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{{Short description|Sliding puzzle with fifteen pieces and one space}}
{{redirect|Magic 15|the numbered grid where each row and column sums to 15|Magic square}}
{{Use British English|date=July 2022}}
[[File:15-puzzle magical.svg|thumb|To solve the puzzle, the numbers must be rearranged into numerical order from left to right, top to bottom.]]
The '''15 puzzle''' (also called '''Gem Puzzle''', '''Boss Puzzle''', '''Game of Fifteen''', '''Mystic Square''' and more) is a [[sliding puzzle]]. It has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 tile positions wide, with one unoccupied position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The goal of the [[puzzle]] is to place the tiles in numerical order (from left to right, top to bottom).

Named after the number of tiles in the frame, the 15 puzzle may also be called a ''"16 puzzle"'', alluding to its total tile capacity. Similar names are used for different sized variants of the 15 puzzle, such as the '''8 puzzle,''' which has 8 tiles in a 3×3 frame.

The ''n'' puzzle is a classical problem for [[Modeling language|modeling]] [[algorithm]]s involving [[heuristic (computer science)|heuristic]]s. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the [[taxicab distance]]s between each block and its position in the goal configuration.<ref name="Korf,2000"/> Note that both are ''[[Admissible heuristic|admissible]]''. That is, they never overestimate the number of moves left, which ensures optimality for certain [[search algorithm]]s such as [[A* search algorithm|A*]].<ref name="Korf,2000"/>

==Mathematics==
=== Solvability ===
[[File:15-Puzzle solved.png|thumb|A solved 15 Puzzle]]
{{harvtxt|Johnson|Story|1879}} used a [[parity (mathematics)|parity]] argument to show 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 binary function of the tile configuration that is [[invariant (mathematics)|invariant]] under any valid move and then using this to partition the space of all possible labelled states into two mutually inaccessible [[equivalence class]]es of the same size. This means that half of all positions are unsolvable, although it says nothing about the remaining half.

The invariant is the [[Parity of a permutation|parity of the permutation]] of all 16 squares plus the parity of the [[taxicab distance]] (number of rows plus number of columns) of the empty square from the lower right corner. This is an invariant because each move changes both the parity of the permutation and the parity of the taxicab distance. In particular, if the empty square is in the lower right corner, then the puzzle is solvable only if the permutation of the remaining pieces is even.

{{harvtxt|Johnson|Story|1879}} also showed that on boards of size ''m'' × ''n'', where ''m'' and ''n'' are both larger or equal to 2, all even permutations ''are'' solvable. It can be proven by induction on ''m'' and ''n'', starting with ''m'' = ''n'' = 2. This means that there are exactly two equivalency classes of mutually accessible arrangements, and that the parity described is the only non-trivial invariant, although equivalent descriptions exist.

{{harvtxt|Archer|1999}} gave another proof, based on defining [[equivalence class]]es via a [[Hamiltonian path]].

{{harvtxt|Wilson|1974}} studied the generalization of the 15 puzzle to arbitrary [[finite graph]]s, the original problem being the case of a 4×4 [[grid graph]]. The problem has some degenerate cases where the answer is either trivial or a simple combination of the answers to the same problem on some subgraphs. Namely, for [[path graph|paths]] and [[cycle graph|polygons]], the puzzle has no freedom; if the graph is [[disconnected graph|disconnected]], only the connected component of the vertex with the "empty space" is relevant; and if there is an [[articulation vertex]], the problem reduces to the same puzzle on each of the [[biconnected graph|biconnected]] components of that vertex. Excluding these cases, Wilson showed that other than one exceptional graph on 7 vertices, it is possible to obtain all permutations unless the graph is [[bipartite graph|bipartite]], in which case exactly the even permutations can be obtained. The exceptional graph is a regular [[hexagon]] with one diagonal and a vertex at the center added; only {{sfrac|1|6}} of its permutations can be attained, which gives an instance of the [[Automorphisms of the symmetric and alternating groups#Exotic map S5 → S6|exotic embedding of S<sub>5</sub> into S<sub>6</sub>]].

