The 24 puzzle is what you get when you take the 15 puzzle and add a row and a column. Twenty-four numbered tiles on a 5×5 board, one empty cell, the same rule: slide neighbouring tiles into the gap until the numbers are in order.
It is the obvious next step after 4×4 — and it is genuinely a different game.
The arithmetic of "one bigger board"
The state space explodes:
| Board | Cells | Tiles | Reachable states | Hardest solve |
|---|---|---|---|---|
| 3×3 (8 puzzle) | 9 | 8 | 181,440 | 31 moves |
| 4×4 (15 puzzle) | 16 | 15 | ~10.46 × 10¹² | 80 moves |
| 5×5 (24 puzzle) | 25 | 24 | ~7.76 × 10²⁴ | 152 moves |
| 6×6 (35 puzzle) | 36 | 35 | ~1.86 × 10⁴¹ | est. 245 moves |
Each step roughly squares the state space. The hardest 24 puzzle takes nearly twice as many moves as the hardest 15 puzzle — and that is the optimal number of moves. A human using the row-and-column method typically takes 200–300 moves.
Solve time, by hand
A confident solver who has mastered the 15 puzzle usually takes 10–20 minutes for a 24 puzzle on first attempts. By the tenth game, the same person is around 6–10 minutes.
The strategy is the same as on smaller boards: solve the top row, solve the left column, recurse into the smaller subpuzzle. The trick is that you now do this twice: solve row 1 and column 1, then solve row 2 and column 2 of the remaining 4×4, then solve the embedded 3×3. Each peel of the onion uses the L-shaped corner manoeuvre.
What gets harder
Two specific things scale badly:
Working memory load. On a 3×3 you track one strategic goal at a time. On a 5×5 you are juggling "I am trying to place 5 in the top-right" while not disturbing tiles 1‑4 and not boxing yourself into a corner. The cognitive load grows.
The "L-shaped manoeuvre" stretches. On a 3×3, putting the last tile of the top row into the corner requires about 5 sub-moves. On a 5×5 the same trick takes 10–12 sub-moves and is two cells deep. New solvers often disturb a row they had locked because they did not account for how far the manoeuvre reaches.
What stays the same
Surprisingly much. The pattern of strategy is exactly the same as the 3×3 game. The endgame — a recursive 3×3 — is identical. If you know the smaller boards, the 5×5 is a longer version of what you already know, not a different puzzle.
That recursive consistency is part of why mathematicians like the slide-puzzle family: the same row-and-column reduction works at every size.
Why solvers struggle here
For optimal solving by computer, the 24 puzzle is where Manhattan distance stops being good enough. A* and IDA* with Manhattan distance can solve any 15 puzzle in seconds but time out on the hardest 24 puzzles in hours.
The solution, found by Korf and Felner in the early 2000s, is additive pattern databases: precompute the optimal cost of permuting subsets of tiles (say, 5+5+5+9), then add the subset costs at search time. With a well-chosen partition, any 24 puzzle solves in seconds even at optimality. Full details live in the 15 puzzle solver guide.
Premium territory
In most apps, 5×5 sits behind a paywall. The reason is simple: it generates much longer sessions than 3×3 or 4×4, so apps treat it as a "value-add" rather than a free-tier feature. Slide Puzzle is the same — 5×5 is part of Premium, alongside 6×6 and unlimited photo imports.
That gating is not malicious. The 5×5 is a different kind of experience — a long, quiet evening with a single board — and it costs more to ship (the artwork has to be sharper, the algorithms have to be more careful about solvability).
When to play the 24 puzzle
A 5×5 board is for the kind of evening when you want to sit still for fifteen minutes and think about nothing else. It is not for the bus, not for the queue, not for the kettle. It is for after the kettle.
It is also the size at which photo slide puzzles become demanding: the photo has to be readable at 25-piece granularity, and many otherwise-good photos break down. The 24 puzzle is the size that taught me which photos work and which don't.
If 15 puzzles feel quick, this is the next step. If 5×5 feels long, the 35 puzzle — 6×6 — is the next one after that.