15 puzzle

The 15 puzzle, also known as the Gem Puzzle, Boss Puzzle, or Game of Fifteen, is a classic sliding puzzle that consists of a 44 grid with 15 numbered square tiles and one empty space. The objective is to rearrange the tiles from a scrambled initial configuration into numerical order (1 to 15, left to right, top to bottom) by sliding them into the empty space. Key Features: Grid Layout: 4 rows 4 columns (16 positions, with 1 empty). Tiles: Numbered 1 through 15, with the 16th position empty. Moves: Only tiles adjacent (horizontally or vertically) to the empty space can slide into it. Mathematical Properties: Solvability: Not all initial configurations are solvable. A puzzle is solvable if: - The number of inversions (pairs of tiles where a higher number precedes a lower number) is even for an even-sized grid (like 44) when the empty space is on an odd row from the bottom (e.g., the 4th row in a 44 grid is row 1 from the bottom). - The number of inversions is odd if the empty space is on an even row from the bottom. God's Number: The maximum number of moves needed to solve any solvable 15 puzzle is 80 (in "half-turn" metric) or 43 (in "quarter-turn" metric). Example of an Unsolvable Initial State: Here, the inversion count for (15, 14) is 1 (odd), and the empty space is on row 4 (odd from the bottom). Since the inversion count is odd, the puzzle is unsolvable. Solving Techniques: Layer-by-Layer: Solve the top row, then the second row, etc. Corner Strategies: Place tiles in the correct order while preserving solved sections. Last Row Challenges: The final row requires specific sequences (e.g., "3-cycle" moves). Historical Note: The puzzle was invented by Noyes Palmer Chapman in the 1870s and popularized by Sam Loyd, who falsely claimed a 1,000 prize for solving an unsolvable configuration. Computational Complexity: The generalized ( n imes n ) version of the 15 puzzle is NP-hard, meaning finding optimal solutions for larger grids is computationally intensive. Would you like help with a specific configuration or solving strategy?