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 by , testing the product of 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.
Train it on Lemma
Invariants unlocks at Level 5 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.
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