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Sudoku (数独 sūdoku^?), also known as Number Place or Nanpure, is a logic-based placement puzzle. The aim of the puzzle is to enter the digits 1 through 9 in each cell of a 9×9 grid made up of 3×3 subgrids (called "regions") so that each row, column, and region contains exactly one instance of each digit. A set of clues, or "givens", constrain the puzzle such that there is only one way to correctly fill in the remainder.

Completed sudoku puzzles are a type of Latin square, with the additional constraint on the contents of individual regions. Leonhard Euler is sometimes cited as the source of the puzzle based on his work with Latin squares.

The modern puzzle Sudoku was invented in Indianapolis in 1979 by Howard Garns. Garns' puzzles appeared in Dell Magazines, which published them under the title "Number Place". Sudoku became popular in Japan in 1986, when puzzle publisher Nikoli discovered the game in older Dell publications. The puzzles became an international hit in 2005.

Introduction

The name "Sudoku" is the Japanese abbreviation of a longer phrase, "Suuji wa dokushin ni kagiru" (数字は独身に限る^?), meaning "the digits must occur only once". It is a trademark of puzzle publisher Nikoli Co. Ltd. in Japan.^[1] In Japanese, the word is pronounced [sɯːdokɯ]; in English, it is usually spoken with an Anglicised pronunciation, [səˈdəʊkuː] (BrE) [səˈdoʊkuː] (AmE) or [ˈsuːdəʊku] (BrE) [ˈsuːdoʊku] (AmE) (See IPA (International Phonetic Alphabet) or IPA chart for English for notation usage.) Other Japanese publishers refer to the puzzle as Number Place, the original U.S. title, or as "Nanpure" for short.^[2] Some non-Japanese publishers spell the title as "Su Doku".

The numerals in Sudoku puzzles are used for convenience; arithmetic relationships between numerals are irrelevant. Any set of distinct symbols will do; letters, shapes, or colours may be used without altering the rules. In fact, ESPN published Sudoku puzzles substituting the positions on a baseball field for the numbers 1-9. Dell Magazines, the puzzle's originator, has been using numerals for Number Place in its magazines since they first published it in 1979.^[2]

The attraction of the puzzle is that the rules are simple, yet the line of reasoning required to solve the puzzle may be complex. The level of difficulty can be selected to suit the audience. The puzzles are often available free from published sources and may be custom-made using software.

Solution methods

The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analyzing.

Scanning

Scanning is performed at the outset and throughout the solution. Scans need to be performed only once in between analyses. Scanning consists of two techniques:

Advanced solvers look for "contingencies" while scanning, narrowing a numeral's location within a row, column, or region to two or three cells. When those cells lie within the same row and region, they can be used for elimination during cross-hatching and counting. Puzzles solved by scanning alone without requiring the detection of contingencies are classified as "easy"; more difficult puzzles cannot be solved by basic scanning alone.

Marking up

Scanning stops when no further numerals can be discovered, making it necessary to engage in logical analysis. One method to guide the analysis is to mark candidate numerals in the blank cells. There are two popular notations: subscripts and dots.

An alternative technique is to "mark up" the numerals that a cell cannot be. A cell will start empty and as more constraints become known, it will slowly fill until only one mark is missing. Assuming no mistakes are made and the marks can be overwritten with the value of a cell, there is no longer a need for any erasures.

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Analysis

The two main approaches to analysis are "candidate elimination"^[3] and "what-if".^[4] In "candidate elimination", progress is made by successively eliminating candidate numerals from cells to leave one choice. After each answer has been achieved, another scan may be performed—usually checking to see the effect of the contingencies. One method works by identifying "matched cells". If precisely two cells within a scope (a particular row, column, or region) contain the same two candidate numerals (p,q), or if precisely three cells within a scope contain the same three candidate numerals (p,q,r), these cells are said to be matched. The placement of these numerals anywhere else within that same scope would make a solution impossible; thus, the candidate numerals (p,q,r) scope can be deleted. When all else fails, ask the question, 'Would entering the eliminated numeral prevent completion of the other necessary placements?' If the answer to the question is 'Yes,' then the candidate numeral in question can be eliminated.

