For the standard 15-puzzle, first check that the starting layout is solvable; then work through overlapping two-row strips, keeping each completed section intact. The method is more reliable than trying to push every tile straight into place, because the last tiles in a row and the final area need room for setup moves.
Know the goal and legal move
The standard 15-puzzle is a 4×4 board with tiles 1–15 and one blank space. The goal is to put the numbered tiles in order from left to right and top to bottom, with the blank in the lower-right corner. A legal move slides one tile next to the blank into that space. The Purdue University puzzle guide notes that not every arrangement can reach the goal.
Check whether the board is solvable
Before spending time on moves, read the numbered tiles left to right and top to bottom, skipping the blank. Count an inversion whenever a larger-numbered tile appears before a smaller-numbered tile. Then count the blank’s row from the bottom, with the bottom row counted as 1.
For a 4×4 board, the solvability test is that the inversion count and blank-row number have opposite parity: one is even and the other odd. This convention matches the goal board, which has zero inversions and the blank in row 1 from the bottom. The University of Alberta’s sliding-tile puzzle resource describes the parity relationship for even-width boards. For boards with an odd number of columns, the corresponding condition is an even inversion count; do not apply the 4×4 blank-row rule unchanged to a different width.
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A board with 14 and 15 swapped and all other tiles already in their goal positions fails the 4×4 test: it has one inversion and the blank is in the bottom row. Legal moves cannot change a board from one solvability class to the other, so repeated trial and error will not fix that arrangement.
Solve the 15-puzzle in overlapping strips
The human method in Purdue’s solution guide works across two-row strips. The overlap gives tiles and the blank space to maneuver as you move down the board. Plan tile placements in sequence, and treat completed sections as fixed: a move that disrupts a solved area can undo more progress than it gains.
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- 15 Number Slide Puzzle.
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- Goal is to get all the numbers to be in order.
- To start, scramble the 15 numbered tiles in the tray then solve the puzzle by sliding the tiles back to the correct numerical order.
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- Arrange rows 1–2. Treat the top two rows as one 2×4 strip and place their tiles in target order. Do not assume every tile can be slid directly home; position other tiles and the blank to make the needed route.
- Arrange rows 2–3. Keep row 1 solved and work on the next overlapping pair. Use row 2 as the shared working space while placing the tiles in rows 2 and 3.
- Arrange rows 3–4. Preserve the upper rows and finish the bottom pair. The final tiles need careful setup rather than a simple straight-line placement.
Use a 2×2 loop to adjust tile order
When the blank is in a 2×2 block, moving it around a small loop brings it back to its starting corner while cycling the three numbered tiles in that block. A loop in the opposite direction cycles them the other way. These local cycles let you change tile order without shifting the blank’s position relative to the block. Use them as part of a plan; they do not mean that every three-tile arrangement can be achieved while keeping all surrounding tiles untouched.
Give the last positions special attention
The last two positions of a row cannot always be filled independently without disturbing the row or the strip below it. Set up those tiles together, using the overlap to reposition them before locking in the row. The same principle applies to the last two rows and the final 2×2 area. The University of Alberta resource describes a constructive approach that places tiles while avoiding fixed tiles, then handles the last elements and final area specially.
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A small worked example shows the pattern
Purdue’s guide demonstrates strip solving on a smaller 2×4 seven-puzzle. Its example starts with top row 1, 3, 4, 7 and bottom row 5, 2, 6, blank. That particular board is solved in six moves, with the top row fixed after move four. The six moves are specific to that starting layout; they are not a general move count for a 15-puzzle.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Choose a hand method or a computer search
For a person solving a physical board, overlapping strips provide a manageable plan: place tiles, preserve solved sections, and use local cycles for adjustments. A computer can instead search the board’s state graph. Breadth-first search finds a shortest path in principle, but the number of states grows quickly.
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A* search prioritizes states using an estimate of how much work remains. The University of Alberta resource identifies misplaced-tile count and summed Manhattan distance as admissible heuristics for the puzzle; used with A* as described there, they allow a shortest solution to be found. A constructive method, by contrast, can aim to finish reliably without minimizing the number of moves.
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- Hand usability: Strip solving is easier to follow at a board than tracking a large search tree.
- Guaranteed completion: A method must account for the last row and final area, not only easy tile placements.
- Shortest solution: Search with a suitable heuristic can target the fewest moves; a constructive method may use more.
- Board size: Search becomes harder as the state space grows, while human strip strategies depend on having room to maneuver around sections being solved.
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