Some problems look active: pieces move, numbers change, diagrams shift. An invariant asks a quieter question: what stays the same no matter what happens?
That one question can turn an impossible-looking puzzle into a short proof.
The Basic Idea
An invariant is a quantity, color, parity, or structure that does not change under the allowed moves.
If the starting position has one invariant value and the target position has a different one, the target is impossible.
COSMO Activity
Color a checkerboard black and white. Remove two opposite corner squares. Can the remaining board be tiled by dominoes?
A domino always covers one black square and one white square. Removing opposite corners removes two squares of the same color. The remaining board has unequal black and white counts, so the tiling is impossible.
What Students Learn
They learn that proof is not always about calculating more. Sometimes it is about choosing the right thing to watch.
Practice Prompt
Create a puzzle where the total number changes but the parity stays fixed. Then ask whether a target state is reachable.