Notes on this puzzle
Fifteen #5312 is a 4×4 board with 15 numbered tiles and one gap, which starts in row 1, column 1. 1 tile starts in its home square.
Add up how far each tile is from its home square and you get 36. 1 pair of tiles already sit in their correct line but in the wrong order, which pushes the minimum to at least 38 moves. Solving it row by row, the method from our guide, takes 86 moves on this board. Tile 13 has the longest trip home: 4 squares.
Where to beginSolve the top row, then the left column, and treat what remains as a 3×3 puzzle.