For larger versions of the ''n'' puzzle, finding a solution is easy. But, the problem of finding the ''shortest'' solution is [[NP-hard]]. It is also NP-hard to [[Approximation algorithm|approximate]] the fewest slides within an additive constant, but there is a polynomial-time constant-factor approximation.<ref name="Ratner1986">{{cite journal |last1=Ratner |first1=Daniel |last2=Warmuth |first2=Manfred |title=Finding a Shortest Solution for the N × N Extension of the 15-PUZZLE Is Intractable |journal=National Conference on Artificial Intelligence |date=1986 |url=https://www.aaai.org/Papers/AAAI/1986/AAAI86-027.pdf |url-status=live |archiveurl=https://web.archive.org/web/20120309151834/https://www.aaai.org/Papers/AAAI/1986/AAAI86-027.pdf |archivedate=2012-03-09}}</ref><ref name="ratner+warmuth">{{cite journal|last=Ratner|first=Daniel|author2=Warmuth, Manfred|title=The (n<sup>2</sup>−1)-puzzle and related relocation problems|journal=Journal of Symbolic Computation|year=1990|volume=10|issue=2|pages=111–137|doi=10.1016/S0747-7171(08)80001-6|doi-access=free}}</ref> For the 15 puzzle, lengths of optimal solutions range from 0 to 80 single-tile moves (there are 17 configurations requiring 80 moves)<ref>Richard E. Korf, [https://dl.acm.org/citation.cfm?id=1455250 Linear-time disk-based implicit graph search], ''Journal of the ACM'' Volume 55 Issue 6 (December 2008), Article 26, pp. 29-30. "For the 4 × 4 Fifteen Puzzle, there are 17 different states at a depth of 80 moves from an initial state with the blank in the corner, while for the 2 × 8 Fifteen Puzzle there is a unique state at the maximum state of 140 moves from the initial state."</ref><ref>A. Brüngger, A. Marzetta, K. Fukuda and J. Nievergelt, [http://www.iro.umontreal.ca/~gendron/Pisa/References/BB/Brungger99.pdf The parallel search bench ZRAM and its applications], ''Annals of Operations Research'' '''90''' (1999), pp. 45–63.<br>:"Gasser found 9 positions, requiring 80 moves...We have now proved that the hardest 15-puzzle positions require, in fact, 80 moves. We have also discovered two previously unknown positions, requiring exactly 80 moves to be solved."</ref> or 43 multi-tile moves;<ref name="multi-tile">[http://cubezzz.duckdns.org/drupal/?q=node/view/223 "The Fifteen Puzzle can be solved in 43 "moves""]. Domain of the Cube Forum</ref> the 8 Puzzle always can be solved in no more than 31 single-tile moves or 24 multi-tile moves (integer sequence [http://oeis.org/A087725 A087725]). The multi-tile metric counts subsequent moves of the empty tile in the same direction as one.<ref name="multi-tile" />

The number of possible positions of the 24 puzzle is {{sfrac|25!|2}} ≈ {{val|7.76e24}}, which is too many to calculate [[God's number]] feasibly using brute-force methods. In 2011, lower bounds of 152 single-tile moves or 41 multi-tile moves had been established, as well as upper bounds of 208 single-tile moves or 109 multi-tile moves.<ref>[http://cubezzz.duckdns.org/drupal/?q=node/view/238/2557#comment-2557 "24 puzzle new lower bound: 152"]. Domain of the Cube Forum</ref><ref>[https://web.archive.org/web/20160307005428/http://juropollo.xe0.ru/stp_results_mxn_en.htm "m × n puzzle (current state of the art)"]. Sliding Tile Puzzle Corner.</ref><ref>[http://cubezzz.duckdns.org/drupal/?q=node/view/241 "208s for 5x5"]. Domain of the Cube Forum.</ref><ref>[http://cubezzz.duckdns.org/drupal/?q=node/view/241/2566#comment-2566 "5x5 can be solved in 109 MTM"]. Domain of the Cube Forum.</ref> In 2016, the upper bound was improved to 205 single-tile moves.<ref>[http://cubezzz.dyndns.org/drupal/?q=node/view/559 "5x5 sliding puzzle can be solved in 205 moves"]. Domain of the Cube Forum.</ref>

===Group theory===
The transformations of the 15 puzzle form a [[groupoid]] (not a group, as not all moves can be composed);<ref>Jim Belk (2008) [https://cornellmath.wordpress.com/2008/01/27/puzzles-groups-and-groupoids/ Puzzles, Groups, and Groupoids], The Everything Seminar</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-1 The 15-puzzle groupoid (1)], Never Ending Books</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-2 The 15-puzzle groupoid (2)], Never Ending Books</ref> this [[Group action (mathematics)#Generalizations|groupoid acts]] on configurations.

