How to Solve 5×5 Sliding Puzzle?

The 5×5 sliding puzzle, or 24-puzzle, consists of twenty-four numbered tiles and one blank space arranged in a 5×5 grid. The goal is to order the tiles in ascending sequence 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 [ ]

Only adjacent orthogonal slides into the blank are permitted. Of the 25! / 2 ≈ 7.8 × 10²⁴ possible states, exactly half are solvable due to permutation parity and blank positioning rules. This article presents a rigorous, incremental strategy that reduces the 5×5 puzzle to a sequence of smaller, manageable sub-problems, achieving human solutions typically in 120–200 moves.

Fundamental Principles

Tile movement is limited to up, down, left, or right. A configuration is solvable if the number of inversions (excluding the blank) is even and the blank’s row distance to the bottom is even (for odd-sized grids). Modern interfaces provide move counters, save states, and automated solvers for validation and study.

Phase 1: Complete the First Three Rows (Tiles 1–15)

Solve the top 5×3 sub-grid before addressing the bottom two rows.

Step 1.1: Position the First Row (Tiles 1–5)

1. Maneuver tile 1 to row 1, column 1.
2. Place tile 2 in row 1, column 2.
3. Continue sequentially to tile 5 in row 1, column 5.

Use 3- or 4-move blank cycles to orbit misplaced tiles. Lock the row upon completion.

Step 1.2: Fill the Second Row (Tiles 6–10)

Position each tile directly beneath its counterpart in row 1, using the bottom three rows as workspace. Insert via vertical insertion loops:

• Move blank to row 4 or 5 beneath target column.
• Slide tile upward into row 2.
• Circulate blank clockwise around the placed tile to secure.

Step 1.3: Complete the Third Row (Tiles 11–15)

Repeat the insertion process, now treating rows 1–2 as immutable. Route the blank exclusively through rows 4–5 to preserve upper structure.

Phase 2: Solve the Left Two Columns of the Bottom Half (Tiles 16, 21)

Reduce the remaining puzzle to a 5×2 area in rows 4–5.

1. Place tile 16 in row 4, column 1.
2. Position tile 21 in row 5, column 1.

Use columns 3–5 in rows 4–5 as maneuvering space. A column-pair strategy prevents row 4 disruption.

Phase 3: Solve Row 4 (Tiles 17–20)

With columns 1–2 in rows 4–5 secured, target row 4, columns 3–5.

1. Insert tile 17 in row 4, column 3.
2. Continue to tile 20 in row 4, column 5.

Employ edge-preserving rotations: keep the blank in row 5 unless lifting a tile. After each placement, execute a 4-move loop to lock the tile and return the blank downward.

Phase 4: Resolve the Final 3×2 Sub-Puzzle

The remaining six positions form a 3×2 rectangle (columns 3–5, rows 4–5) containing tiles 22, 23, 24, and the blank, with row 4 partially solved. Reduce this to a 2×3 endgame.

Step 4.1: Position Tile 22

Place tile 22 in row 5, column 3 (beneath tile 17). Use row 5, columns 4–5 as temporary storage.

Step 4.2: Solve the 2×3 Corner

The final six tiles (22–24, blank, and two from row 4) must be arranged as:

17 18 19 20
21 22 23 24 [ ]

Common configurations include:

• Linear row misalignment: Slide along row 5, using column 5 as pivot.
• Diagonal swap (23 ↔ 24): Execute a double-T cycle:

1. Blank left → 24 up → blank right → 23 right → 24 down.
2. Reverse to restore.

• Full 3-cycle: Rotate blank, 23, 24 counterclockwise using row 4 column 5 as anchor.

All 2×3 configurations are solvable in ≤18 moves with practiced sequences.

Optimization Strategies

Optimal human solutions range 120–160 moves; theoretical maximum is ~200.

• Row-Pair Solving: Position tiles in vertical pairs (1-6, 2-7, etc.) to halve vertical search.
• Enhanced Heuristics: Combine Manhattan distance with linear conflicts and pattern databases for mental planning.
• Blank Corridor: Maintain a movable blank path along unsolved edges.
• Macro Patterns: Recognize 2×2 solved blocks as single units.

Automated solvers reveal minimal paths for post-solution analysis.

Common Errors and Corrective Measures

• Premature Row 4 Commitment: Complete left columns before filling row 4 horizontally.
• Blank Isolation: Avoid trapping the blank in solved rows without extraction plan.
• Over-Cycling: Cap insertion loops at 6 moves; reassess if exceeded.
• Endgame Overlook: Treat final 3×2 as independent—solve methodically.

Theoretical and Practical Significance

The 5×5 puzzle exponentially increases state space, making it a standard for advanced search algorithms (e.g., IDA* with pattern databases). Human success relies on hierarchical reduction: 5×3 → 5×2 → 3×2. Mastery develops chunking, foresight, and algorithmic intuition, directly applicable to 6×6 and higher grids.

Consistent practice with move tracking and save states accelerates pattern recognition and efficiency.

Conclusion

Solving the 5×5 sliding puzzle requires disciplined progression: secure the top three rows, anchor the bottom-left columns, complete row 4, and resolve the final 3×2 endgame. Vertical pair insertion, edge-safe rotations, and standardized 2×3 resolutions transform complexity into structured logic. This layered methodology ensures reliable solutions within competitive move counts, establishing expertise scalable across all sliding puzzle dimensions.