Sliding Tile Puzzles, Explained

What it is

The classic version is the 15 puzzle, a four by four frame holding fifteen numbered tiles and one gap. You slide a tile into the gap to move it, and the goal is to arrange the numbers in order. Because you can only move tiles into the single empty space, getting one tile home often means disturbing others.

Solve it in layers

The trick is to solve the puzzle in order and never break what you have finished. Complete the top row first, then the left column, then treat the rest as a smaller puzzle. Working outside in shrinks the problem until only a small corner remains.

The tricky corners

• When placing the last two tiles of a row, set them up together in a rotation rather than one at a time.
• Keep the empty space where you want to work, and plan two or three moves ahead.
• The final two by two corner is solved with a simple cycle of rotations.

A note on solvability

Not every scramble of a sliding puzzle can be solved. Exactly half of all arrangements are reachable, a fact rooted in a neat piece of mathematics about swaps. A well made app only ever hands you a solvable board, so you can always trust that a solution exists.