Because the combinations of the 15 puzzle can be generated by [[Permutation#Definition|3-cycles]], it can be proved that the 15 puzzle can be represented by the [[alternating group]] <math>A_{15}</math>.<ref>{{cite web | access-date=26 December 2020 | first1=Robert | last1=Beeler | url=https://faculty.etsu.edu/beelerr/fifteen-supp.pdf | title=The Fifteen Puzzle: A Motivating Example for the Alternating Group | publisher=East Tennessee State University | website=faculty.etsu.edu/ | archive-date=7 January 2021 | archive-url=https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf | url-status=dead }}</ref> In fact, any <math>2 k - 1</math> sliding puzzle with square tiles of equal size can be represented by <math>A_{2 k - 1}</math>.
==History==
[[Image:15-puzzle-loyd.svg|thumb|right|[[Sam Loyd]]'s unsolvable 15 Puzzle, with tiles 14 and 15 exchanged. This puzzle is not solvable as it would require a change of the invariant to move it to the solved state.]]
[[Image:Great presidential puzzle2.jpg|thumb|U.S. political cartoon about finding a Republican presidential candidate in 1880]]
The puzzle was "invented" by Noyes Palmer Chapman,<ref name="slocum-sonneveld" /> 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 (see [[magic square]]). Copies of the improved 15 puzzle made their way to [[Syracuse, New York]], by way of Chapman's son, Frank, and from there, via sundry connections, to [[Watch Hill, Rhode Island]], and finally to [[Hartford]], [[Connecticut]], where students in the [[American School for the Deaf]] started manufacturing the puzzle. By December 1879, these were sold both locally and in [[Boston]], [[Massachusetts]]. Shown one of these, Matthias Rice, who ran a 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, Charles Pevey, a dentist in [[Worcester, Massachusetts|Worcester]], Massachusetts, garnered some attention by offering a cash reward for a solution to the 15 Puzzle.<ref name="slocum-sonneveld"/>

The game became a [[Fad|craze]] in the U.S. in 1880.<ref>{{harvtxt|Slocum|Singmaster|2009|p=15}}</ref>

Chapman applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, this 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.<ref name="slocum-sonneveld">''The 15 Puzzle'', by Jerry Slocum & Dic Sonneveld, 2006. {{ISBN|1-890980-15-3}}</ref>

=== Sam Loyd ===
[[File:Sam Loyd - The 14-15 Puzzle in Puzzleland.jpg|thumb|left|Sam Loyd's 1914 illustration of the unsolvable variation.]]
From 1891 until his death in 1911, [[Sam Loyd]] claimed that he had invented the puzzle. However, Loyd had no connection to the invention or initial popularity of the puzzle. Loyd's first article about the puzzle was published in 1886, and it was not until 1891 that he first claimed to be the inventor.<ref name="slocum-sonneveld" /><ref>Barry R. Clarke, ''Puzzles for Pleasure'', pp.10-12, Cambridge University Press, 1994 {{ISBN|0-521-46634-2}}.</ref>

Some later interest was fueled by Loyd's offer of a $1,000 prize ({{Inflation|US|1000|1891|fmt=eq}}) to anyone who could provide a solution for achieving a particular combination specified by Loyd, namely reversing the 14 and 15, which Loyd called the '''14-15 puzzle'''.<ref name="Korf,2000">{{Citation | first=R. E. | last=Korf | editor1-first=B. Y. | editor2-last=Walsh | editor2-first=T. | chapter-url=https://www.aaai.org/Papers/AAAI/2000/AAAI00-212.pdf | doi=10.1007/3-540-44914-0_3 | series=SARA 2000. Lecture Notes in Computer Science | pages=45–55 | publisher=Springer, Berlin, Heidelberg | year=2000 | isbn=978-3-540-67839-7 | access-date=2010-04-26 | chapter=Recent Progress in the Design and Analysis of Admissible Heuristic Functions | volume=1864 | editor1-last=Choueiry | title=Abstraction, Reformulation, and Approximation | url=http://www.cs.iastate.edu/~honavar/korf2000.pdf | archive-date=2010-08-16 | archive-url=https://web.archive.org/web/20100816065953/http://www.cs.iastate.edu/~honavar/korf2000.pdf | url-status=dead }}</ref> This is impossible, as had been shown over a decade earlier by {{harvtxt|Johnson|Story|1879}}, because it requires a transformation from an even to an odd permutation.

