View source for Sudoku

{{Short description|Logic-based number-placement puzzle}}
{{Distinguish|text=[[Sodoku]] (the disease)}}
{{Use mdy dates |date=August 2013 |cs1-dates=sy}}
{{Multiple image
| direction = vertical
| align = right
| image1 = Sudoku Puzzle by L2G-20050714 standardized layout.svg
| image2 = Sudoku Puzzle by L2G-20050714 solution standardized layout.svg
| width = 250
| caption1 = A typical Sudoku puzzle
| caption2 = The solution to the puzzle above
| alt1 = A typical Sudoku puzzle, with nine rows and nine columns that intersect at square spaces. Some of the cells are filled with a number; others are blank cells to be solved.
| alt2 = The previous puzzle, showing its solution.
}}
'''Sudoku''' ({{IPAc-en|s|uː|ˈ|d|oʊ|k|uː|,_|-|ˈ|d|ɒ|k|-|,_|s|ə|-}}; {{langx|ja|数独|sūdoku|digit-single}}; originally called '''Number Place''')<ref name="time magazine">{{cite web |last=Grossman |first=Lev |title=The Answer Men |url= http://www.time.com/time/magazine/article/0,9171,2137423,00.html |archive-url= https://web.archive.org/web/20130301013815/http://www.time.com/time/magazine/article/0,9171,2137423,00.html |url-status= dead |archive-date= March 1, 2013 |newspaper=[[Time (magazine)|Time]] |location= New York |access-date=March 4, 2013 |date=March 11, 2013}}{{registration required}}</ref> is a [[logic puzzle|logic-based]],<ref>{{cite news |last=Arnoldy |first=Ben |title=Sudoku Strategies|work=[[The Christian Science Monitor]]}}</ref><ref>{{cite news|last=Schaschek|first=Sarah|title=Sudoku champ's surprise victory |date=March 22, 2006 |newspaper=The Prague Post |url= http://www.praguepost.com/P03/2006/Art/0323/news5.php |access-date=February 18, 2009 |archive-url= https://web.archive.org/web/20060813145953/http://www.praguepost.com/P03/2006/Art/0323/news5.php |archive-date = August 13, 2006}}</ref> [[combinatorics|combinatorial]]<ref>{{cite conference
 | last1 = Gradwohl | first1 = Ronen
 | last2 = Naor | first2 = Moni
 | last3 = Pinkas | first3 = Benny
 | last4 = Rothblum | first4 = Guy N.
 | editor1-last = Crescenzi | editor1-first = Pierluigi
 | editor2-last = Prencipe | editor2-first = Giuseppe
 | editor3-last = Pucci | editor3-first = Geppino
 | contribution = Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles
 | doi = 10.1007/978-3-540-72914-3_16
 | pages = 166–182
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Fun with Algorithms, 4th International Conference, FUN 2007, Castiglioncello, Italy, June 3-5, 2007, Proceedings
 | volume = 4475
 | year = 2007| isbn = 978-3-540-72913-6
 }}</ref> number-placement [[puzzle]]. In classic Sudoku, the objective is to fill a 9&nbsp;×&nbsp;9 grid with digits so that each column, each row, and each of the nine 3&nbsp;×&nbsp;3 subgrids that compose the grid (also called "boxes", "blocks", or "regions") contains all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which for a [[well-posed problem|well-posed]] puzzle has a single solution.

French newspapers featured similar puzzles in the 19th century, and the modern form of the puzzle first appeared in 1979 [[puzzle book]]s by [[Dell Magazines]] under the name Number Place.<ref name=Smith /> However, the puzzle type only began to gain widespread popularity in 1986 when it was published by the Japanese puzzle company [[Nikoli (publisher)|Nikoli]] under the name Sudoku, meaning "single number".<ref>{{cite journal |title=Unwed Numbers |first=Brian |last=Hayes |journal=American Scientist |volume=94 |issue=1 |year=2006 |pages=12–15 |doi=10.1511/2006.57.3475}}</ref> In newspapers outside of Japan, it first appeared in ''[[The Conway Daily Sun]]'' (New Hampshire) in September 2004, and then ''[[The Times]]'' (London) in November 2004, both of which were thanks to the efforts of the Hong Kong judge [[Wayne Gould]], who devised a [[computer program]] to rapidly produce unique puzzles.

