15 Puzzle guide

Why some boards cannot be solved.

The 15 Puzzle looks like a simple random shuffle, but not every arrangement can reach the solved state. This guide explains the useful idea without requiring a proof-heavy detour.

Why solvability matters

A solved 4×4 board has the tiles in order and the empty space in the bottom-right corner. When a board is scrambled by moving a neighboring tile into the gap, every step is reversible and stays inside the same reachable set of positions.

A random permutation does not have that guarantee. The usual check counts inversions—the pairs of numbered tiles that appear in the wrong order—and also considers the empty square's row on an even-width board. If the parity does not match the solved state, the position is impossible rather than merely difficult.

A practical solve plan

1. Lock the first row. Place the first two tiles, then prepare the last pair together so you do not break the row while finishing it.
2. Work in layers. Repeat the same idea for the second row and then the left side of the bottom half.
3. Protect the gap. Treat the empty square as the control point; move it around the unfinished area instead of disturbing completed tiles.
4. Measure cleanly. Compare time, moves, and TPS only after a complete solve. A fast-looking start is not useful if it creates many corrections later.

Further reading

These references provide broader background on the puzzle and its mathematical structure:

• 15 Puzzle — WikipediaHistory and the classic solvability problem.
• 15 Puzzle — Wolfram MathWorldMathematical terminology and parity context.