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, also known as ''Number Place'', is a Logic -based placement Puzzle . The aim of the puzzle is to enter a numerical digit from 1 through 9 in each cell of a 9×9 Grid made up of 3×3 subgrids (called "regions"), starting with various digits given in some cells (the "givens"); each row, column, and region must contain only one instance of each numeral. Completing the puzzle requires patience and logical ability. Although the puzzle was first published in a U.S. puzzle Magazine in 1979, it initially caught on in Japan in 1986 and attained international popularity in 2005. INTRODUCTION The name "''Sudoku''" is the Japanese abbreviation of a longer phrase, , meaning "the digits must remain single". It is a trademark of puzzle publisher Nikoli Co. Ltd in Japan. In Japanese, the word is pronounced in English, it is usually spoken with an Anglicised pronunciation, [ ( BrE ) ( AmE ) or [ ( BrE ) [] ( AmE ) (See IPA, International Phonetic Alphabet for notation usage.) Other Japanese publishers refer to the puzzle as ''Number Place'', the original U.S. title, or as "Nanpure" for short. 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 ( Penny Press ' ''Scramblets'' and Knight Features Syndicate 's ''Sudoku Word'' both use letters). Dell Magazines , the puzzle's originator, has been using numerals for ''Number Place'' in its magazines since they first published it in 1979. Numerals are used throughout this article. The attraction of the puzzle is that the rules are simple, yet the line of reasoning required to reach the solution may be complex. ''Sudoku'' is recommended by some teachers as an exercise in Logical Reasoning . The level of difficulty of the puzzles can be selected to suit the Audience . The puzzles are often available free from published sources and may also be custom-generated using software. GAMEPLAY The puzzle is most frequently a 9×9 grid, made up of 3×3 subgrids called "regions" (other terms include "boxes", "blocks", and the like when referring to the standard variation; even "quadrants" is sometimes used, despite this being an inaccurate term for a 9×9 grid). Some cells already contain numerals, known as "givens" (or sometimes as "clues"). The goal is to fill in the empty cells, one numeral in each, so that each column, row, and region contains the numerals 1–9 exactly once. Each numeral in the solution therefore occurs only once in each of three "directions" or "scopes", hence the "single numbers" implied by the puzzle's name. SOLUTION METHODS The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. Scanning Scanning is performed at the outset and throughout the solution. Scans only have to be performed one time in between analysis periods. Scanning consists of two basic techniques:
Advanced solvers look for "contingencies" while scanning—that is, narrowing a numeral's location within a row, column, or region to two or three cells. When those cells all lie within the same row (or column) ''and'' region, they can be used for elimination purposes during cross-hatching and counting ( Contingency example at Puzzle Japan ). Particularly challenging puzzles may require multiple contingencies to be recognized, perhaps in multiple directions or even intersecting—relegating most solvers to marking up (as described below). Puzzles which can be solved by scanning alone without requiring the detection of contingencies are classified as "easy" puzzles; more difficult puzzles, by definition, cannot be solved by basic scanning alone. Marking up Scanning stops when no further numerals can be discovered. From this point, it is necessary to engage in some logical analysis. Many find it useful to guide this analysis by marking candidate numerals in the blank cells. There are two popular notations: subscripts and dots.
An alternative technique, that some find easier, is to "mark up" those numerals that a cell ''cannot'' be. Thus a cell will start empty and as more constraints become known it will slowly fill. When only one mark is missing, that has to be the value of the cell. One advantage to this method of marking is that, 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. When using marking, additional analysis can be performed. For example, if a digit appears only one time in the mark-ups written inside one region, then it is clear that the digit should be there, even if the cell has other digits marked as well. Analysis The two main approaches to analysis are "candidate elimination" and "what-if".
::One method of candidate elimination works by identifying "matched cells". Cells are said to be matched within a particular row, column, or region (scope) if two cells contain the same pair of candidate numerals (''p'',''q'') and no others, or if three cells contain the same triplet of candidate numerals (''p'',''q'',''r'') and no others. The placement of these numerals anywhere else within that same scope would make a solution for the matched cells impossible; thus, the candidate numerals (''p'',''q'',''r'') appearing in unmatched cells in that same row, column or region (scope) can be deleted. ::This principle also works with candidate numeral subsets, that is, if three cells have candidates (''p'',''q'',''r''), (''p'',''q''), and (''q'',''r'') or even just (''p'',''r''), (''q'',''r''), and (''p'',''q''), all of the set (''p'',''q'',''r'') elsewhere within that same scope can be deleted. The principle is true for all quantities of candidate numerals. ::A second related principle is also true. If, within any set of cells (row, column or region), a set of candidate numerals can only appear within a number of cells equal to the quantity of candidate numerals, the cells and numerals are matched and only those numerals can appear in the matched cells. Other candidates in the matched cells can be eliminated. For example, if the 2 numerals (''p'',''q'') can only appear in 2 cells within a specific set of cells (row, column or region), all other candidates in those 2 cells can be eliminated. ::The first principle is based on cells where only matched numerals appear. The second is based on numerals that appear only in matched cells. The validity of either principle is demonstrated by posing 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. Advanced techniques carry these concepts further to include multiple rows, columns, and regions.
Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numerals into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The proverbial Holy Grail is to find a technique which minimizes counting, marking up, and rubbing out. Computer solutions Although a simple Ariadne's Thread ( Depth-first Search ) algorithm can solve (or prove invalid) any ''Sudoku'' puzzle, it is computationally inefficient and as such rarely employed on its own. There are two general approaches taken in the creation of serious ''Sudoku''-solving programs:
Either way, a computer program is capable of ''exhaustively'' searching a ''Sudoku'' puzzle for solutions, thereby determining whether it is valid (has exactly one solution) or not, with great ease relative to a human attempting the same. Human-style solvers will typically operate by maintaining a mark-up matrix identical to that a human solver may use (see "Marking up" under "Solution methods" above), and search for contingencies, matched cells, and other elements a human solver can utilize in order to determine and exclude cell values, resorting to Ariadne's thread only as a last resort. Each type of operation performed can be assigned a difficulty value; the sum of these values can be construed as a difficulty level of the puzzle. Many rapid-style solvers still employ Backtracking searches, but with various shortcuts and optimizations to reduce the depth of the search tree; which techniques are superior is under frequent debate. 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 may be applied when alternate values cannot otherwise be excluded. Although for standard ''Sudoku'' problems highly optimized and sophisticated backtracking programs are the most efficient, another popular way of solving such constraint problems is Donald Knuth 's Dancing Links Algorithm for solving the Exact Matrix Cover problem, of which the ''Sudoku'' problems are a special case. Knuth's algorithm can be applied by converting the ''Sudoku'' puzzle to a matrix cover problem, solve this problem instead, and convert the solution obtained back to a completed ''Sudoku'' grid. This method is now preferred by many ''Sudoku'' programmers, by virtue of its execution speed, simplicity and ease of implementation, and the availability of documentation and reference source code. 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 Published puzzles often are ranked in terms of difficulty. Surprisingly, the number of givens has little or no bearing on a puzzle's difficulty. A puzzle with a minimum number of givens may be very easy to solve, and a puzzle with more than the average number of givens can still be extremely difficult to solve. The difficulty of a puzzle is based on the relevance and the positioning of the given numbers rather than the quantity of the numbers. Computer solvers can estimate the difficulty for a human to find the solution, based on the complexity of the solving techniques required. This estimation allows publishers to tailor their ''Sudoku'' puzzles to audiences of varied solving experience. Some online versions offer several difficulty levels. Most publications sort their ''Sudoku'' puzzles into four rating levels, although the actual cut-off points of the levels and indeed the names of the levels themselves can vary widely. Typically, however, the titles are some set of synonyms of "easy", "intermediate", "hard", and "challenging". CONSTRUCTION It is possible to set starting grids with more than one solution and to set grids with no solution, but such are not considered proper ''Sudoku'' puzzles; as in most other pure-logic puzzles, a unique solution is expected. Building a ''Sudoku'' puzzle by hand can be performed efficiently by pre-determining the locations of the givens and assigning them values only as needed to make deductive progress. Such an undefined given can be assumed to not hold any particular value as long as it is given a different value before construction is completed; the solver will be able to make the same deductions stemming from such assumptions, as at that point the given is very much defined as something else. 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. (This technique is adaptable to composing puzzles other than ''Sudoku'' as well.) 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. It is commonly believed that Dell ''Number Place'' puzzles are computer-generated; they typically have over 30 givens placed in an apparently random scatter, some of which can possibly be deduced from other givens. They also have no authoring credits — that is, the name of the constructor is not printed with any puzzle. Wei-Hwa Huang claims that he was commissioned by Dell to write a ''Number Place'' puzzle generator in the winter of 2000; prior to that, he was told, the puzzles were hand-made. The puzzle generator was written with Visual C++ , and although it had options to generate a more Japanese-style puzzle, with Symmetry Constraints and fewer numbers, Dell opted not to use those features, at least not until their recent publication of ''Sudoku''-only magazines. Nikoli ''Sudoku'' are hand-constructed, with the author being credited; the givens are always found in a symmetrical pattern. 