Notes on this puzzle
Fifteen #5204 is a 4×4 board with 15 numbered tiles and one gap, which starts in row 1, column 2. 3 tiles start in their home squares.
Add up how far each tile is from its home square and you get 17. 2 pairs of tiles already sit in their correct line but in the wrong order, which pushes the minimum to at least 21 moves. The shortest possible solution is exactly 31 moves. Matching it is a real achievement. Solving it row by row, the method from our guide, takes 59 moves on this board. Tile 6 has the longest trip home: 2 squares.
Where to beginTiles are not far from home after this short shuffle; fix one corner, then grow outwards from it.