==History==
{{more sources section|date=April 2025}}
[[File:Sudoku.jpg|thumb|upright|From ''La France'' newspaper, July 6, 1895: The puzzle instructions read, "Use the numbers 1 to 9 nine times each to complete the grid in such a way that the horizontal, vertical, and two main diagonal lines all add up to the same total."]]

===Predecessors===
Number puzzles appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from [[magic square]]s. ''[[Le Siècle]]'', a Paris daily, published a partially completed 9×9 magic square with 3×3 subsquares on November 19, 1892.<ref name="Le_Siècle_185672">{{cite journal
|last=Boyer
|first=Christian
|date=May 2006
|title=Supplément de l'article "Les ancêtres français du sudoku"
|journal=Pour la Science
|pages=1–6
|url=http://cboyer.club.fr/multimagie/SupplAncetresSudoku.pdf
|access-date=August 3, 2009
|archive-url=https://web.archive.org/web/20061210103525/http://cboyer.club.fr/multimagie/SupplAncetresSudoku.pdf
|archive-date=December 10, 2006
|language=fr
}}</ref> It was not a Sudoku because it contained double-digit numbers and required arithmetic rather than logic to solve, but it shared key characteristics: each row, column, and subsquare added up to the same number.

On July 6, 1895, ''Le Siècle''{{'s}} rival, ''[[La France (French newspaper)|La France]]'', refined the puzzle so that it was almost a modern Sudoku and named it {{lang|fr|carré magique diabolique}} ('diabolical magic square'). It simplified the 9×9 magic square puzzle so that each row, column, and [[broken diagonal]]s contained only the numbers 1–9, but did not mark the subsquares. Although they were unmarked, each 3×3 subsquare did indeed comprise the numbers 1–9, and the additional constraint on the broken diagonals led to only one solution.<ref>{{cite web|access-date=August 3, 2009|archive-date=October 10, 2007|archive-url=https://web.archive.org/web/20071010081626/http://cboyer.club.fr/multimagie/English/SudokuAncestors.htm|first=Christian|language=fr|last=Boyer|publisher=(personal webpage)|title=Sudoku's French ancestors|url=http://cboyer.club.fr/multimagie//English/SudokuAncestors.htm|year=2007}}</ref>

These weekly puzzles were a feature of French newspapers such as ''[[L'Écho de Paris]]'' for about a decade, but disappeared about the time of [[World War I]].<ref name=L>{{cite news |first=Jack |last=Malvern |title=Les fiendish French beat us to Su Doku |newspaper=[[The Times]]|date=June 3, 2006 |url=https://www.thetimes.com/article/les-fiendish-french-beat-us-to-su-doku-mrqqshlsjcg |access-date=2024-10-02}}</ref>

===Modern Sudoku===
The modern Sudoku was most likely designed anonymously by [[Howard Garns]], a 74-year-old retired architect and freelance puzzle constructor from [[Connersville, Indiana]], and first published in 1979 by [[Dell Magazines]] as Number Place (the earliest known examples of modern Sudoku).<ref name=" time magazine"/> Garns' name was always present on the list of contributors in issues of ''Dell Pencil Puzzles and Word Games'' that included Number Place and was always absent from issues that did not.<ref name="Garns">{{cite web |url=http://www.mathpuzzle.com/MAA/41-Sudoku%20Variations/mathgames_09_05_05.html |title=Ed Pegg Jr.'s Math Games: Sudoku Variations |access-date=October 3, 2006 |last=Pegg |first=Ed Jr. |date=September 15, 2005 |work=MAA Online |publisher=[[The Mathematical Association of America]]}}</ref> He died in 1989 before getting a chance to see his creation as a worldwide phenomenon.<ref name="Garns"/> Whether or not Garns was familiar with any of the French newspapers listed above is unclear.

