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.

Articles Game modes Solver & solutions Custom & photo puzzles Who it is for Math & theory
15 Puzzle Algorithm — From BFS to Pattern Databases

A history of 15-puzzle solving algorithms: BFS, A* with Manhattan distance, IDA*, walking distance, and additive pattern databases.
Manhattan Distance Heuristic — Why It Works

How the Manhattan distance heuristic works, why it is admissible and consistent, and where it falls short for slide-puzzle solving.
Sliding Puzzle Strategy — The Techniques That Scale

The four techniques that solve any sliding tile puzzle by hand: row-and-column reduction, the L-shaped corner manoeuvre, empty-cell routing, endgame.
Why Some 15 Puzzles Are Unsolvable — A Plain-Language Check

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.
15 Puzzle Parity — The Proof of Why Some Configurations Are Unsolvable

The authoritative 15 puzzle parity proof: count inversions plus the blank's row to test any configuration. Worked examples for the 14–15 swapped and 13-15-14 boards, the knight (Guarini) cousin, and an FAQ.