In the "what-if" approach (also called "guess-and-check", "bifurcation", "backtracking" and "Ariadne's thread"), a cell with two candidate numerals is selected, and a guess is made. The steps are repeated unless a duplication is found or a cell is left without a possible candidate, in which case the alternative candidate must be the solution. For each cell's candidate, the question is posed: 'will entering a particular numeral prevent completion of the other placements of that numeral?' If the answer is 'yes', then that candidate can be eliminated. The what-if approach requires a pencil and eraser or a good layout memory.

Computer solutions

Computers are, of course, capable of exhaustively searching a Sudoku puzzle for its solution(s). However, in general this sort of approach will only be achievable in reasonable time if (a) the puzzle is relatively small in size (order-3 or 4, for example), and/or (b) the puzzle has a relatively high proportion of the cells already filled in correctly. Indeed, from a Computer Science perspective, Sudoku has been proven to belong in the class of NP-Complete problems, implying that we cannot hope to find a polynomially bounded algorithm (i.e. a program that will always be able to return the correct answer in a reasonable amount of time) for all possible puzzles.

Given the above, there are two general approaches taken in the creation of serious Sudoku-solving programs: Human solving methods and rapid-style methods. Human-style solvers will typically operate by maintaining a mark-up matrix, and search for contingencies, matched cells, and other elements that a human solver can utilize in order to determine and exclude cell values.

Many rapid-style solvers employ backtracking searches, with various pruning techniques also being used in order to help reduce the size of the search tree. Note, however, that regardless of whether pruning methods are used or not, it is still the case that this sort of methodology might still take an unreasonable amount of time for many larger Sudoku problems. One alternative to this method is to employ stochastic-based optimisation methods instead such as Simulated Annealing, which has recently been shown to perform well on various instances of orders 3, 4 and 5.

Finally, another alternative uses finite domain constraint programming. A constraint program specifies the constraints of the puzzle (the fact that every number in each row, each column, and each 3×3 region must be unique, and the provided "givens"); a finite-domain solver applies the constraints successively to narrow down the solution space until a solution is found. Backtracking or stochastic optimisation techniques may be applied when alternate values cannot be excluded.

Rapid solvers are preferred for trial-and-error puzzle-creation algorithms, which allow for testing large numbers of partial problems for validity in a short time; human-style solvers can be employed by hand-crafting puzzlesmiths for their ability to rate the challenge of a created puzzle and show the actual solving process their target audience can be expected to follow.

Difficulty ratings

The difficulty of a puzzle is based on the relevance and the positioning of the given numbers rather than their quantity. Surprisingly, the number of givens does not always reflect a puzzle's difficulty. Computer solvers can estimate the difficulty for a human to find the solution, based on the complexity of the solving techniques required. Some online versions offer several difficulty levels.

Most publications sort their Sudoku puzzles into four or five rating levels, although the actual cut-off points and the names of the levels themselves can vary widely. Typically, however, the titles are synonyms of "easy", "intermediate", "hard", and "challenging". Another approach is to rely on the experience of a group of human test solvers. Puzzles can be published with a median solving time rather than an algorithmically defined difficulty level.

Construction

Building a Sudoku puzzle can be performed by pre-determining the locations of the givens and assigning them values only as needed to make deductive progress. This technique gives the constructor greater control over the flow of puzzle solving, leading the solver along the same path the compiler used in building the puzzle. Great caution is required, however, as failing to recognize where a number can be logically deduced at any point in construction—regardless of how tortuous that logic may be—can result in an unsolvable puzzle when defining a future given contradicts what has already been built. Building a Sudoku with symmetrical givens is a simple matter of placing the undefined givens in a symmetrical pattern to begin with.

Nikoli Sudoku are hand-constructed, with the author being credited; the givens are always found in a symmetrical pattern.^[5] Dell Number Place Challenger (see Variants below) puzzles also list authors. The Sudoku puzzles printed in most UK newspapers are apparently computer-generated but employ symmetrical givens; The Guardian famously claimed that because they were hand-constructed, their puzzles would contain "imperceptible witticisms" that would be very unlikely in computer-generated Sudoku.

Variants

Even though 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 previously 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, with Daily SuDoku's 16×16-grid Monster SuDoku ^[6], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths. Even larger sizes, for instance 100×100,^[7] have been claimed.

