Sudoku (数独 sūdoku, ; English pronunciation: /suːˈdoʊkuː/ soo-doh-koo) is a logic-based,[1][2] combinatorial[3] 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 sub-grids that compose the grid (also called "boxes", "blocks", "regions", or "sub-squares") contains all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which typically has a unique solution.
Completed puzzles are always a type of Latin square with an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice in the same 9×9 playing board row or column or in any of the nine 3×3 subregions of the 9×9 playing board.[4]
The puzzle was popularized in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning single number.[5] It became an international hit in 2005.[6]
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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-based daily, published a partially completed 9×9 magic square with 3×3 sub-squares on November 19, 1892.[7] 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 sub-square added up to the same number.
On July 6, 1895, Le Siècle's 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 sub-squares. Although they are unmarked, each 3×3 sub-square does indeed comprise the numbers 1–9 and the additional constraint on the broken diagonals leads to only one solution.[8]
These weekly puzzles were a feature of French newspapers such as L'Echo de Paris for about a decade but disappeared about the time of World War I.[9]
According to Will Shortz, the modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from 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.[10] He died in 1989 before getting a chance to see his creation as a worldwide phenomenon.[10] It is unclear if Garns was familiar with any of the French newspapers listed above.
The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984[10] as Sūji wa dokushin ni kagiru (数字は独身に限る), which 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(數獨) by Maki Kaji (鍜治 真起 Kaji Maki), taking only the first kanji of compound words to form a shorter version.[10] 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.
Although the 9×9 grid with 3×3 regions is by far the most common, variations abound. 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. The Times offers a 12×12-grid Dodeka sudoku with 12 regions of 4×3 squares. Dell regularly publishes 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal). Nikoli offers 25×25 Sudoku the Giant behemoths. Sudoku-zilla,[11] a 100x100-grid was published in print in 2010.
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 also to 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.
A variant named "Mini Sudoku" appears in the American newspaper USA Today and elsewhere, which is played on a 6×6 grid with 3×2 regions. The object is the same as standard Sudoku, but the puzzle only uses the numbers 1 through 6.
Another variant is the combination of Sudoku with Kakuro on a 9×9 grid, called Cross Sums Sudoku, in which clues are given in terms of cross sums. The clues can also be given by cryptic alphametics in which each letter represents a single digit from 0 to 9. An example is NUMBER+NUMBER=KAKURO which has a unique solution 186925+186925=373850. Another example is SUDOKU=IS*FUNNY whose solution is 426972=34*12558.
The Killer sudoku variant combines elements of sudoku and kakuro.
Photo Sudoku is a variation that allows users to play sudoku with pictures.
Alphabetical variations have emerged, sometimes called Wordoku; there is no functional difference 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 other words, other than the main word. Like in the example to the left, the words "Kari", "Park" and "Per" could also be found in the solution. This might be avoided by e.g. substituting the character "R" with e.g. a "Q".
A typical problem is shown on the right, using colors to define the groups of cells. More often, puzzles are printed in black and white, with thin dotted lines used to outline the "cages" (see below for terminology).
Hypersudoku is one of the most popular variants. It is published by newspapers and magazines around the world and is also known as "NRC Sudoku", "Windoku", "Hyper-Sudoku" and "4 Square Sudoku". The layout is identical to a normal Sudoku, but with additional interior areas defined in which the numbers 1 to 9 must appear. The solving algorithm is slightly different from the normal Sudoku puzzles because of the leverage on the overlapping squares. This overlap gives the player more information to logically reduce the possibilities in the remaining squares. The approach to playing is similar to Sudoku but with possibly more emphasis on scanning the squares and overlap rather than columns and rows.
Puzzles constructed from multiple Sudoku grids are common. Five 9×9 grids which 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 to be found in overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.
Str8ts shares the Sudoku requirement of uniqueness in the rows and columns but the third constraint is very different. Str8ts uses black cells (some with clue numbers) to divide the board into compartments. These must be filled with a set of numbers that form a 'straight', like the poker hand. A straight is a set of numbers with no gaps in them, such as '4,3,6,5' – and the order can be non-sequential. 9×9 is the traditional size but with suitable placement of black cells any size board is possible.