The puzzle was introduced in Japan by {{nihongo|[[Maki Kaji]]|鍜治 真起|Kaji Maki}}, president of the [[Nikoli (publisher)|Nikoli]] puzzle company, in the paper ''Monthly Nikolist'' in April 1984<ref name="Garns"/> as {{nihongo|'' Sūji wa dokushin ni kagiru''|数字は独身に限る|}}, which can be translated as "the digits must be single", or as "the digits are limited to one occurrence" (In Japanese, ''dokushin'' means an "unmarried person"). The name was later abbreviated to ''Sudoku'' (数独), taking only the first [[kanji]] of compound words to form a shorter version.<ref name="Garns"/> "Sudoku" is a registered trademark in Japan<ref name="jp_trademark_5056856">{{cite web |url=https://www.j-platpat.inpit.go.jp/web/TR/JPT_5056856/F9A8AC1B402D5E9F6E7B4E7246A6CE85 |title=Reg. No. 5056856 |work=Japanese Trademark 5056856 |access-date=October 3, 2018 |publisher=Japan Platform for Trademark Information}}</ref> and the puzzle is generally referred to as {{nihongo|Number Place|ナンバープレース|Nanbāpurēsu}} or, more informally, a shortening of the two words, {{nihongo|Num(ber) Pla(ce)|ナンプレ|'' Nanpure''}}. In 1986, Nikoli introduced two innovations: the number of givens was restricted to no more than 32, and puzzles became "symmetrical" (meaning the givens were distributed in [[Rotational symmetry|rotationally symmetric cells]]). It is now published in mainstream Japanese periodicals, such as the ''[[Asahi Shimbun]]''.

=== Spread outside Japan ===

In 1997, Hong Kong judge [[Wayne Gould]] saw a partly completed puzzle in a Japanese bookshop. Over six years, he developed a computer program to produce unique puzzles rapidly.<ref name=Smith />

The first newspaper outside of Japan to publish a Sudoku puzzle was ''[[The Conway Daily Sun]]'' (New Hampshire), which published a puzzle by Gould in September 2004.<ref>{{cite news|url=https://www.nytimes.com/2007/03/21/business/worldbusiness/21sudoku.html?pagewanted=2&_r=1&sq&st=cse%22Conway%20Daily%20Sun&scp=1%22%202004%20gould |title=Correction attached to "Inside Japan's Puzzle Palace" |newspaper=[[The New York Times]] |date=March 21, 2007 |url-access=subscription}}</ref><ref>{{cite web |url=https://www.independent.co.uk/news/uk/this-britain/the-rise-and-rise-of-sudoku-5336007.html |title=The rise and rise of Sudoku |website=[[The Independent]] |author=<!-- No byline --> |date=April 10, 2006}}</ref>

Gould pitched the idea of publishing Sudoku puzzles to newspapers, offering the puzzles for free in exchange for the newspapers' attributing them to him and linking to his website for solutions and other puzzles.

Knowing that British newspapers have a long history of publishing [[crossword]]s and other puzzles, he promoted Sudoku to ''[[The Times]]'' in Britain, which launched it on November 12, 2004 (calling it Su Doku). The first letter to ''The Times'' regarding Su Doku was published the following day on November 13 from Ian Payn of [[Brentford]], complaining that the puzzle had caused him to miss his stop on the [[London Underground|tube]].<ref>{{cite web|url=https://www.thetimes.com/sport/football/article/deep-in-thought-l2wwrwqqw9k |title=Deep in thought |date=November 13, 2004 |first=Ian |last=Payn |website=[[The Times]]}}</ref> Sudoku puzzles rapidly spread to other newspapers as a regular feature.<ref name=Smith>{{cite news|url=https://www.theguardian.com/media/2005/may/15/pressandpublishing.usnews |title=So you thought Sudoku came from the Land of the Rising Sun ...|first=David |last=Smith |newspaper=The Observer |date=May 15, 2005 |access-date=June 13, 2008 |quote=The puzzle gripping the nation actually began at a small New York magazine}}</ref><ref>{{cite news |last=Devlin |first=Keith |title=The Numbers Game (book review of ''Taking Sudoku Seriously'' by Jason Rosenhouse et al.) |newspaper=[[The Wall Street Journal]] |page=C5 |date=January 28–29, 2012}}</ref>

