Math & theory
Everything we know about slide puzzles, organised by what you want to do — play a particular size, solve one, turn a photo into one, or read the maths behind why some boards are impossible.
A history of 15-puzzle solving algorithms: BFS, A* with Manhattan distance, IDA*, walking distance, and additive pattern databases.
The 15 puzzle parity theorem: check if a board is solvable by counting inversions plus the blank's row. Clean proof and the Sam Loyd story.
How the Manhattan distance heuristic works, why it is admissible and consistent, and where it falls short for slide-puzzle solving.
The four techniques that solve any sliding tile puzzle by hand: row-and-column reduction, the L-shaped corner manoeuvre, empty-cell routing, endgame.
How to check if a 15 puzzle is solvable, in plain language: count inversions, add the blank's row, check parity. What to do if you get a bad board.