How to Solve a 9×9 Sliding Puzzle?

The 9×9 sliding puzzle, or 80-puzzle, contains eighty numbered tiles and one blank space within a 9×9 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 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 [ ]

Only orthogonal slides into the adjacent blank are permitted. Of the 81! / 2 ≈ 1.2 × 10¹⁵³ possible states, exactly half are solvable due to permutation parity and blank row parity (odd grid). This article provides a rigorous, hierarchical reduction strategy that solves any solvable 9×9 instance in 900–1500 moves for advanced human solvers.

Fundamental Principles

Movement is restricted to up, down, left, or right. Solvability requires an even inversion count (excluding the blank) and an even row distance for the blank to the bottom. Interfaces typically feature move counters, state saving, and automated solvers for validation and sequence study.

Phase 1: Complete the Top Seven Rows (Tiles 1–63)

Solve a 9×7 sub-grid, leaving a 9×2 workspace.

Step 1.1: Solve Row 1 (Tiles 1–9)

Position tiles sequentially using the lower eight rows as maneuvering space. Lock the row upon completion.

Step 1.2: Solve Rows 2–7 (Tiles 10–63)

For each row r (2 ≤ r ≤ 7):

1. Vertical column insertion: Place tile (9(r-1)+c)* beneath tile (9(r-2)+c)* for c = 1 to 9.
2. Blank confinement: Restrict blank to rows r+1 through 9.
3. 7-move lock cycle after insertion:
   • Slide tile upward.
   • Blank right × 3 → down → left × 3 → up (extended perimeter) to secure.

Route the blank along the unsolved border to preserve upper structure.

Phase 2: Anchor the Bottom-Left 5×2 Block (Tiles 64, 73)

Reduce the remaining puzzle to a 9×2 band.

1. Place tile 64 in row 8, column 1.
2. Position tile 73 in row 9, column 1.

Use columns 6–9 in rows 8–9 as workspace. Complete column 1 fully before horizontal expansion.

Phase 3: Solve Row 8 (Tiles 65–72)

With column 1 secured in rows 8–9, fill row 8, columns 2–9.

1. Insert tile 65 in row 8, column 2.
2. Proceed to tile 72 in row 8, column 9.

Boundary-safe protocol:

• Maintain blank in row 9 unless lifting a tile.
• After placement, execute an 8-move edge loop (right × 4 → down → left × 4 → up) to return blank downward.

Phase 4: Resolve the Final 7×2 Endgame Band

The remaining twenty-two positions form a 7×2 rectangle (columns 3–9, rows 8–9) containing tiles 74–80, blank, and seven from row 8. Reduce to a 2×7 linear band after anchoring column 2.

Step 4.1: Secure Column 2

1. Place tile 74 in row 9, column 2.
2. Confirm tile 65 above it.

Step 4.2: Solve the 2×7 Band

Target configuration:

64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 [ ]

Apply 2×n band resolution techniques:

• Linear left-shift: Slide tiles along row 9, using row 8 as buffer.
• Adjacent swap (e.g., 79 ↔ 80):
  1. Blank left → 80 up → blank right → 79 right → 80 down.
  2. Reverse cycle.
• 3-cycle: Blank → 78 → 79 → blank (via column 9).
• 6-cycle for distant misalignment: Use column 9 as pivot for multi-tile rotation.

All 2×7 configurations resolve in ≤70 moves with memorized sequences.

Optimization Strategies

Human solutions average 1000–1300 moves; theoretical bounds exceed 1500.

• Macro-Block Solving: Treat 3×2 or 4×2 solved units as single entities.
• Pattern Database Integration: Reference 2×6 and 2×7 endgame tables mentally.
• Triple Highways: Preserve columns 7–9 as vertical blank transit lanes.
• Atomic Sequences: Bundle 7–8 move lock cycles into single mental operations.

Automated solvers demonstrate minimal paths for post-game refinement.

Common Errors and Corrective Measures

• Premature Row 8 Fill: Complete left columns before row 8 horizontal progression.
• Blank Entrapment: Never isolate blank in rows 1–7 without ≥9-move escape.
• Loop Inflation: Cap insertion sequences at 9 moves; restructure if breached.
• Endgame Misreduction: Treat final band as 2×7, not 7×2—solve linearly.

Theoretical and Practical Significance

The 9×9 puzzle represents the pinnacle of human-solvable sliding puzzles, testing advanced search algorithms (e.g., enhanced pattern databases, parallel IDA*). Human success hinges on hierarchical decomposition: 9×7 → 9×2 → 2×7. Mastery of band-solving, macro-blocks, and perimeter routing provides a universal template for all grid sizes.

Regular practice with move tracking and state persistence sharpens pattern recognition and strategic efficiency.

Conclusion

Solving the 9×9 sliding puzzle requires disciplined progression: secure the top seven rows via vertical column insertion, anchor the bottom-left 5×2 block, complete row 8 with extended edge loops, and resolve the final 2×7 band using linear shifts and cyclic corrections. This layered methodology transforms astronomical complexity into a structured sequence of sub-problems, enabling consistent solutions within competitive move ranges. Expertise in these principles equips solvers for the full spectrum of sliding puzzle challenges.

Practice these techniques on SlidoPuzzle.com.