The rapid rise of Sudoku in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as when ''The Guardian''{{'s}} ''G2'' section advertised itself as the first newspaper supplement with a Sudoku grid on every page).<ref name=G2>{{cite news |title=G2, home of the discerning Sudoku addict |url=https://www.theguardian.com/g2/story/0,,1482817,00.html |work=The Guardian |date=May 13, 2005 |access-date=September 16, 2006 |location=London}}</ref> Recognizing the different psychological appeals of easy and difficult puzzles, ''The Times'' introduced both, side by side, on June 20, 2005. From July 2005, [[Channel 4]] included a daily Sudoku game in their [[teletext]] service. On August 2, the BBC's program guide ''[[Radio Times]]'' featured a weekly Super Sudoku with a 16×16 grid.

[[File:SudokuLive2.jpg|thumb|The world's first live TV Sudoku show, held on July 1, 2005, ''[[Sky One]]'']]
The world's first live TV Sudoku show, ''Sudoku Live'', was a [[puzzle contest]] first broadcast on July 1, 2005, on the British pay-television channel [[Sky One]]. It was presented by [[Carol Vorderman]]. Nine teams of nine players (with one celebrity in each team) representing geographical regions competed to solve a puzzle. Each player had a hand-held device for entering numbers corresponding to answers for four cells. Phil Kollin of [[Winchelsea|Winchelsea, England]], was the series grand prize winner, taking home over £23,000 over a series of games. The audience at home was in a separate interactive competition, which was won by Hannah Withey of [[Cheshire]].

Later in 2005, the [[BBC]] launched ''[[SUDO-Q]]'', a [[game show]] that combined Sudoku with general knowledge. However, it used only 4×4 and 6×6 puzzles. Four seasons were produced before the show ended in 2007.

[[File:Sudokujf.JPG|thumb|right|175px|A Sudoku competition at [[SM City Baliuag]]]]

An annual [[World Sudoku Championship]] series has been organized by the [[World Puzzle Federation]] since 2006, except in 2020 and 2021 during the [[COVID-19 pandemic]].

In 2006, a Sudoku website published a tribute song by Australian songwriter Peter Levy, but the song download was later removed due to heavy traffic. The Japanese Embassy nominated the song for an award, and Levy claimed he was in discussions with [[Sony]] in Japan to release the song as a single.<ref>{{cite news |url=http://www.heraldsun.com.au/news/hit-song-has-the-numbers/story-e6frf7jo-1111112155196 |archive-url=https://archive.today/20121230192938/http://www.heraldsun.com.au/news/hit-song-has-the-numbers/story-e6frf7jo-1111112155196 |url-status=dead |archive-date=December 30, 2012 |title=Hit Song Has the Numbers |date=August 17, 2006 |work=[[The Herald Sun]] |access-date=October 5, 2008}}</ref>

Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions of [[Linux]]. The software has also been released on video game consoles, such as the [[Nintendo DS]], [[PlayStation Portable]], the [[Game Boy Advance]], [[Xbox Live Arcade]], the [[Nook]] e-book reader, Kindle Fire tablet, several [[iPod]] models, and the [[iPhone]]. Many [[Nokia]] phones also had Sudoku. In fact, just two weeks after [[Apple Inc.]] debuted the online [[App Store (iOS)|App Store]] within its [[iTunes Store]] on July 11, 2008, nearly 30 different Sudoku games were already in it, created by various [[software developers]], specifically for the iPhone and iPod Touch. Sudoku games also rapidly became available for [[web browser]] users and for basically all gaming, cellphone, and computer platforms.

In June 2008, an Australian drugs-related jury trial costing over [[Australian dollar|A$]] 1 million was aborted when it was discovered that four or five of the twelve jurors had been playing Sudoku instead of listening to the evidence.<ref>{{cite news |url=http://www.smh.com.au/news/national/jurors-get-1-million-trial-aborted/2008/06/10/1212863636766.html |title=The game's up: jurors playing Sudoku abort trial |last=Knox |first=Malcolm |date=June 11, 2008 |work=[[The Sydney Morning Herald]] |access-date=June 11, 2008}}</ref>

