The fifteen tiles
The 4×4 sliding puzzle, often called the 15-puzzle, has been around since the 1870s. It caused a real craze around 1880, and it has stayed a standard brain teaser ever since. Its smaller cousin, the 3×3 version, is a common example in computer science for testing search methods. The game here offers both, plus 5×5 and 6×6 boards.
The key idea is that you never move a tile on its own. You move the gap. Every slide swaps the gap with one of the tiles touching it. So to send a tile somewhere, you walk the gap around it.
Moving one tile where you want it
Say you want to move a tile one square to the left. First bring the gap to the square on the tile's left, without disturbing tiles you have already placed. Slide the tile in. Now the gap is on the tile's right, the wrong side. To push the tile left again, the gap has to circle round it: down, left, left, up, or the same over the top. Then you slide the tile once more. Each further step left costs five slides.
That five-slide loop is the basic building block of every solve. Practise it on the 3×3 board until you do it without thinking. Choose which side to circle on so the gap passes through unsolved tiles, never through a finished row.
The same loop works in any direction. To move a tile diagonally, alternate. Move it one step across, then one step down, and the gap will already be close to where it needs to be next.
Shrinking the board
Large boards don't need new techniques. They need the same ones, applied in layers. Solve the top row, then the left column. What remains is a smaller board of the same shape. A solved top row and left column on the 5×5 leave a 4×4 problem. Do it again and you have a 3×3. On the 6×6, peel off three rows and columns this way. Picture the board shrinking, and you'll never face a big puzzle all at once.
Here is a short example on the 4×4 board. With the top row done, place 5 and 9, the rest of the left column. Treat them like the last two tiles of a row, turned on their side. Bring 9 into the square where 5 belongs, then bring 5 to its right. Now drop both down at once with the gap. The left column is done, and only a 3×3 block with tiles 6, 7, 8, 10, 11, 12, 14 and 15 remains.
The last 2×3
The end of every solve is a 2×3 or 2×2 area in the bottom-right corner. In a 2×2 area, the three tiles just rotate in a circle. Turn the gap around until they line up. You don't need to think, just count. For a 2×3 area, place the left column of the two rows first, as you did above. Then finish the 2×2.
Mistakes that cost moves
- Breaking finished rows. It is tempting to borrow a solved tile to get the gap through. That costs you the slides to fix it again. Route the gap through unsolved space, even if that's a little longer.
- Chasing sage tiles. A tile turns sage when it sits in its home spot. Early in a solve, a tile that happens to be home may have to move. Don't protect it until its row or column is the one you're working on.
- Placing the last tile of a row directly. It almost never fits without disturbing its neighbour. Use the drop-in trick: stack the last two tiles, then swing them in together.
Practice and settings
Settings has four sizes: 3×3, 4×4, 5×5 and 6×6. Start on 3×3 to learn the five-slide loop. Move to 4×4 when you can solve it without undoing your own rows. The Moves counter is your score, and the clock only starts on your first slide, so you can study the board first. For a change, switch the tile face from Numbers to Picture and use a photo of your own. Without numbers you'll read edges and colours instead, which is a good test of whether you really know where each tile goes.
