Fifteen puzzle
From Wikipedia, the free encyclopedia
The n-puzzle is known in various versions, including the 8 puzzle, the 15 puzzle, and with various names. 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.
The n-puzzle is a classical problem for modelling algorithms involving heuristics. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the Manhattan distances between each block and its position in the goal configuration. Note that both are admissible, i.e., they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*.
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[edit] Solvability
A simple parity argument shows that half of the starting positions for the n-puzzle are impossible to resolve, no matter how many moves are made. This is done by considering a function of the tile configuration that is invariant under any valid move, and then using this to partition the space of all possible labeled states into two equivalence classes of reachable and unreachable states.
The invariant is the parity of permutations of all 16 squares (15 pieces plus empty square) plus the parity of the taxicab distance moved by the empty square. This is an invariant because each move changes the parity of the permutation and the parity of the taxicab distance. In particular if the empty square is not moved the permutation of the remaining pieces must be even. The same argument works for all rectangular boards, or more generally for all boards with no odd cycles.
[edit] Noyes Chapman's Fifteen Puzzle
In its most famous version, the Fifteen Puzzle, initially known as the Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square, and many others, has a 4×4 grid, where in the initial configuration the pieces are in ascending order, except that pieces 14 and 15 are in reverse order. This puzzle is not solvable because it would require a change of the invariant.
[edit] History
Sam Loyd claimed from 1891 until his death in 1911 that he invented the puzzle. But he had nothing to do with the invention or popularity of the puzzle.[1] The puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34. Copies of the improved Fifteen Puzzle made their way to Syracuse, New York by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, RI, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle and, by December 1879, selling them both locally and in Boston (Massachusetts). Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late-January 1880, Dr. Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle. The game became a craze in the U.S. in February 1880, Canada in March, Europe in April, but that craze had pretty much dissipated by July. Apparently the puzzle was not introduced to Japan until 1889. Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey.[1]
For larger versions of the n-puzzle, finding a solution is easy, but the problem of finding the shortest solution is NP-hard.[2]
In the USSR the Minus Cube was manufactured, a 3D variant of the 15-puzzle.
Bobby Fischer was an expert at solving the 15-Puzzle, provided that it was in a configuration that could be solved. He had been timed to be able to solve it within 25 seconds; Fischer demonstrated this on November 8, 1972 on The Tonight Show Starring Johnny Carson.
[edit] See also
[edit] Notes
- ^ a b The 15 Puzzle, by Jerry Slocum & Dic Sonneveld. ISBN 1-890980-15-3
- ^ Daniel Ratner, Manfred K. Warmuth. Finding a Shortest Solution for the N × N Extension of the 15-PUZZLE Is Intractable. National Conference on Artificial Intelligence, 1986.
[edit] References
- Archer, Aaron (1999). "A Modern Treatment of the 15 Puzzle", American Mathematical Monthly 106, pp. 793-799.
