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Sudoku (数独) (,, , originally called Number Place) is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 subgrids that compose the grid (also called "boxes", "blocks", or "regions") contain all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a single solution.

Completed games are always an example of a Latin square which include an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice in the same row, column, or any of the nine 3×3 subregions of the 9×9 playing board.

French newspapers featured variations of the Sudoku puzzles in the 19th century, and the puzzle has appeared since 1979 in puzzle books under the name Number Place. However, the modern Sudoku only began to gain widespread popularity in 1986 when it was published by the Japanese puzzle company Nikoli under the name Sudoku, meaning "single number". It first appeared in a U.S. newspaper, and then The Times (London), in 2004, thanks to the efforts of Wayne Gould, who devised a computer program to rapidly produce unique puzzles.

Predecessors
Number puzzles appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from magic squares. Le Siècle, a Paris daily, published a partially completed 9×9 magic square with 3×3 subsquares on November 19, 1892. 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 rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row, column, and broken diagonals contained only the numbers 1–9, but did not mark the subsquares. Although they are unmarked, each 3×3 subsquare does indeed comprise the numbers 1–9 and the additional constraint on the broken diagonals leads to only one solution.

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.

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). Garns's 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. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon. Whether or not Garns was familiar with any of the French newspapers listed above is unclear.

The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984 as Sūji wa dokushin ni kagiru (数字は独身に限る), which also can be translated as "the digits must be single" or "the digits are limited to one occurrence" (In Japanese, dokushin means an "unmarried person"). At a later date, the name was abbreviated to Sudoku (数独) by Maki Kaji (鍜治 真起), taking only the first kanji of compound words to form a shorter version. "Sudoku" is a registered trademark in Japan and the puzzle is generally referred to as Number Place (ナンバープレース) or, more informally, a portmanteau of the two words, Num(ber) Pla(ce) (ナンプレ). 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 rotationally symmetric cells). It is now published in mainstream Japanese periodicals, such as the Asahi Shimbun.

Cognitive scientist Jeremy Grabbe found that Sudoku involved an area of cognition called working memory. A subsequent experiment by Grabbe showed that routine Sudoku playing could improve working memory in older people.

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. Knowing that British newspapers have a long history of publishing crosswords 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 tube. Sudoku puzzles rapidly spread to other newspapers as a regular feature.

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 G2 section advertised itself as the first newspaper supplement with a Sudoku grid on every page). 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.

In the United States, the first newspaper to publish a Sudoku puzzle by Wayne Gould was The Conway Daily Sun (New Hampshire), in 2004.

The world's first live TV Sudoku show, Sudoku Live, was a puzzle contest first broadcast on July 1, 2005, on 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, 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.

In 2006, a Sudoku website published songwriter Peter Levy's Sudoku tribute song, but quickly had to take down the MP3 file due to heavy traffic. British and Australian radio picked up the song, which is to feature in a British-made Sudoku documentary. The Japanese Embassy also nominated the song for an award, with Levy doing talks with Sony in Japan to release the song as a single.

Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions of Linux. 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 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. One of the most popular video games featuring Sudoku is Brain Age: Train Your Brain in Minutes a Day!. Critically and commercially well-received, it generated particular praise for its Sudoku implementation  and sold more than 8 million copies worldwide. Due to its popularity, Nintendo made a second Brain Age game titled Brain Age2, which has over 100 new Sudoku puzzles and other activities.

In June 2008, an Australian drugs-related jury trial costing over A$ 1 million was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to evidence.

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: Suguru,Tectonic,jigsaw Sudoku...). 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.

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
The Killer Sudoku variant combines elements of Sudoku and Kakuro.

Alphabetical Sudoku
Alphabetical variations have emerged, sometimes called Wordoku; no functional difference exists in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid. A Wordoku might contain words other than the main word.

"Quadratum latinum" is a Sudoku variation with Roman numerals (I, II, III, IV, ..., IX) proposed by Hebdomada aenigmatum, a monthly magazine of Latin puzzles and crosswords. Like the Wordoku, it presents no functional difference from a normal Sudoku, but adds the visual difficulty of using Roman numerals.

Hyper Sudoku
Hyper Sudoku uses the classic 9×9 grid with 3×3 regions, but defines four additional interior 3×3 regions in which the numbers 1–9 must appear exactly once. It was invented by Peter Ritmeester and first published by him in Dutch Newspaper NRC Handelsblad in October 2005, and since April 2007 on a daily basis in The International New York Times (International Herald Tribune). The first time it was called Hyper Sudoku was in Will Shortz's Favorite Sudoku Variations (February 2006). It is also known as Windoku because with the grid's four interior regions shaded, it resembles a window with glazing bars.

Twin Sudoku
In Twin Sudoku two regular grids share a 3×3 box. This is one of many possible types of overlapping grids. The rules for each individual grid are the same as in normal Sudoku, but the digits in the overlapping section are shared by each half. In some compositions neither individual grid can be solved alone – the complete solution is only possible after each individual grid has at least been partially solved.

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.

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. Some are different shapes in the arrangement of overlapping 9×9 grids, such as butterfly, windmill, or flower. 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. Another variant on the logic of 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 form a tenth Sudoku puzzle without any cell completed; hence, "clueless". A new variant mixes Sudoku with the sliding tile puzzle in Sudoku Slide Extreme. In this variant all all of the positions are filled in. Tiles are moved to the proper position to solve the puzzle. This variant contains power-ups and a campaign mode. Examples and other variants can be found in the Glossary of Sudoku.

Algorithms
Algorithmic techniques include backtracking, stochastic search, and constraint programming.

