15 puzzle

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Revision as of 15:24, 5 May 2020 by RedstoneTim (talk | contribs) (Added notation and methods)
15 puzzle
Alternative names: 16 puzzle, Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square
Shape: Square
Internal Mechanism: None
Inventor: Noyes Palmer Chapman
Year: 1874
Produced by: QiYi

The 15 puzzle is a sliding puzzle made out of 15 numbered square pieces in a 4x4 grid, with one empty spot which allows for moving the pieces around. The objective is to solve the puzzle by bringing the pieces with numbers on them in the correct order. Instead of numbers, images are sometimes used as well.


Due to the puzzle's properties, 15 puzzle notation is quite easy and intuitive.

There are four basic types of moves: U, R, D and L, referring to Up, Right, Down and Left, respectively. The moves describe the movement of the pieces, not the empty spot. That means that when a piece moves to the right and the empty spot moves to the left, this is referred to as R. When more than one of the same move is done in a row, the number of moves is appended instead of writing out the move that many times. E.g. when R is performed twice, this is referred to as R2.


Lots of different approaches to solving the 15 puzzle exist, some of which are listed here.


Layer by layer (sometimes also called Rows) is one of easiest methods to come up with and therefore most commonly used among non-speedsolvers. The puzzle is done layer by layer or in rows for most of the solve. The steps are the following:

  1. Solve one row.
    1. Solve the two upper-left pieces.
    2. Place one of the last two pieces in the correct layer but wrong position.
    3. Place the other piece on top.
    4. Solve the first piece using L or R.
    5. Solve the second piece using U.
  2. Solve another row.
  3. Solve the two left-most pieces (similar to solving the last two pieces of one layer, but vertically).
  4. Solve the next two left-most pieces.
  5. Solve the remaining three pieces intuitively.

For solving the last two pieces of a layer, one can alternatively also solve the piece that touches one of the other two pieces (i.e. not the one that belongs in the corner), move the first piece of the layer out of the way, move the other three pieces (including the random edge) in its place, move the unsolved edge up so it becomes solved, take the random edge out of the layer and resolve the second, third and first piece.


The fringe method is a method for speedsolving the 15 puzzle and roughly similar to LBL in speed. It solves the puzzle by solving the fringes of the puzzle, gradually reducing it to smaller puzzles until the puzzle is solved.

  1. Solve the top and left row to reduce the puzzle to an 8 puzzle.
  2. Repeat again to reduce to a 3 puzzle.
  3. Solve the remaining three pieces intuitively.

External links

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