Slide Puzzle Guide: Solve the 15 Puzzle

The one rule and the one trap

The rule: a tile next to the gap can slide into it. That is all. The trap: you cannot swap two tiles without disturbing others, because every move is a slide. So the method is to solve the board in layers, making each layer permanent, and to shrink the problem until it is a 2 × 2 corner that falls to a few moves.

Every daily board can be solved: the scramble is made by sliding, so no impossible arrangement is ever shown.

Step 1: the top row

Place tile 1 in the top-left corner, then 2 beside it, then 3. Moving a tile around the board without disturbing solved tiles is the basic skill: bring the gap next to the tile on the side you want it to move toward, slide the tile, then walk the gap round (through unsolved tiles only) to the other side and slide again. Three or four of these "walk the gap round" cycles move any tile anywhere.

The last tile of a row, 4, needs the trick. If you place 4 straight into the corner you will find 3 has been pushed out. Instead:

1. Put 3 into the top-right corner (where 4 belongs).
2. Put 4 directly beneath it.
3. Move the gap to where 3 belongs (third square of the row).
4. Slide 3 left into its place, then 4 up into the corner.

The row is now permanent. Never touch it again.

Step 2: the second row

Solve 5, 6, 7 and 8 exactly the same way, using the bottom two rows as working space. The last two tiles of the row use the same trick: 7 into the end square, 8 beneath it, gap into 7's home, 7 slides across, 8 slides up.

On a 3 × 3 board (easy days), after the top row you go straight to step 3 with a 2 × 3 area left.

Step 3: the left column of what is left

With two rows solved you have a 2 × 4 area. Now switch direction: solve the left column of the remaining area from the top down, that is 9 then 13. The same end-of-line trick applies, rotated: put 13 where 9 belongs, 9 to its right, gap into 13's home, 13 slides down, 9 slides left.

Then 10 and 14 in the next column the same way. You are left with a 2 × 2 corner holding 11, 12, 15 and the gap.

Step 4: the final 2 × 2

Three tiles and a gap in a 2 × 2 rotate round. Slide them round the little square until 11 and 12 are on top and 15 is bottom-left. It takes at most a handful of moves, and if the tiles seem to go round forever, you have mis-placed a tile in an earlier step (usually 14 or 10). Check those first.

Doing it in fewer moves

The method above is reliable but not minimal. The leaderboard ranks by fewest slides, then time, so once the method is comfortable:

• Plan the gap's route before each cycle so it takes the short way round.
• Place two tiles with one journey when they are near each other.
• On 3 × 3 boards, look for positions where the final rotation can start early.
• Avoid moving a tile away from its home and back; every such round trip is two wasted slides.
• Use swipes or the arrow keys: the arrow moves the tile in that direction into the gap, which is faster than tapping.

Common mistakes

1. Placing the last tile of a row directly. It evicts its neighbour. Use the trick.
2. Disturbing solved rows. The gap must only travel through unsolved tiles.
3. Solving the bottom rows row by row. After two rows, go column by column.
4. Rotating the last 2 × 2 hoping for luck. If it will not come, a tile elsewhere is wrong.
5. Random sliding at the start. The first move should already be bringing tile 1 toward its corner.

Glossary

• Gap: the empty space a tile slides into.
• Cycle: moving a tile one step by sliding it, then walking the gap round to the other side.
• Layer: a row or column solved and never touched again.
• Parity: the mathematical property that makes half of all arrangements unsolvable; daily boards are always solvable because they are scrambled by sliding.