NEW CHALLENGE: This challenge was a suggestion by jgens. I am very thankful that he provided me with this neat problem. A 15-puzzle has the following form: The puzzle above is solved. The object of the game is to take an unsolved puzzle, such as and to make a combination of moves to bring the puzzle in the solved position. An allowable move here is to slide 11 to the right or to slide 12 down. After you slide 11 to the right, you have the options to slide 11 to the left, to slide 3 to the right, or to slide 15 down. Etc. The challenge question now is to take the following unsolved puzzle: Prove that you cannot solve this puzzle. That is, there is no finite combinations of moves that can bring this position into the solved position. Furthermore, try to devise a criterion to decide which puzzles are solvable and which are not.