Take a solved 15-puzzle, swap 14 and 15, and try to fix it with legal slides. You cannot, and the reason is a single conserved number that every legal move leaves untouched.
The proof: read the board left to right, top to bottom, and track inversions plus the blank's row from the bottom, mod 2. Horizontal slides flip neither. Vertical slides flip both. Their sum is locked. The solved board sits at one value; the swapped board sits at the other. No path connects them.
A clean parity-invariant argument hiding inside a children's puzzle.
00:00 The impossible swap
00:46 Read the board as a sentence
01:12 Inversions and parity
02:03 The candidate invariant
02:39 Vertical slides break it
03:12 The hidden invariant
04:01 Half the universe is unreachable
04:35 The other side of a wall
#15Puzzle #ParityArgument #Invariant #Permutations #MathProofs