Human solving techniques
The simplest technique is called naked singles. A naked single is a cell which has exactly one possibility, due to restrictions in its row, column, or 3×3 box (all of these shapes are called "houses"). For example, a cell with 2 and 3 in its row, 4, 5, and 6 in its column, and 7, 8, and 9 in its box must be a 1.

Group
A naked single is the trivial case of a group, which is a set of n cells entirely contained within one house, each cell of which contains some subset of a set of numbers of size n. A naked single is a set of cells of size 1, containing 1 cell with 1 possibility. A pair, the first non-trivial case, is a set of two cells in one house, which can only contain two different digits, and similarly to a naked single, rule out every other possible occurrence of either of those two digits within the relevant house. For example, a 1-2 pair in a row is found when two different cells in that row can be no other digit besides 1 or 2, which prevents any other cell in that row from being either a 1 or a 2. A triple is the next larger group. A 1-2-3 triple in a house similarly prevents every other possible occurrence of 1, 2, or 3 in that house.

The groups technique can be expanded to the size of the grid, but note that every size group has a reciprocal; since groups in the same house prevent other occurrences of their constituent digits, groups in houses are mutually exclusive and thus in a 9×9 sudoku, the existence of a quintuple on 5 digits proves the existence of a quadruple on 4 digits.

X-wing
An x-wing pattern exists when there exists a rectangle of cells in the grid, exactly two corners of which must contain a given digit. For example, if it is known that in two different rows, a 1 must be in the same two columns, then the x-wing shape is created. For example, an x-wing on 2's in rows proves that no other 2 may be placed anywhere else in the same columns, and an x-wing on 2's in columns proves that no other 2 may be placed anywhere else in the same rows. The name derives from the fact that within the rectangle's corners, the only possible orientations of the relevant digit are on opposing diagonal corners, since two of the same digit may not be placed in the same row or column. These two possible diagonals, which create an X shape, always prevent every other possible occurrence of the relevant digit in the same rows and columns.

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). The relationship between the two theories is known, after it was proven that a first-order formula that does not mention blocks is valid for Sudoku if and only if it is valid for Latin squares.

The general problem of solving Sudoku puzzles on n2×n2 grids of n×n blocks is known to be NP-complete. Many computer algorithms, such as backtracking and dancing links can solve most 9×9 puzzles efficiently, but combinatorial explosion occurs as n increases, creating 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. 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 (proven January 2012, and confirmed September 2013). Over 49,000 Sudokus with 17 clues have been found, many by Japanese enthusiasts. Sudokus with 18 clues and rotational symmetry have been found, and there is at least one Sudoku that has 18 clues, exhibits two-way diagonal symmetry and is automorphic. The maximum number of clues that can be provided while still not rendering a unique solution is four short of a full grid (77); if two instances of two numbers each are missing from cells that occupy the corners of an orthogonal rectangle, and exactly two of these cells are within one region, the numbers can be assigned two ways. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum.

The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960, or around $6.67$. This is roughly $1.2$ times the number of 9×9 Latin squares. Various other grid sizes have also been enumerated—see the main article for details. The number of essentially different solutions, when symmetries such as rotation, reflection, permutation, and relabelling are taken into account, was shown to be just 5,472,730,538.

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 uniqueness of the solution.) However, statistical techniques combined with a puzzle generator show that about (with 0.065% relative error) 3.10 × 1037 minimal puzzles and 2.55 × 1025 nonessentially equivalent minimal puzzles exist.

Competitions



 * The first World Sudoku Championship was held in Lucca, Italy, from March 10 to 12, 2006. The winner was Jana Tylová of the Czech Republic. The competition included numerous variants.
 * The second World Sudoku Championship was held in Prague, Czech Republic, from March 28 to April 1, 2007. The individual champion was Thomas Snyder of the USA. The team champion was Japan.
 * The third World Sudoku Championship was held in Goa, India, from April 14 to 16, 2008. Thomas Snyder repeated as the individual overall champion, and also won the first ever Classic Trophy (a subset of the competition counting only classic Sudoku). The Czech Republic won the team competition.
 * The fourth World Sudoku Championship was held in Žilina, Slovakia, from April 24 to 27, 2009. After past champion Thomas Snyder of the USA won the general qualification, Jan Mrozowski of Poland emerged from a 36-competitor playoff to become the new World Sudoku Champion. Host nation Slovakia emerged as the top team in a separate competition of three-membered squads.
 * The fifth World Sudoku Championship was held in Philadelphia, Pennsylvania, from April 29 to May 2, 2010. Jan Mrozowski of Poland successfully defended his world title in the individual competition, while Germany won a separate team event. The puzzles were written by Thomas Snyder and Wei-Hwa Huang, both past U.S. Sudoku champions.
 * The 12th World Sudoku Championship (WSC) was held in Bangalore, India, from October 15 to 22, 2017. Kota Morinishi of Japan won the Individual WSC and China won the team event.
 * The 13th World Sudoku Championship took place in the Czech Republic.
 * In the United States, The Philadelphia Inquirer Sudoku National Championship has been held three times, each time offering a $10,000 prize to the advanced division winner and a spot on the U.S. National Sudoku Team traveling to the world championships. The winners of the event were Thomas Snyder (2007), Wei-Hwa Huang (2008), and Tammy McLeod (2009). In the 2009 event, the third-place finalist in the advanced division, Eugene Varshavsky, performed quite poorly onstage after setting a very fast qualifying time on paper, which caught the attention of organizers and competitors including past champion Thomas Snyder, who requested organizers reconsider his results due to a suspicion of cheating. Following an investigation and a retest of Varshavsky, the organizers disqualified him and awarded Chris Narrikkattu third place.