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 '' licenses and publishes Nikoli-constructed ''Sudoku'' puzzles, though it does not include credits. ''The Guardian'' famously claimed that because they were hand-constructed, their puzzles would contain "imperceptible Witticism s" that would be very unlikely in computer-generated ''Sudoku''. The challenge to ''Sudoku'' programmers is teaching a program how to build ''clever'' puzzles, such that they may be indistinguishable from those constructed by humans; Wayne Gould required six years of tweaking his popular program before he believed he achieved that level. VARIANTS Although the 9×9 grid with 3×3 regions is by far the most common, numerous variations abound: sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with ), and Nikoli proffering 25×25 ''Sudoku the Giant'' behemoths. 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 '''') or otherwise (see final puzzle of "Sudoku Variations" article in References). Chess -themed variants have also arisen, all of which fit this category; certain numerals are replaced with chess pieces, whose attack ranges are translated into extra regions and/or additional placement restrictions. Any of these means of increasing restrictions tends to reduce the number of givens required to render a puzzle valid (has exactly one solution). Other kinds of extra restrictions can be arithmetical in nature, such as requiring the numbers in delineated segments of the grid to have specific sums or products (an example of the former being '' Killer Su Doku '' in '' The Times ''), demarcating all places arithmetically adjacent digits appear orthogonally adjacent in the grid, providing the parity of all cells, requiring the Lo Shu Square to appear in the solution, and so on. Some such variants forsake standard givens entirely. 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''. {Link without Title} 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'' devised by Steve Schaefer has an entire sentence embedded into the puzzle; the ''Super Wordoku'' [http://sudoku.top-notch.co.uk/superwordoku.asp 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 claim this as a feature designed to defeat solving programs. Here are some of the more notable single-instance variations:
MATHEMATICS OF SUDOKU See Also: Mathematics of Sudoku The general problem of solving ''Sudoku'' puzzles on ''n''&2 × ''n''&2 boards of ''n'' × ''n'' blocks is known to be that knows the entire Game Tree . Solving ''Sudoku'' puzzles (as well as any other NP-hard problem) can be expressed as a Graph Colouring problem. The aim of the puzzle in its standard form is to construct a proper 9-colouring of a particular graph, given a partial 9-colouring. The graph in question has 81 vertices, one vertex for each cell of the grid. The vertices can be labelled with the ordered pairs , where x and y are integers between 1 and 9. In this case, two distinct vertices labelled by and are joined by an edge if and only if:
The puzzle is then completed by assigning an integer between 1 and 9 to each vertex, in such a way that vertices that are joined by an edge do not have the same integer assigned to them. A valid ''Sudoku'' solution grid is also a . The result was derived through logic and Brute Force Computation . The derivation of this result was considerably simplified by analysis provided by Frazer Jarvis and the figure has been confirmed independently by Ed Russell. Russell and Jarvis also showed that when symmetries were taken into account, there were 5,472,730,538 solutions {Link without Title} . The number of valid ''Sudoku'' solution grids for the 16×16 derivation is not known. The maximum number of givens that can be 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 [http://www.csse.uwa.edu.au/~gordon/sudokumin.php , and 18 with the givens in rotationally symmetric cells. HISTORY The puzzle was designed anonymously by Howard Garns , a 74-year-old retired architect and freelance puzzle constructor, and first published in 1979. Although likely inspired by the Latin Square invention of Leonhard Euler , Garns added a third dimension (the regional restriction) to the mathematical construct and (unlike Euler) presented the creation as a puzzle, providing a partially-completed grid and requiring the solver to fill in the rest. The puzzle was first published in New York by the specialist puzzle publisher Dell Magazines in its magazine ''Dell Pencil Puzzles and Word Games'', under the title ''Number Place'' (which we can only assume Garns named it). The puzzle was introduced in Japan by ''. Within Japan, Nikoli still holds the trademark for the name ''Sudoku''; other publications in Japan use alternative names. In 1989 , Loadstar / Softdisk Publishing published ''DigitHunt'' on the Commodore 64 , which was apparently the first home computer version of ''Sudoku''. At least one publisher still uses that title. Yoshimitsu Kanai published his computerized puzzle generator under the name ''Single Number'' for the . Bringing the process full-circle, reprints Nikoli ''Sudoku'' in '' GAMES Magazine '' under the name ''Squared Away''; the '' New York Post '', '' USA Today '', '' The Boston Globe '', '' Washington Post '', '' The Examiner '', and '' San Francisco Chronicle '' now also publish the puzzle. It is also often included in puzzle anthologies, such as ''The Giant 1001 Puzzle Book'' (under the title ''Nine Numbers''). Within the context of puzzle history, parallels are often cited to popular in the 1980s . ''Sudoku'' has been called the " Rubik's cube of the 21st century ". Popularity in the media In 1997 , retired Hong Kong judge Wayne Gould , 59, a New Zealand er, saw a partly completed puzzle in a Japanese bookshop. Over 6 years he developed a computer program to produce puzzles quickly. Knowing that British Newspaper s have a long history of publishing Crossword s and other puzzles, he promoted ''Sudoku'' to '' The Times '' in Britain, which launched it on 12 November 2004 (calling it ''Su Doku''). The puzzles by Pappocom, Gould's software house, have been printed daily in the ''Times'' ever since. Three days later '' The Daily Mail '' began to publish the puzzle under the name "Codenumber". '' The Daily Telegraph '' introduced its first ''Sudoku'' by its puzzle compiler Michael Mepham on 19 January 2005 , and other Telegraph Group newspapers took it up very quickly. Nationwide News Pty Ltd began publishing the puzzle in '' The Daily Telegraph '' of Sydney on 20 May 2005 ; five puzzles with solutions were printed that day. The immense surge in popularity of ''Sudoku'' in British newspapers and internationally has led to it being dubbed in the world media in 2005 the "fastest growing puzzle in the world". It was not until the British ''Daily Telegraph'' introduced the puzzle on a daily basis on 23 February 2005 with the full front-page treatment advertising the fact, that the other UK national newspapers began to take real interest. The ''Telegraph'' continued to splash the puzzle on its front page, realizing that it was gaining sales simply by its presence. Until then the ''Times'' had kept very quiet about the huge daily interest that its daily ''Sudoku'' competition had aroused. That newspaper already had plans for taking advantage of their market lead, and a first ''Sudoku'' book was already on the stocks before any other national UK papers had realised just how popular ''Sudoku'' might be. By April and May 2005 the puzzle had become popular in these publications and it was rapidly introduced to several other national British newspapers including '' The Independent '', '' The Guardian '', '' The Sun '' (where it was labelled ''Sun Doku''), and '' The Daily Mirror ''. As the name ''Sudoku'' became well-known in Britain, the ''Daily Mail'' adopted it in place of its earlier name "Codenumber". Newspapers competed to promote their ''Sudoku'' puzzles, with ''The Times'' and the ''Daily Mail'' each claiming to have been the first to feature ''Sudoku''. The rapid rise of ''Sudoku'' from relative obscurity in Britain to a front-page feature in national newspapers attracted commentary in the media (see ''References'' below) and parody (such as when ''The Guardian'''s ''G2'' section advertised itself as the first newspaper supplement with a ''Sudoku'' grid on every page {Link without Title} ). ''Sudoku'' became particularly prominent in newspapers soon after the 2005 General Election leading some commentators to suggest that it was filling the gaps previously occupied by election coverage. A simpler explanation is that the puzzle attracts and retains readers—''Sudoku'' players report an increasing sense of satisfaction as a puzzle approaches completion. 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 (at page 391). On 2 August 2005, the BBC's programme guide '' Radio Times '' started to feature a weekly Super Sudoku. The Dutch company Mobile Excellence International developed together with their Vietnamese partner the first mobile I-mode Sudoku game. The game was launched throughout Europe in September 2005. {Link without Title} As a one-off, 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. Conferring was permitted although the lack of acquaintance of the players with each other inhibited an analytical discussion. The audience at home was in a separate interactive competition. A Sky One publicity stunt to promote the programme with the world's largest ''Sudoku'' puzzle went awry when the 275 foot (84 m) square puzzle was found to have 1,905 correct solutions. The puzzle was carved into a hillside in Chipping Sodbury , near Bristol , England , in view of the M4 Motorway . The stunt was cleverly timed to coincide with a major road expansion, where an imposed 40 mph speed restriction allowed drivers to safely view the puzzle whilst driving. United States broadcaster CBS has run several stories concerning ''Sudoku'', including on the '' Early Show '' in summer 2005 , and on the CBS Evening News that autumn, on October 26 . Dr. House was clearly seen working on a ''Sudoku'' puzzle on his office computer in one scene of the December 13 , 2005 episode of '' House, M. D. ''; ''Sudoku'' is supposedly now banned on the studio set due to the cast constantly playing it. During February 7th's episode of '' The Daily Show '', correspondent Jason Jones suggested that to ease the conflict over the Jyllands-Posten Muhammed caricatures, newspapers should be stripped down to only featuring ''Sudoku'' puzzles. COMPETITIONS The first world championship was held in , a 31-year-old accountant from the Czech Republic . The competition included variants; a full list can be found in the PDF here . The United States Sudoku Association Inc. {Link without Title} is another corporation hosting tournaments across the United States. Currently, they are sponsoring a tournament for charity for the American Legion . Their website also includes a forum. SEE ALSO
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