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The Sliding Puzzle — often called the 15-puzzle — is a 4×4 grid containing 15 numbered tiles and one empty space. Any tile adjacent to the empty space can slide into it. There are no pieces to lift, no tiles to swap arbitrarily — only one move type, repeated over and over. The constraint is what makes the puzzle so deep: simple rules, surprisingly intricate problem-solving.
Your goal is to slide the tiles into ascending numerical order: 1, 2, 3, 4 on the top row, 5, 6, 7, 8 on the second row, 9, 10, 11, 12 on the third row, and 13, 14, 15 plus the empty space on the bottom. Once the arrangement is in order with the empty cell in the bottom-right corner, the puzzle is solved. Strong players can solve any scramble in under a minute; the goal for casual players is just to get there at all.
The sliding tile puzzle dates back to 1874 and is generally credited to Noyes Chapman, an American postmaster who likely created the original 15-puzzle. It became a worldwide craze in 1880 — newspapers ran prize competitions and people obsessed over it the way Rubik's Cube would obsess a later generation. One famous unsolvable variant designed by puzzle promoter Sam Loyd offered a thousand-dollar reward that no one could collect, because the configuration was mathematically impossible. The proof of why some configurations are unsolvable was worked out shortly after by Johnson and Story.
Start by placing tiles 1 and 2 in their final positions. These are easy — you can usually move them directly without breaking anything. The hard part is placing tiles 3 and 4 without disturbing 1 and 2, because doing it naively is impossible. The solution is the corner trick: instead of putting 3 in column 3 and then 4 in column 4, you first place tile 4 in the top-right corner (column 4) and then place tile 3 in the cell directly below it (column 3, row 2). Then a short sequence rotates them into their final spots together.
Without the corner trick, the top row simply cannot be completed. Every experienced sliding-puzzle solver learns this pattern first because it's used for every row and column of the puzzle. It is the foundational technique of the entire solving method.
The corner trick has a short cyclic move sequence that's worth memorizing once. Once you know it, placing the last two tiles of any row takes seconds. Practice it five or six times in a row on a fresh puzzle until it becomes automatic — that single skill cuts your average solve time roughly in half.
Once your top row is locked in (1, 2, 3, 4 across the top), the same logic applies to the left column. Place tile 5 and tile 9 — but again, you cannot place them independently. Use the corner trick rotated 90 degrees: put tile 9 in the bottom-left corner of the current working area first, then put tile 5 above it, and rotate them into place together.
After the top row and left column are done, you have only a 3×3 region remaining. The 3×3 subpuzzle is much easier to think about than the full 4×4 because it has fewer tiles and a smaller move space. The whole strategy is about shrinking the problem until what's left is small enough to solve directly.
Inside the 3×3 subpuzzle, do the same thing: solve the top row of the 3×3 (tiles 6, 7, 8) with the corner trick, then solve the left column of the 3×3 (tiles 6, 10). At this point you've reduced the puzzle to a 2×3 block containing the last few tiles and the empty cell.
The discipline of "always shrink the working area" is what makes the row-by-row method scale to bigger sliding puzzles. The same approach works on the 24-puzzle (5×5) and even the 35-puzzle (6×6) — the only difference is that you have more rows and columns to peel off before reaching the small final block.
The final 2×3 block is the one place where the row-by-row method has to change. You cannot solve the bottom row of the 2×3 independently because of a property called parity — every move you make in the tiny remaining region permutes the tiles in a constrained way, and trying to lock one row leaves the other unsolvable.
The solution is to treat the 2×3 as a single unit and apply a known cyclic pattern that rotates the six remaining tiles into their final positions all at once. This pattern is worth memorizing alongside the corner trick — together they give you a complete recipe that can solve any solvable scramble of the 15-puzzle.
Desktop: Click any tile adjacent to the empty space to slide it into that space. Tiles not adjacent to the empty cell cannot move directly — you have to clear a path first.
Mobile / Tablet: Tap any tile adjacent to the empty cell to slide it, or swipe in the direction you want a tile to move. The whole row or column between your finger and the empty cell may shift together depending on settings.
You can usually choose the board size before each game. The classic is 4×4 (the 15-puzzle); 3×3 is much easier and a good practice ground.
The sliding puzzle scales naturally to different board sizes and themes:
The 15-puzzle remains the canonical version. Master it first, then scale up to 24 or down to 8 depending on your appetite.
The standard method is to solve the puzzle row by row and column by column. Place the top row (tiles 1, 2, 3, 4) first, using the corner trick to position 3 and 4 together. Then solve the left column (tiles 5 and 9) the same way. Once the top row and left column are locked, you only have a 3×3 subpuzzle left. Solve another row and column of that, and finally the last 2×3 block as a single unit using a fixed cycle pattern.
No. Only half of all possible tile arrangements on a 15-puzzle are solvable. The other half are mathematically impossible to solve no matter how many moves you make. This was proved by Johnson and Story shortly after the puzzle craze of the 1880s. It is determined by a property called permutation parity. Sam Loyd famously offered a 1000 dollar prize for solving one of the unsolvable configurations, knowing no one could ever claim it.
The corner trick is the standard technique for placing the last two tiles of a row or column without disturbing what you've already solved. For the top row, you place tile 4 in the top-right corner first and tile 3 in the cell directly below it. You then use a short sequence of moves to rotate them into their final positions together. The same trick works on the left column for tiles 5 and 9. Without the corner trick you cannot finish the row, because placing the last two tiles independently is geometrically impossible.
The 15-puzzle dates to around 1874 and is generally credited to Noyes Chapman, an American postmaster. It became a worldwide craze in 1880, with newspapers running prize competitions and crowds obsessed with solving it. Sam Loyd was a famous puzzle promoter of the era and is sometimes mistakenly credited as the inventor — he actually popularized an unsolvable variant. The mathematics behind solvability was published shortly afterward by Johnson and Story.
Any solvable configuration of the 15-puzzle can be solved in at most 80 single-tile moves. This was proved by computer search. The exact minimum for a given scramble varies — easy scrambles may need only 20-40 moves, while the hardest scrambles require all 80. The number is often quoted as the diameter of the 15-puzzle graph.
Once you have solved the top row, the left column, and one more row and column, you are left with a small 2×3 block. This last region cannot be solved tile by tile in the same way because moving any tile within it disturbs the others. Instead you have to treat the block as a single unit and apply a fixed cyclic pattern that rotates the remaining tiles into place. Most beginners get stuck here precisely because they keep trying to solve the bottom row independently and run into a parity problem.
Put the corner trick and row-by-row method into practice now.
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