Parity Invariants
📋 What it is
An invariant is something that never changes no matter what moves you make; parity (odd/even) is a powerful one.
🗣️ Coach says
Some puzzles are “impossible” and you can PROVE it with an invariant — a quantity that never changes. If the start is “even” and the goal is “odd” and every move keeps parity the same, you can never get there. One idea proves impossibility instantly.
🧠 Memory hook
If a move never changes parity, you can’t go from even to odd — impossible, proven.
😂 Giggle
Why did the trick question refuse to be rushed?
Because the fast answer was usually the wrong one!
😲 Whoa!
This is how the impossible 15-puzzle scramble is proven unsolvable — a parity invariant stays fixed under every legal slide, so half of all scrambles simply can’t reach the goal.
✅ Quick check: Every move keeps a puzzle’s parity EVEN, but the goal is ODD. Can you solve it?
Say your answer out loud first — then reveal.
No — if parity never changes and start (even) ≠ goal (odd), the goal is unreachable; the invariant proves it impossible.
An invariant preserved by every move separates reachable from unreachable states.
🧪 Try it! (2 minutes)
Flip a row of coins where each move flips two at once. Can you ever reach an odd number of heads? Test the invariant.