15 game

''The n-puzzle is known in various versions, including the 8 puzzle, the 15 puzzle, and with various names (Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others). It is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. If the size is 3×3, the puzzle is called the 8-puzzle or 9-puzzle, and if 4×4, the puzzle is called the 15-puzzle or 16-puzzle. The object of the puzzle is to place the tiles in order (see diagram) by making sliding moves that use the empty space.'' [source]

Puzzles, Groups, and Groupoids [source]

The Unapologetic Mathematician [source]

''The random arrangement of the 15-puzzle has an interesting property. If all rows/columns summed parity is odd, the puzzle is not solvable. So as long as the empty cell is situated in a corner and the summed row/column parity is even, the puzzle is solvable.

The probability to randomly place the pieces for a solvable/unsolvable puzzle in the 4×4 slots – distributes to an even 50-50%''[source]. More theory here. And the solution to the 8 puzzle: here.

SIMULATOR(S) [interactive simulation]

Variation:
[minus cube, animation]