Another common variant is for additional restrictions to be enforced on the placement of numbers beyond the usual row, column, and region requirements. Often the restriction takes the form of an extra "dimension"; the most common is for the numbers in the main diagonals of the grid to also be required 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.

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 and The Sydney Morning Herald this form of puzzle is known as Samurai SuDoku.^[8] Puzzles with twenty or more overlapping grids are not uncommon in some Japanese publications. 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.

Alphabetical variations have also emerged; 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. The Code Doku^[9] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku^[10] from Top Notch embeds two 9-letter words, one on each diagonal. It is debatable whether these are true Sudoku puzzles: although they purportedly have a single linguistically valid solution, they cannot necessarily be solved entirely by logic, requiring the solver to determine the embedded words. Top Notch claims this as a feature designed to defeat solving programs.

A new variant without any given numbers for start comes up this summer (2006) from India by Yoogi Games,^[13] named Greater-Than or Comparison Sudoku. All rules of a standard Sudoku are the same, but you have to find out by comparisons the sequence of the digits (or letters if you want, but the symbols used must be in ascending order) from one through nine for each region. Therefore, each cell's border line inside every region is notched in the meaning of < (less than) – or vice versa.

Mathematics of Sudoku

A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any 3×3 block. The number of classic 9×9 Sudoku solution grids was shown in 2005 by Bertram Felgenhauer and Frazer Jarvis to be 6,670,903,752,021,072,936,960 ^[14] (sequence A107739 in OEIS) : this is roughly 0.00012% 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 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). Both results have been confirmed by independent authors.^{citation needed]}

The maximum number of givens provided while still not rendering the solution unique 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—is unsolved, 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,^[16] ^[17] and 18 with the givens in rotationally symmetric cells.

History

Number puzzles first 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 9x9 magic square with 3x3 sub-squares in 1892.^[18] 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.

Within three years Le Siècle's rival, La France, refined the puzzle so that it was almost a modern Sudoku. It simplified the 9x9 magic square puzzle so that each row and column contained only the numbers 1-9, but did not mark the sub-squares. Although they are unmarked, each 3x3 sub-square does indeed comprise the numbers 1-9. However, the puzzle cannot be considered the first Sudoku because, under modern rules, it has two solutions. The puzzle setter ensured a unique solution by requiring 1-9 to appear in both diagonals.

These weekly puzzles were a feature of newspaper titles including L'Echo de Paris for about a decade but disappeared about the time of the First World War.^[19]

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, and first published in 1979 by Dell Magazines as Number Place, because Garns' name was always present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included also Number Place, and was always absent from issues that did not and those are the earliest known examples of modern Sudoku.^[2] Sadly, he died in 1989 before getting a chance to see his creation as a worldwide phenomenon.^[2]

The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984^[2] as Suuji wa dokushin ni kagiru (数字は独身に限る^?), which can be translated as "the numbers must be single" or "the numbers must occur only once." At a later date, the name was abbreviated to Sudoku by Maki Kaji (鍜治真起 Kaji Maki^?), taking only the first kanji of compound words to form a shorter version.^[2] In 1986, Nikoli introduced two innovations: the number of givens restricted to no more than 32^{citation needed]} and puzzles became "symmetrical" (meaning the givens were distributed in rotationally symmetric cells).^[1] It is now published in mainstream Japanese periodicals, such as the Asahi Shimbun.

Popularity in the media

In 1997, retired Hong Kong judge Wayne Gould, 59, a New Zealander, saw a partly completed puzzle in a Japanese bookshop. Over six years he developed a computer program to produce puzzles quickly.^[20] 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).

It was rapidly introduced by The Daily Telegraph, the Daily Mail and The Independent. By April and May 2005 the puzzle became a national phenomenon and was introduced to several other national British newspapers including The Guardian, The Sun (where it was labelled Sun Doku), and The Daily Mirror.^{citation needed]}

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.

The world's first live TV Sudoku show, Sudoku Live, was 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. The audience at home was in a separate interactive competition.

Sudoku is now also a very popular part of the games available for mobile phones. Gameloft, Guardian Unlimited, World Sudoku League and Playcom are some of the distributors.

Competitions

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