Solving a 1000-Piece Jigsaw Puzzle Maze

[dariusmartin]

A few days ago I was watching a youtube video of someone solving a jigsaw puzzle and a thought came to my head; is it possible to make a maze with only one entrance and only one exit that is printed out on a jigsaw puzzle, where the only way the maze is traversable is if the puzzle is solved in 1 of 2 different configurations? (1000 pieces including edges)

 

[FactChecker]

I think so. Make a maze with one solution and with one path (call it the "dead path") that is not connected to the start or exit and with a wall on the boundary of the puzzle. Then determine a jigsaw puzzle for it. Now, cut out a uniquely shaped piece at the exit and an identically shaped piece on the exterior boundary of the dead path. Those two pieces should be swappable. With the two pieces in their original positions, there is a solution. With the two pieces swapped, the exit is blocked and the prior exit is now moved to an exit of the dead path that does not connect to the start.