The 15 Puzzle
Fifteen tiles, one gap, four rows. The 15 puzzle is small enough to hold in one hand and large enough that pushing hopefully at it will not work: there are more than ten trillion positions it could be showing you, and the only way home is a plan followed in a fixed order. The board below is scrambled by legal slides from a finished frame, so whatever it has dealt you can definitely be finished. What follows is the order to work in, and the one pair of tiles that refuses to be placed one at a time.
Eighty slides is the worst the frame can do to you
The furthest any reachable position sits from a finished board is 80 single-tile slides, settled by exhaustive computer search rather than by argument, and it is the number worth carrying around. The count of positions is not: it is enormous, it is the subject of the page on why half of them cannot be finished at all, and it tells you nothing whatever about what to do next.
Eighty does tell you something. Whatever this frame has dealt you is nowhere near it, which means a board that has resisted a hundred hopeful pushes is not a hard board – it is a board being worked in no particular order. The order is below, and it is the same order every time, on every scramble, for the whole of the rest of your life with this puzzle.
The order that works, tile by tile
Finish the board from the outside in and never disturb what is already done. Place tile 1, then tile 2, then tiles 3 and 4 as a pair. That completes the top row and you never touch it again. Repeat on the second row: tile 5, tile 6, then 7 and 8 together. You are left with a two-row strip along the bottom, and that strip is worked in columns rather than rows – 9 and 13 together, then 10 and 14 together. Three tiles remain in a two-by-two box in the corner, and they only rotate: keep turning them the same way round and they arrive.
Why the strip flips from rows to columns is worth knowing, because it is where most people lose a row they had already finished. Once the top two rows are locked, the gap can only travel inside the bottom two, and a tile down there can only be brought home by circling it within the strip. Finishing the third row left to right in the ordinary way needs working space above it that no longer exists.
The two tiles you cannot place one at a time
Try it the obvious way and the trap is immediate. Put tile 3 in its home square at the top, then try to bring tile 4 in beside it: the only route the gap has into tile 4’s corner runs through the square tile 4 itself has to occupy, so the two of them circle each other indefinitely. The same dead end waits at the bottom of every column.
The way out is to place both in one manoeuvre. For the end of the top row, park tile 3 in the top-right corner – tile 4’s home, not its own – and park tile 4 directly underneath it. Bring the gap round to tile 3’s home square, slide tile 3 left into it, then slide tile 4 up. Two moves, both home. The column version is that shape turned ninety degrees: for tiles 9 and 13, park 9 in the bottom-left cell with 13 immediately to its right, bring the gap to 9’s home square, slide 9 up and then 13 left.
Why the hint sometimes shows a number in brackets
Press Hint and one tile takes a dashed outline, with its number printed under the board. Sometimes a second number follows in brackets, and that one is a promise rather than decoration: it is how many moves remain in a complete shortest solution, and it is printed only when a shortest solution was genuinely computed. On a four-by-four that search is capped at 22 moves deep, because beyond that it would lock the tab up for seconds at a time. A badly scrambled frame therefore takes its hint from the layered plan described above instead – a real move on a real route home, just not the shortest one, and no bracketed count beside it. One scrambled board played on nothing but those hints, move after move without a single decision, came home in 104 slides. When the bracketed number does appear you are within 22 moves of a finished frame, and it drops by one for every hint you follow.