Sliding-Tile & Solvability
📋 What it is
In a sliding-tile puzzle you slide tiles into the gap; some scrambles can be solved and some can NEVER be.
🗣️ Coach says
Here’s the surprise: not every scrambled 15-puzzle can be solved! There’s a hidden rule (based on how many pairs are out of order) that decides it. So before grinding for hours, you can CHECK whether a scramble is even solvable.
🧠 Memory hook
Some scrambles are impossible — check solvability BEFORE you grind.
😂 Giggle
Why did the lateral thinker turn the problem sideways?
Because the answer was hiding at an angle!
😲 Whoa!
In the 1880s a puzzle-maker offered $1,000 to anyone who could solve one specific 15-puzzle scramble — it was safe money, because that scramble was mathematically impossible.
✅ Quick check: Two identical 15-puzzles are scrambled differently. Are both guaranteed solvable?
Say your answer out loud first — then reveal.
No — exactly half of all scrambles are unsolvable; a parity rule decides which, so you must check.
Sliding-tile puzzles have an invariant that makes some positions unreachable.
🧪 Try it! (2 minutes)
Scramble a small sliding puzzle (or app) and try it — if it feels impossible, it might genuinely be.