A tabletop version of Sudoku can be played with a standard 81-card Set deck (see Set game). A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. The Times also publishes a three-dimensional version under the name Tredoku. There is a Sudoku version of the Rubik's Cube named Sudoku Cube.
There are many other variants. Some are different shapes in the arrangement of overlapping 9×9 grids, such as butterfly, windmill, or flower.[12] 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.[10] Another variant on the logic of solution is Clueless Sudoku, in which nine 9×9 Sudoku grids are themselves placed in a three-by-three 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".[12]
A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any of the 9 blocks of contiguous 3×3 cells. The relationship between the two theories is now completely known, after Denis Berthier proved in his book The Hidden Logic of Sudoku (May 2007) that a first-order formula that does not mention blocks (also called boxes or regions) is valid for Sudoku if and only if it is valid for Latin Squares (this property is trivially true for the axioms and it can be extended to any formula). (Citation taken from p. 76 of the first edition: "any block-free resolution rule is already valid in the theory of Latin Squares extended to candidates" – which is restated more explicitly in the second edition, p. 86, as: "a block-free formula is valid for Sudoku if and only if it is valid for Latin Squares").
The first known calculation of the number of classic 9×9 Sudoku solution grids was posted on the USENET newsgroup rec.puzzles in September 2003[13] and is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS), or approximately 6.67 x 1021. This is roughly 1.2×10−6 times the number of 9×9 Latin squares. A detailed calculation of this figure was provided by Bertram Felgenhauer and Frazer Jarvis in 2005.[14] 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 by Ed Russell and Frazer Jarvis to be just 5,472,730,538[15] (sequence A109741 in OEIS).
The maximum number of givens provided while still not rendering a unique solution is four short of a full grid; if two instances of two numbers each are missing and the cells they are to occupy form the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the numbers can be assigned. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum. The inverse problem—the fewest givens that render a solution unique—was recently proven by Gary McGuire to be 17[16] , although the lowest number yet found for the standard variation without a symmetry constraint is 17, a number of which have been found by Japanese puzzle enthusiasts,[17][18] and 18 with the givens in rotationally symmetric cells. Over 48,000 examples of Sudokus with 17 givens resulting in a unique solution are known.
Early in 2009, Puerto Rican mathematician Francisco Javier Rodriguez, lead researcher at the Centro Chaguin de Investigación Matematicas y Filosoficas in Mayaguez, discovered a variation, called Sudoku de Calne, based on the conbíf algorithm.
In 2010 mathematicians Paul Newton and Stephen DeSalvo of the University of Southern California showed that the arrangement of numbers in Sudoku puzzles are more random than the number arrangements in randomly generated 9x9 matrices. This is because the rules of Sudoku exclude some random arrangements that have an innate symmetry.[19]
In 1997, New Zealander and retired Hong Kong judge Wayne Gould, then in his early 50s, saw a partly completed puzzle in a Japanese bookshop. Over six years he developed a computer program to produce puzzles quickly. 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 12 November 2004 (calling it Su Doku). The first letter to The Times regarding Su Doku was published the following day on 13 November from Ian Payn of Brentford, complaining that the puzzle had caused him to miss his stop on the tube.[20]
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).[21] Recognizing the different psychological appeals of easy and difficult puzzles, The Times introduced both side by side on 20 June 2005. From July 2005, Channel 4 included a daily Sudoku game in their Teletext service. On 2 August, the BBC's programme guide Radio Times featured a weekly Super Sudoku which features 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.[22]
The world's first live TV Sudoku show, Sudoku Live, was a puzzle contest first broadcast on 1 July 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 combines Sudoku with general knowledge. However, it uses 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,[23] but quickly had to take down the mp3 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.[24]
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, several iPod models, and the iPhone. In fact, just two weeks after Apple Inc. debuted the online App Store within its iTunes store on July 11, 2008, there were already nearly 30 different Sudoku games, 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[25][26][27] and sold more than 8 million copies worldwide.[28] 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$1,000,000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to evidence.[29]
|title= specified when using {{Cite web}}". Solving the Sudoku Minimum Number of Clues Problem. http://www.math.ie/McGuire_V1.pdf. Retrieved 01/02/2012.