"Sliding tile puzzle" is the umbrella name. Underneath it sit half a dozen named puzzles, all sharing one rule — a tile can move only into an adjacent empty cell — and varying everything else: board shape, tile count, number of empty cells, whether tiles can rotate. This guide walks the family tree.
The square-grid branch: N puzzles
The mainstream of the family. A square board with N² − 1 numbered (or pictured) tiles and one empty cell.
| Name | Board | Tiles | Hardest optimal solve |
|---|---|---|---|
| 8 puzzle | 3×3 | 8 | 31 moves |
| 15 puzzle | 4×4 | 15 | 80 moves |
| 24 puzzle | 5×5 | 24 | 152 moves |
| 35 puzzle | 6×6 | 35 | est. 245 moves |
| 48 puzzle | 7×7 | 48 | unknown (research) |
| 80 puzzle | 9×9 | 80 | unknown |
The same strategy — solve the top row, solve the left column, recurse — works at every size in this column. Bigger boards mean longer solve times, not different puzzles.
These are the puzzles most people mean when they say "slide puzzle". They are also the ones computer scientists benchmark on.
Non-square grids
Less common but interesting:
Rectangular — 3×4, 4×5, 5×6 boards. Some commercial versions, including 19th-century wooden ones, were rectangular. The same strategy works; the corner-L manoeuvre is slightly different at the asymmetric edge.
Hexagonal — tiles on a hexagonal grid, six possible neighbours per cell instead of four. Mathematically more permissive (more move options per state), psychologically more confusing. A niche.
Triangular — tiles on a triangle grid. Even rarer. The math is fine; the gameplay is awkward.
More than one empty cell
The single-empty-cell version is the standard. There are commercial puzzles with two or more empty cells — most famously Klotski and its relatives, where tiles of different sizes (1×1, 1×2, 2×2) slide around a board with a few empty cells. Klotski is a different game, strategically: you are not trying to put tiles in order but to manoeuvre a particular tile to an exit.
Klotski is sometimes lumped under "sliding tile puzzle". It is not really part of the family — different goal, different strategy, different mathematical structure.
Sliding plus rotation
Combine the slide rule with a rotation mechanic and you get a wide variety of physical puzzles: the Hungarian Rings, certain Rubik's-Cube-adjacent products, and the so-called "loopover" puzzles. These cross-pollinate with the slide-puzzle family but are again strategically different.
For a player coming from the 15 puzzle, the closest such cousin is Rubik's 15 — a small physical toy with the 15-puzzle layout but the constraint that adjacent pairs can also be swapped.
What ties the family together
Two mathematical facts make the slide-puzzle family coherent:
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State-graph structure. Every variant can be modeled as a graph: nodes are board states, edges are legal moves. Optimal solving is shortest-path on this graph. The graph is enormous but well-behaved, which is why heuristic search works so well.
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Parity invariants. Most variants — including all the standard N puzzles — have a parity rule that splits the reachable states into two halves. Half are reachable from the goal; half are not. Apps that generate random starting positions either pre-screen with the parity test or generate by walking backwards from the goal.
What to play, by mood
If you have never tried any of them:
- Start with the 8 puzzle for ten minutes. It is fast enough that you can absorb the rule and try the strategy.
- Move to the 15 puzzle when 8 feels trivial. This is the canonical experience.
- Try the 24 puzzle when you want a longer sit-down.
- Try Klotski if you want a different kind of slide puzzle — same mechanic, different goal.
- Try hexagonal grids if you want to feel like a novice again. They will humble you.
Most modern apps, ours included, ship the 8, 15, 24, and 35 puzzles in one place. That is the central trunk of the family, and the central trunk is where almost everyone plays.