A 4×4 sliding puzzle with one tile missing. Scramble it by sliding and it always solves — but start from Loyd's famous 14-15 swap and no sequence of slides will ever fix it. Every slide flips two hidden switches in lockstep, and that lockstep is the whole story.
Every board hides two switches. Switch one: is the count of out-of-order tile pairs — read the board like a book, skip the blank — even or odd? Switch two: counting the bottom row as row 1, is the blank's row even or odd? A sideways slide touches neither switch. An up-or-down slide of a single tile flips both at once — always both, never just one.
So the two switches are welded together: their sum can wiggle, but its parity never changes, no matter how long you slide. Solved has the sum odd. Loyd's board — one pair out of order, blank at home — has it even. Even can't slide its way to odd, so the $1,000 was never in danger. The only thing between his board and victory is a move the rules don't allow.