How to Solve a 6×6 Sliding Puzzle?

The 6×6 sliding puzzle, known as the 35-puzzle, contains thirty-five numbered tiles and one blank space within a 6×6 grid. The objective is to arrange the tiles in ascending order from left to right and top to bottom, with tile 1 in the top-left corner and the blank in the bottom-right position:

 1  2  3  4  5  6
 7  8  9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 [ ]

Movement is restricted to orthogonal slides into the adjacent blank. Of the 36! / 2 ≈ 3.7 × 10⁴¹ states, exactly half are solvable due to permutation parity and blank parity constraints for even grids. This article details a hierarchical, row-by-row reduction strategy that systematically solves any solvable 6×6 instance in 200–350 moves for skilled human solvers.

Fundamental Principles

Tiles slide only into the adjacent blank space. Solvability requires an even inversion count (excluding the blank) and an even taxicab distance for the blank to the bottom-right corner. Interfaces typically include move counters, state saving, and automated solvers for verification.

Phase 1: Complete the Top Four Rows (Tiles 1–24)

Treat the grid as a 6×4 solved section atop a 6×2 workspace.

Step 1.1: Solve Row 1 (Tiles 1–6)

Position tiles sequentially using the entire lower grid as maneuvering space. Lock the row upon completion.

Step 1.2: Solve Rows 2–4 (Tiles 7–24)

For each row r (2 ≤ r ≤ 4):

1. Vertical pair insertion: Place tile (6(r-1)+c)* beneath tile (6(r-2)+c)* for columns c = 1 to 6.
2. Blank corridor: Maintain the blank in rows r+1 to 6.
3. 4-move lock cycle after each tile:

• Slide tile up into position.
• Blank right → down → left → up (clockwise) to secure.

Route the blank along the unsolved perimeter to avoid disturbing solved rows.

Phase 2: Secure the Bottom-Left 2×2 Anchor (Tiles 25, 31)

Reduce the remaining puzzle to a 6×2 band (rows 5–6).

1. Place tile 25 in row 5, column 1.
2. Position tile 31 in row 6, column 1.

Use columns 3–6 in rows 5–6 as workspace. Complete column 1 fully before proceeding horizontally.

Phase 3: Solve Row 5 (Tiles 26–30)

With column 1 locked in rows 5–6, fill row 5, columns 2–6.

1. Insert tile 26 in row 5, column 2.
2. Continue sequentially to tile 30 in row 5, column 6.

Edge-safe protocol:

• Keep blank in row 6 unless lifting a tile.
• After placement, execute a 5-move perimeter loop (right → down → left → up → right) to return blank downward without displacement.

Phase 4: Resolve the Final 4×2 Endgame Band

The remaining ten positions form a 4×2 rectangle (columns 3–6, rows 5–6) containing tiles 32–35, blank, and four from row 5. Reduce to a 2×4 linear band after anchoring column 2.

Step 4.1: Anchor Column 2

1. Place tile 32 in row 6, column 2.
2. Secure tile 26 above it (already in row 5).

Step 4.2: Solve the 2×4 Band

Target configuration:

25 26 27 28 29 30
31 32 33 34 35 [ ]

Standard 2×n band resolutions apply:

• Parity-aligned linear sort: Slide tiles leftward along row 6, using row 5 as temporary storage.
• Swap correction (e.g., 34 ↔ 35):

1. Blank left → 35 up → blank right → 34 right → 35 down.
2. Mirror cycle to restore.

• 3-cycle rotation: Blank → 34 → 35 → blank (counterclockwise via column 6).

All 2×4 configurations resolve in ≤28 moves with memorized sequences.

Optimization Strategies

Human solutions average 250–300 moves; theoretical bounds exceed 400.

• Quad-Pair Solving: Position tiles in 2×2 blocks (1-2-7-8, etc.) to reduce cognitive load.
• Pattern Database Mental Lookup: Memorize 2×3 and 2×4 endgame tables.
• Blank Highway: Preserve a movable column (e.g., column 6) as a vertical blank path.
• Macro Sequencing: Treat 4-move lock cycles as atomic operations.

Automated solvers provide optimal paths for post-game analysis.

Common Errors and Corrective Measures

• Early Row 5 Horizontal Fill: Complete left columns before row 5 expansion.
• Blank Entrapment: Never isolate blank in rows 1–4 without multi-move escape.
• Excessive Micro-Moves: Limit insertion sequences to ≤6 moves; replan if breached.
• Endgame Misreduction: Treat final band as 2×4, not 4×2—solve linearly.

Theoretical and Practical Significance

The 6×6 puzzle amplifies state-space complexity, serving as a testbed for advanced AI techniques (e.g., bidirectional search, corner pattern databases). Human mastery hinges on progressive reduction: 6×4 → 6×2 → 2×4. Expertise in band-solving and macro patterns scales directly to 7×7 and 8×8 grids.

Routine practice with move tracking and state saving accelerates proficiency and move efficiency.

Conclusion

Solving the 6×6 sliding puzzle demands structured hierarchy: secure the top four rows via vertical pair insertion, anchor the bottom-left 2×2, complete row 5 with edge-safe loops, and resolve the final 2×4 band using linear and cyclic sequences. This layered methodology converts exponential complexity into a finite series of sub-problems, enabling consistent solutions within competitive move ranges. Mastery of these principles establishes a robust framework for all even-dimensional sliding puzzles.

Practice these techniques on SlidoPuzzle.com.