== Variants ==
{{Multiple image|direction=horizontal|align=right|image1=A nonomino sudoku.svg|image2=A nonomino sudoku solution.svg|width=150|caption1=A ''[[nonomino]]'' or [[Jigsaw puzzle|jigsaw]] Sudoku, as seen in ''[[The Sunday Telegraph]]''...|caption2=...and its solution (red numbers)|alt1=A Sudoku puzzle grid with many colors, with nine rows and nine columns that intersect at square spaces. Some of the spaces are filled with a digit; others are blank spaces to be solved.|alt2=The previous puzzle, solved with digits in the blank spaces.}}

=== Variations of grid sizes or region shapes ===
Although the 9×9 grid with 3×3 regions is by far the most common, many other variations exist. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with ''[[pentomino]]'' regions have been published under the name Logi-5; the [[World Puzzle Championship]] has featured a 6×6 grid with 2×3 regions and a 7×7 grid with six ''[[heptomino]]'' regions and a disjoint region. Larger grids are also possible, or different irregular shapes (under various names such as ''Suguru'', ''Tectonic'', ''Jigsaw Sudoku'' etc.). ''The Times'' offers a 12×12-grid "Dodeka Sudoku" with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 "Number Place Challenger" puzzles (using the numbers 1–16 or the letters A-P). Nikoli offers 25×25 "Sudoku the Giant" behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010.<ref>{{cite book |last=Eisenhauer |first=William |title=Sudoku-zilla |year=2010 |publisher=CreateSpace |page=220 |isbn=978-1-4515-1049-2}}</ref>

==== Mini Sudoku ====

Under the name "Mini Sudoku", a 6×6 variant with 3×2 regions appears in the American newspaper ''[[USA Today]]'' and elsewhere. The object is the same as that of standard Sudoku, but the puzzle only uses the numbers 1 through 6. A similar form, for younger solvers of puzzles, called "The Junior Sudoku", has appeared in some newspapers, such as some editions of ''The Daily Mail''.

=== Imposing additional constraints ===
Another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often, the limit takes the form of an extra "dimension"; the most common is to require the numbers in the main diagonals of the grid to also be unique. The aforementioned "Number Place Challenger" puzzles are all of this variant, as are the Sudoku X puzzles in ''[[The Daily Mail]]'', which use 6×6 grids.

=== Killer sudoku ===
{{Multiple image|direction=horizontal|align=right|image1=Killersudoku color.svg|image2=Killersudoku color solution.svg|width=150|caption1=A Killer Sudoku puzzle|caption2=And its solution}}
{{Main|Killer sudoku}}

The killer sudoku variant combines elements of sudoku and [[kakuro]]. A killer sudoku puzzle is made up of 'cages', typically depicted by boxes outlined with dashes or colours. The sum of the numbers in a cage is written in the top left corner of the cage, and numbers cannot be repeated in a cage.

=== Other variants ===
Puzzles constructed from more than two grids are also common. Five 9×9 grids that overlap at the corner regions in the shape of a ''[[quincunx]]'' is known in Japan as ''Gattai'' 5 (five merged) Sudoku. In ''The Times'', ''[[The Age]]'', and ''[[The Sydney Morning Herald]]'', this form of puzzle is known as Samurai Sudoku. ''[[The Baltimore Sun]]'' and the ''[[Toronto Star]]'' publish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are placed in the overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.

[[File:Comparison Sudoku.png|thumb|250px|An example of Greater Than Sudoku]]
A tabletop version of Sudoku can be played with a standard 81-card Set deck (see [[Set game]]). A three-dimensional Sudoku puzzle was published in ''[[The Daily Telegraph]]'' in May 2005. ''The Times'' also publishes a three-dimensional version under the name Tredoku. Also, a Sudoku version of the [[Rubik's Cube]] is named [[Sudoku Cube]].

