15 Puzzle
strategy.

Start with 1 and 2

Bring 1 to the top left, then 2 beside it. To move a tile in a chosen direction, the gap must first be on that side of it. Trace that route before moving the number. You may need to send a target away from home to get the gap around it.

Once 1 and 2 are set, try to keep them there. Work in the remaining cells to stage 3 at the top-right corner and 4 directly beneath it. Bring the gap to the top row’s third cell. You now have a two-slide finish: 3 left, then 4 up.

The diagram shows just the top two rows. Lower cells remain available for staging; the numbers 5, 6 and 7 shown here are context, not a second row you must already have solved.

Repeat the idea for 5 through 8

With the first row complete, use the three rows below it. Put 5 and 6 in the second row’s first two cells. Stage 7 at the far right of that row and 8 underneath. Bring the gap to the left of 7; slide 7 left and 8 up. Do not route the gap through the completed first row just because it looks shorter.

“Protect a row” is a planning rule, not a lock in the engine. You can still move those tiles. If you disturb one, make sure you have a deliberate way to restore it before continuing.

For the last two rows, turn your thinking sideways

There is no spare row underneath the bottom row, so the same row-closing trick has no room to work. Instead, finish the leftmost column of the remaining two-row strip as a pair: 9 above 13, then 10 above 14. Keep the unfinished cells to the right free for staging and gap circulation.

As each column is completed, treat it as protected. Work toward the final 2×2 square containing 11, 12, 15 and the gap. Rotate those remaining pieces around the square to reach the finish. On a solvable board with the earlier edges correct, the final area can be brought into order by this cycle.

Why the closest tile is not always the next tile

A tile one cell from its goal can be blocked because the gap cannot reach the useful side without breaking a pair. Moving a different tile away can open that route. Counting straight-line distances is helpful for comparing starts, but does not tell you the sequence of slides to make.

If the last two numbers of a row are reversed, stop trying to exchange them directly. Return to staging the pair, using the unfinished area below. A direct swap is not a legal move and one missing tile can undo many rushed slides.

Check the premise before blaming the strategy

A hand-entered 4×4 arrangement can be impossible. Read the numbered tiles left to right and count inversions, ignoring the gap. Add the gap’s row counted from the bottom. An odd sum is solvable; an even sum is not. Our board starts by legal shuffles, so an unsolved game here always has a route. This parity test is different from the 3×3 rule, which needs an even inversion count alone.

Use the controls as a practice notebook

Touch a tile beside the gap to slide it. For keyboard practice, focus the board and use the direction the tile should travel. Restart gives you the exact same shuffle, so you can rehearse a better staging route. Undo adds a move; it is useful for learning, not a way to reduce a recorded move count.

Hint reveals a legal next tile from a verified complete route. It is not an optimal solver, and its route need not match the row you were working on. Once shown, the attempt stays Assisted, even after a restart of that shuffle. New puzzle begins a fresh Solo attempt. Timer and separate local bests remain visible beside your practice board.