How to Solve a 4×4 Sliding Puzzle?

The 4×4 sliding puzzle, known as the 15-puzzle, comprises fifteen numbered tiles and one blank space within a 4×4 grid. The objective is to arrange the tiles in ascending order from left to right and top to bottom, concluding with tile 1 in the top-left corner and the blank in the bottom-right position. This configuration represents the standard goal state:

 1  2  3  4
 5  6  7  8
 9 10 11 12
13 14 15 [ ]

The puzzle permits only orthogonal slides into the adjacent blank space. Of the 16! / 2 ≈ 10.5 trillion reachable states, exactly half are solvable due to permutation parity constraints. This article outlines a rigorous, layer-by-layer method to solve any solvable 4×4 instance efficiently, typically in 60–100 moves for human solvers.

Fundamental Principles

Movement is restricted to up, down, left, or right into the blank. Solvability requires an even number of inversions (pairs where a higher tile precedes a lower one in row-major order, ignoring the blank) plus an even taxicab distance for the blank to the bottom-right corner in even-sized grids. Interfaces generally provide move counters, progress saving, and automated solvers for verification.

Phase 1: Solve the First Two Rows

Treat the puzzle as two 4×2 sub-grids stacked vertically. Complete the top eight positions (tiles 1–8) before addressing the bottom.

Step 1.1: Position the First Row (Tiles 1–4)

1. Locate tile 1 and maneuver it to row 1, column 1.
2. Place tile 2 in row 1, column 2.
3. Insert tile 3 in row 1, column 3.
4. Position tile 4 in row 1, column 4.

Use the blank to orbit around misplaced tiles via 3- or 4-move cycles. Once the row is complete, designate it as immutable.

Step 1.2: Complete the Second Row (Tiles 5–8)

1. Position tile 5 directly beneath tile 1 (row 2, column 1).
2. Place tile 6 beneath tile 2 (row 2, column 2).
3. Insert tile 7 beneath tile 3 (row 2, column 3).
4. Position tile 8 beneath tile 4 (row 2, column 4).

To insert a tile without disturbing the first row, execute a vertical pair maneuver:

• Move the blank beneath the target column in row 3 or 4.
• Slide the target tile upward into row 2.
• Rotate the blank clockwise around the newly placed tile to lock it.

Repeat for each tile, routing the blank along the bottom two rows to avoid the solved section.

Phase 2: Solve the Left Column of the Bottom Half (Tiles 9 and 13)

With the top eight tiles fixed, reduce the effective puzzle to a 4×2 area (rows 3–4) while preserving the upper boundary.

1. Place tile 9 in row 3, column 1 (beneath tile 5).
2. Position tile 13 in row 4, column 1 (beneath tile 9).

Use the blank in columns 2–4 of rows 3–4 as workspace. A column-first strategy prevents premature disruption of row 3.

Phase 3: Solve the Third Row (Tiles 10–12)

Now target row 3, columns 2–4.

1. Insert tile 10 in row 3, column 2.
2. Place tile 11 in row 3, column 3.
3. Position tile 12 in row 3, column 4.

Employ edge-safe rotations:

• Keep the blank in row 4 unless moving a tile upward.
• After placing each tile, circulate the blank counterclockwise around it to secure position.

At this stage, the grid resembles:

 1  2  3  4
 5  6  7  8
 9 10 11 12
13  ?  ? [ ]

Phase 4: Resolve the Final 2×2 Corner

The remaining tiles (14, 15, and blank) occupy the bottom-right 2×3 area, but only a 2×2 sub-puzzle requires resolution. Four configurations are possible; all are solvable in ≤14 moves.

Case A: Tiles 14 and 15 swapped, blank correct

Execute a T-shape cycle twice:

1. Blank left → 15 up → blank right → 14 right → 15 down → blank left.
2. Repeat inverted sequence.

Case B: Cyclic misalignment (14 → 15 → blank)

Use a 3-cycle resolution:

1. Move 15 left.
2. Blank down, right, up sequence to rotate positions.
3. Restore blank to bottom-right.

Case C: Linear misalignment

Slide tiles along row 4, using row 3 column 4 as temporary storage.

Case D: Diagonal swap

Combine a vertical lift of 15 with horizontal shift of 14, then reverse.

These sequences are standard and documented in puzzle theory; practice recognizes patterns instantly.

Optimization Strategies

Human-optimal solutions average 60–80 moves; God’s algorithm upper bound is 80 for any position.

• Pair Solving: Position tiles in vertical pairs (1-5, 2-6, etc.) to reduce search space.
• Manhattan + Linear Conflict Heuristic: Add penalty for tiles in correct row/column but blocking each other.
• Parity-Aware Blank Routing: Maintain even blank row distance when possible.
• Macro Moves: Treat 2×2 rotations as single operations mentally.

Automated solvers demonstrate minimal paths for study.

Common Errors and Corrective Measures

• Early Bottom-Row Commitment: Delay row 3 until column 1 is complete.
• Blank Trapping: Never isolate the blank in a solved row without escape plan.
• Over-Rotation: Limit cycles to 4 moves maximum per insertion.
• Ignoring Parity: Verify solvability if progress stalls (rare in random generation).

Theoretical and Practical Significance

The 4×4 puzzle scales complexity exponentially, serving as a benchmark for search algorithms (IDA*, pattern databases). Human mastery requires chunking—recognizing 2×2 solved blocks as units—transferable to 5×5 and larger grids.

Regular practice with move tracking sharpens strategic foresight. The layered method ensures consistent solvability while inviting refinement.

Conclusion

Solving the 4×4 sliding puzzle demands disciplined progression: complete the top two rows, secure the left column, fill the third row, and resolve the final 2×2. Rotational maneuvers, pair positioning, and endgame patterns transform apparent chaos into structured logic. This systematic framework enables reliable solutions within competitive move counts, building expertise applicable to all sliding puzzle variants.

Practice these methods on SlidoPuzzle.com.