, b$ by $a + b + ab$, testing the product of $(1 + x)$ over all numbers reveals the hidden invariant.

The sibling notion, a monovariant, only ever moves one way (say, strictly decreases). Since it cannot decrease forever over integers, the process must halt, the standard weapon for termination arguments.

More techniques

• InductionTwo finite steps climb an infinite ladder.
• Power of a PointOne number controls every line through P.
• All 29 techniques →