The 15 puzzle — sometimes the "fifteen puzzle", "Mystic Square", or just "the 4×4 slider" — is the canonical slide puzzle. A 4×4 board with fifteen numbered square tiles and one empty cell. You slide tiles into the empty cell, one move at a time, until the numbers run in order: 1‑2‑3‑4 across the top, 5‑6‑7‑8, 9‑10‑11‑12, 13‑14‑15 and the empty in the bottom-right.
Where it came from
The puzzle was invented in 1874 by a postmaster from Canastota, New York named Noyes Palmer Chapman. By 1880 it had reached Boston schools, by mid-1880 it was a craze in the United States, and by autumn it had crossed the Atlantic and gripped Britain, France, and Germany. Newspapers ran columns of solutions. Workmen left their jobs to puzzle. Authorities in some German towns reportedly banned it from public spaces because government clerks were doing nothing else.
Sam Loyd — the New York puzzle-maker, never short of a self-promoting story — claimed to have invented it, which he hadn't, and offered $1000 in cash to anyone who could solve a specific 14-and-15-swapped variant. That variant turned out to be mathematically unsolvable (see 15 puzzle parity for the proof). Nobody won. Loyd was happy.
How to play
Three rules:
- You can only slide a tile that is adjacent to the empty cell — directly above, below, left, or right of it. No diagonals.
- The slide moves the tile into the empty cell. The empty cell is now where the tile used to be.
- You're done when the tiles are in numerical order.
That's it. There is no time limit, no scoring, no second goal. The whole game is in those three sentences.
How hard is it?
The 15 puzzle has 16!/2 ≈ 10.46 trillion reachable arrangements (half of all 16!, because of the parity rule). The shortest solution from the hardest starting position is 80 moves. The median is about 52.
A new player solves their first 15 puzzle in 15–30 minutes by trial and error. A player who has learned the row-and-column method from the 8 puzzle solves any 15 puzzle in 3–7 minutes. A speed-solver, racing optimally chosen moves, can do it in under 90 seconds.
What makes the 4×4 the sweet spot
There is a reason the 15 puzzle, and not the 8 or the 24, became the world phenomenon. The 4×4 board is:
- Long enough to be satisfying. The 8 puzzle solves so quickly that there is no real arc to it. The 15 takes long enough to give you the small relief of a finished board.
- Short enough not to drag. The 24 puzzle takes 15+ minutes for a confident solver. The 35 puzzle takes 30+. The 4×4 fits inside a coffee break.
- Just hard enough to think about. The strategy is not obvious at first contact. The "L-shaped corner manoeuvre" needs to be discovered or taught — and once it is, the whole puzzle opens up.
That sweet spot is why the 15 puzzle is the size that is used as shorthand for the whole family. People say "slide puzzle" and picture a 4×4.
In computer science
For decades the 15 puzzle was the standard benchmark for heuristic search algorithms. Richard Korf's 1985 paper introducing IDA* used 100 random 15 puzzles to demonstrate the speedup over A*. Modern pattern-database papers still report results on the 15. The puzzle is just small enough to be tractable, just large enough that bad algorithms time out.
If you have ever taken an "Introduction to AI" course, you have solved a 15 puzzle as a programming exercise. Most likely with A* and the Manhattan distance heuristic. (For deeper material, see the solver guide.)
Variants in modern apps
The 15 puzzle has a few common modern variants:
- Number tiles — the original, plain digits on tiles.
- Picture tiles — a photograph sliced into N×N replaces the numbers. You solve by reassembling the image.
- Hybrid — picture tiles with optional small numbers overlaid, for players who want a hint about where each tile belongs.
Some apps go further: hexagonal grids, larger boards (5×5 = 24 puzzle, 6×6 = 35 puzzle), rotation puzzles where the tiles can rotate, and so on. The 4×4 numeric version is the lingua franca.
Where to play
On a phone, native apps are the smoother experience — no upload, no analytics, no ads in the middle of a tile slide. Slide Puzzle ships with the classic numbered version plus 300 photographic covers and four sizes (3×3, 4×4, 5×5, 6×6).
On the web, browser-based 15 puzzles exist in the dozens. They are convenient but usually upload imported photos and run analytics. If you only want a couple of games and never want to import a photo of your own, they are fine.
If you have not solved one before, start with the 8 puzzle for thirty seconds — just to feel the move — then come back to the 15. The strategy you pick up on the smaller board scales directly.