This is done by considering a function of the tile configuration that is invariant under any valid move, and then using this to partition the space of all possible labeled states into two equivalence classes of reachable and unreachable states. Johnson & Story (1879) used a parity argument to show that half of the starting positions for the n puzzle are impossible to resolve, no matter how many moves are made. That is, they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the taxicab distances between each block and its position in the goal configuration.

The n puzzle is a classical problem for modelling algorithms involving heuristics. Similar names are used for different sized variants of the 15 puzzle, such as the 8 puzzle which has 8 tiles in a 3×3 frame.

Named for the number of tiles in the frame, the 15 puzzle may also be called a 16 puzzle, alluding to its total tile capacity. The goal of the puzzle is to place the tiles in numerical order. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tile positions high and 4 positions wide (for a total of 16 positions), leaving one unoccupied position.

To solve the puzzle, the numbers must be rearranged into numerical order.

My Slider Puzzle downloading