==Varieties of the 15 puzzle==
The [[Minus Cube]], manufactured in the [[Soviet Union|USSR]], is a [[Three-dimensional space|3D]] puzzle with similar operations to the 15 Puzzle.

Versions of the 15 puzzle include a different number of tiles, such as the 8-puzzle or 24-puzzle.

== Pop culture ==
[[World Chess Championship|Chess world champion]] [[Bobby Fischer]] was an expert at solving the 15 puzzle.<ref>Clifford A. Pickover, ''The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics'', p. 262, Sterling Publishing, 2009 {{ISBN|1402757964}}.</ref> He had been timed to be able to solve it within 25 seconds; Fischer demonstrated this on [[List of The Tonight Show Starring Johnny Carson episodes (1972)#November|November 8, 1972]], on ''[[The Tonight Show Starring Johnny Carson]]''.<ref>[https://www.youtube.com/watch?v=QxvnEwvgfeI "Bobby Fischer solves a 15 puzzle in 17 seconds on Carson Tonight Show - 11/08/1972"], ''The Tonight Show'', 8 November 1972, Johnny Carson Productions, retrieved 1 August 2021.</ref><ref>Adam Spencer, ''Adam Spencer's Big Book of Numbers: Everything you wanted to know about the numbers 1 to 100'', p. 58, Brio Books, 2014 {{ISBN|192113433X}}.</ref>

==See also==
* [[Combination puzzles]]
* [[Jeu de taquin]], an operation on skew Young tableaux similar to the moves of the 15 puzzle
* [[Klotski]]
* [[Mechanical puzzles]]
* [[Pebble motion problems]]
* [[Rubik's Cube]]
* [[Three cups problem]]

==Notes==
{{reflist|30em}}

==References==
{{refbegin}}
*{{Citation | last1=Archer | first1=Aaron F. | title=A modern treatment of the 15 puzzle | mr=1732661  | year=1999 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=106 | issue=9 | pages=793–799 | doi=10.2307/2589612| jstor=2589612 | citeseerx=10.1.1.19.1491 }}
*{{Citation | last1=Johnson | first1=Wm. Woolsey | last2=Story | first2=William E. | title=Notes on the "15" Puzzle | year=1879 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=2 | issue=4 | pages=397–404 | jstor=2369492 | doi=10.2307/2369492}}
* Edward Kasner & James Newman (1940) [[Mathematics and the Imagination]], pp 177&ndash;80, [[Simon & Schuster]].
* {{ cite book | last1 = Slocum | first1 = Jerry | last2 = Singmaster | first2 = David | title = The Cube: The Ultimate Guide to the World's Best-Selling Puzzle—Secrets, Stories, Solutions | publisher = [[Black Dog & Leventhal Publishers|Black Dog & Leventhal]] | year = 2009 | ISBN = 978-1579128050 }}
*{{Citation | last1=Wilson | first1=Richard M. | title=Graph puzzles, homotopy, and the alternating group | doi=10.1016/0095-8956(74)90098-7  | mr=0332555  | year=1974 | journal=Journal of Combinatorial Theory, Series B | issn=0095-8956 | volume=16 | pages=86–96| doi-access=free }}
{{refend}}

==External links==
{{Commons category}}
* [https://hc11.home.xs4all.nl/15puzzle/15puzzen.htm The history of the 15 puzzle]
* [https://15puzzle.codingbroz.com/ 15 Puzzle Game]
* [http://www.chessandpoker.com/fifteen-puzzle-solution.html Fifteen Puzzle Solution]
* [http://oeis.org/A151944 Maximal number of moves required for the m X n generalization of the 15 puzzle]
* [http://kociemba.org/themen/fifteen/fifteensolver.html 15-Puzzle Optimal Solver] with download (from Herbert Kociemba)

[[Category:Mechanical puzzles]]
[[Category:Combination puzzles]]
[[Category:NP-complete problems]]
[[Category:Permutations]]
[[Category:19th-century fads and trends]]