Many other variants have been developed.<ref>{{cite book |last1=Snyder |first1=Thomas |last2=Huang |first2=Wei-Hwa |title=Mutant Sudoku |publisher=Puzzlewright Press |year=2009 |isbn=978-1-402765025}}</ref><ref>{{cite book |last=Conceptis |first=Puzzles |title=Amazing Sudoku Variants |publisher=Puzzlewright |year=2013 |isbn=978-1454906520 |oclc=700343731}}</ref><ref>{{cite book |last=Murali |first=A V |title=A Collection of Fascinating Games and Puzzles |publisher=CreateSpace Independent Publishing |year=2014 |isbn=978-1500216429 |oclc=1152132274}}</ref> Some are different shapes in the arrangement of overlapping 9×9 grids, such as butterfly, windmill, or flower.<ref name="ja1">{{cite web |url=http://www.janko.at/Raetsel/Zahlenraetsel.htm |title=Zahlenraetsel |website=janko.at}}</ref> Others vary the logic for solving the grid. One of these is "Greater Than Sudoku". In this, a 3×3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers.<ref name="Garns"/> Another variant on the logic of the solution is "Clueless Sudoku", in which nine 9×9 Sudoku grids are each placed in a 3×3 array. The center cell in each 3×3 grid of all nine puzzles is left blank and forms a tenth Sudoku puzzle without any cell completed; hence, "clueless".<ref name="ja1"/> Examples and other variants can be found in the [[Glossary of Sudoku]].

== Mathematics of Sudoku ==
[[File:Sudoku Puzzle (an automorphic puzzle with 18 clues).svg|thumb|220px|A Sudoku with 18 clues and two-way diagonal symmetry]]

{{Main|Mathematics of Sudoku}}

This section refers to classic Sudoku, disregarding jigsaw, hyper, and other variants. A completed Sudoku grid is a special type of [[Latin square]] with the additional property of no repeated values in any of the nine blocks (or ''boxes'' of 3×3 cells).<ref>{{cite journal
 | last = Keedwell | first = A. D.
 | date = November 2006
 | doi = 10.1017/s0025557200180234
 | issue = 519
 | journal = [[The Mathematical Gazette]]
 | jstor = 40378190
 | pages = 425–430
 | title = Two remarks about Sudoku squares
 | volume = 90}}</ref>

The general problem of solving Sudoku puzzles on ''n''<sup>2</sup>×''n''<sup>2</sup> grids of ''n''×''n'' blocks is known to be [[NP-complete]].<ref>{{cite journal
 | last1 = Yato | first1 = Takayuki
 | last2 = Seta | first2 = Takahiro
 | issue = 5
 | journal = IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
 | pages = 1052–1060
 | title = Complexity and completeness of finding another solution and its application to puzzles
 | archive-url = https://web.archive.org/web/20200303233817/http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL87-2.pdf
 | archive-date = 2020-03-03
 | url = http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL87-2.pdf
 | volume = E86-A
 | year = 2003}}</ref> Many [[Sudoku solving algorithms]], such as [[Brute-force search|brute force]]-backtracking and [[dancing links]] can solve most 9×9 puzzles efficiently, but [[combinatorial explosion]] occurs as ''n'' increases, creating practical limits to the properties of Sudokus that can be constructed, analyzed, and solved as ''n'' increases. A Sudoku puzzle can be expressed as a [[graph coloring]] problem.<ref name="Lewis2015">{{cite book |last=Lewis |first=R. |title=A Guide to Graph Colouring: Algorithms and Applications |publisher=Springer |year=2015 |isbn=978-3-319-25728-0 |doi=10.1007/978-3-319-25730-3 |s2cid=26468973 |oclc=990730995 }}</ref> The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring.

The fewest clues possible for a proper Sudoku is 17.<ref>{{cite journal |first1=G. |last1=McGuire |first2=B. |last2=Tugemann |first3=G. |last3=Civario |pages=190–217 |year=2014 |doi=10.1080/10586458.2013.870056 |title=There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem |journal=Experimental Mathematics |volume=23 |issue=2 |arxiv=1201.0749}}</ref> Tens of thousands of distinct Sudoku puzzles have only 17 clues.<ref name=seventeen3>{{cite web |url=http://www.csse.uwa.edu.au/~gordon/sudokumin.php |title=Minimum Sudoku |access-date=February 28, 2012 |last=Royle |first=Gordon | author-link = Gordon Royle |archive-date=November 26, 2006 |archive-url=https://web.archive.org/web/20061126162713/http://www.csse.uwa.edu.au/~gordon/sudokumin.php |url-status=dead }}</ref>

