The 15 puzzle is a famous puzzle where you slide the numbered tiles from 1 to 15 into order. But this puzzle has a hidden secret — if you scramble the tiles randomly (by picking them up and rearranging them), exactly half of all possible arrangements can never be solved, no matter how much you slide the tiles. Find out why for yourself.
The secret: Mathematically, sliding a tile has the same effect as "swapping the positions of two tiles" (a move involving the empty space is equivalent to an even number of swaps). So only arrangements that can be reached from the solved state through an even number of swaps are solvable — these arrangements are called "even permutations."