Fifteen puzzle

The fifteen puzzle is a game in which 15 of the 16 squares of a 4x4 frame are filled with numbered sliding pieces, leaving one space in which to slide one piece at a time. The object is to slide the 15 pieces into numerical order.

Thus, the goal is to produce this configuration:

If the pieces 14 and 15 are exchanged, resulting in the following initial configuration:

then the puzzle is not soluble.

The 'hook' in the fifteen puzzle is that there are two distinct sets of positions which can be assembled from the pieces, depending on the parity (the parity in this context is the number of pairs of pieces that must be swapped to obtain the solution), and there is no way of moving between them using the allowed moves, as they preserve parity.

This allowed Sam Loyd, who invented the puzzle in the 1870s, to offer a prize of $1000 which could never be claimed, fuelling the popularity of the game. The game became a craze in both the USA and in Europe.

For larger versions of the fifteen puzzle, the problem of finding a solution is easy. The problem of finding the shortest solution is NP-complete.

External links:

• Article explaining parity in the fifteen puzzle