The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960, or around {{val|6.67|e=21}}.<ref>{{cite OEIS|A107739|Number of (completed) sudokus (or Sudokus) of size n^2 X n^2}}</ref> The number of essentially different solutions, when [[symmetries]] such as rotation, reflection, permutation, and relabelling are taken into account, is much smaller, 5,472,730,538.<ref>{{cite OEIS|A109741|Number of inequivalent (completed) n^2 X n^2 sudokus (or Sudokus)}}</ref>

Unlike the number of complete Sudoku grids, the number of minimal 9×9 Sudoku puzzles is not precisely known. (A minimal puzzle is one in which no clue can be deleted without losing the uniqueness of the solution.) However, statistical techniques combined with a puzzle generator show that about (with 0.065% relative error) 3.10&nbsp;×&nbsp;10<sup>37</sup> minimal puzzles and 2.55&nbsp;×&nbsp;10<sup>25</sup> nonessentially equivalent minimal puzzles exist.<ref name="UnbiasedStat">{{cite book | first = Denis | last = Berthier | chapter-url = http://hal.archives-ouvertes.fr/hal-00641955 |date=December 4, 2009 |title=Innovations in Computing Sciences and Software Engineering |editor= Elleithy, Khaled |pages=165–70 |chapter=Unbiased Statistics of a CSP – A Controlled-Bias Generator | doi = 10.1007/978-90-481-9112-3 | bibcode = 2010iics.book.....S | isbn = 978-90-481-9111-6 |publisher=Springer |access-date = December 4, 2009 }}</ref>

== See also ==
{{div col|colwidth=22em}}
* [[36 Cube]]
* [[Blendoku]]
* [[Constraint satisfaction problem]]
* [[Cracking the Cryptic]]
* [[Futoshiki]]
* [[Glossary of Sudoku]]
* [[Hashiwokakero]]
* [[Hidato]]
* [[KenKen]]
* [[List of Nikoli puzzle types]]
* [[Nonogram]]
* [[Str8ts]]
{{div col end}}

== References ==
{{Reflist}}

== Further reading ==
* {{cite journal |last=Delahaye |first=Jean-Paul |url=http://www.cs.virginia.edu/~robins/The_Science_Behind_SudoKu.pdf |title=The Science Behind Sudoku |journal=[[Scientific American]] |date=June 2006 |volume=294 |issue=6 |pages=80–87 |pmid=16711364 |doi=10.1038/scientificamerican0606-80 |jstor=26061494|bibcode=2006SciAm.294f..80D }}
* {{cite journal |last=Provan |first=J. Scott |title=Sudoku: Strategy Versus Structure |journal=American Mathematical Monthly |date=October 2009 |volume=116 |issue=8 |pages=702–707 |doi=10.4169/193009709X460822|s2cid=38433481 |url=https://cdr.lib.unc.edu/downloads/gt54kx559 }} Also as [http://stat-or.unc.edu/research/Current%20Reports/techpdf/TR_08_04.pdf UNC/STOR/08/04].

== External links ==
{{Commons category|Sudoku}}
* [http://news.bbc.co.uk/2/hi/asia-pacific/6745433.stm 'Father of Sudoku' puzzles next move] ([[BBC]])
<!-- ===========================(NoMoreLinks)=============================== -->
<!-- | DO NOT ADD MORE LINKS TO THIS ARTICLE. WIKIPEDIA IS NOT A COLLECTION OF | -->
<!-- | LINKS. If you think that your link might be useful, do not add it here, | -->
<!-- | but put it on this article's discussion page first or submit your link  | -->
<!-- | to the appropriate category at the Open Directory Project (www.dmoz.org)| -->
<!-- | and link back to that category using the {{dmoz}} template.             | -->
<!-- |                                                                         | -->
<!-- |           Links that have not been verified WILL BE DELETED.            | -->
<!-- |  See [[Wikipedia:External links]] and [[Wikipedia:Spam]] for details    | -->
<!-- ===========================(NoMoreLinks)=============================== -->

[[Category:Sudoku| ]]
[[Category:Abstract strategy games]]
[[Category:Puzzles]]
[[Category:Latin squares]]
[[Category:Logic puzzles]]
[[Category:NP-complete problems]]
[[Category:Computer-assisted proofs]]