Process Problems
Concept
Invariant
Find a quantity — a parity, a sum mod , a coloring — that a process never changes. If start and target differ in it, the target is unreachable.
In plain terms. Spot something that stays the same no matter what moves you make; if the goal has a different value, you can never get there.
Example. On a chessboard, a domino always covers one black and one white square, so you can never tile a board with two same-color corners removed.
Concept
Monovariant
A quantity that only increases (or only decreases) each step proves the process must terminate.
In plain terms. Find something that always moves one direction; since it can't do so forever, the process has to stop.
Example. If each move strictly lowers a positive whole-number "energy," the moves must eventually end.