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. Mathematics Solvability A solved 15 puzzle 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 after 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 (from left to right, top to bottom). 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 which has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 positions wide, with one unoccupied position. To solve the puzzle, the numbers must be rearranged into numerical order from left to right, top to bottom. For the numbered grid where each row and column sums to 15, see Magic square.
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