Sudoku (Japanese: 数独, sЕ«doku), sometimes spelled Su Doku, is a logic-based placement puzzle, also known as Number Place in the United States. The aim of the canonical 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 first published in 1979, Sudoku initially caught on in Japan in 1986 and attained international popularity in 2005.

Introduction

The term Sudoku implies "numbers singly", or loosely translates to "all the numbers must remain unmarried", in Japanese; it is a registered trademark of puzzle publisher Nikoli Co. Ltd in Japan. Other Japanese publishers generally refer to the puzzle as Nanpure (Number Place), its original title. Sudoku is pronounced as the English words "SUE-dough-coo", with the first syllable accented.

The numerals in Sudoku puzzles are used for convenience; arithmetic relationships between numerals are absolutely 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 completion rules are simple, yet the line of reasoning required to reach the completion may be difficult. 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 also may be custom-generated using software.

Rules and terminology

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). Some cells already contain numbers, known as "givens" (or sometimes as "clues"). The goal is to fill in the empty cells, one number in each, so that each column, row, and region contains the numbers 1–9 exactly once. Each number in the solution therefore occurs only once in each of three "directions", 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 periodically throughout the solution. Scans may have to be performed several times in between analysis periods. Scanning consists of two basic techniques:

• Cross-hatching: the scanning of rows (or columns) to identify which line in a particular region may contain a certain number by a process of elimination. This process is then repeated with the columns (or rows). For fastest results, the numbers are scanned in order of their frequency. It is important to perform this process systematically, checking all of the digits 1–9.

• Counting 1–9 in regions, rows, and columns to identify missing numbers. Counting based upon the last number discovered may speed up the search. It also can be the case (typically in tougher puzzles) that the value of an individual cell can be determined by counting in reverse—that is, scanning its region, row, and column for values it cannot be to see which is left.

Advanced solvers look for "contingencies" while scanning—that is, narrowing a number'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 comes to a halt when no further numbers 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 numbers in the blank cells. There are two popular notations: subscripts and dots.

• In the subscript notation the candidate numbers are written in subscript in the cells. The drawback to this is that original puzzles printed in a newspaper usually are too small to accommodate more than a few digits of normal handwriting. If using the subscript notation, solvers often create a larger copy of the puzzle or employ a sharp or mechanical pencil.
• The second notation is a pattern of dots with a dot in the top left hand corner representing a 1 and a dot in the bottom right hand corner representing a 9. The dot notation has the advantage that it can be used on the original puzzle. Dexterity is required in placing the dots, since misplaced dots or inadvertent marks inevitably lead to confusion and may not be easy to erase without adding to the confusion. Using a pencil would then be recommended.

An alternative technique that some find easier is to mark up those numbers that a cell cannot be. Thus a cell will start empty and as more constraints become known it will slowly fill. When only one marking is missing, that has to be the value of the cell.

Analysing

The two main approaches to analysis are "candidate elimination" and "what-if".

• In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed—usually checking to see the effect of the latest number. There are a number of elimination tactics, all of which are based on the simple rules given above, which have important and useful corollaries, including:
  1. A given set of n cells in any particular block, row, or column can only accommodate n different numbers. This is the basis for the "unmatched candidate deletion" technique, discussed below.
  2. Each set of candidate numbers, 1–9, must ultimately be in an independently self-consistent pattern. This is the basis for advanced analysis techniques that require inspection of the entire set of possibilities for a given candidate number. Only certain "closed circuit" or "n×n grid" possibilities exist (which have acquired peculiar names such as "X-wing" and "Swordfish", among others). If these patterns can be identified, elimination of candidate possibilities external to the grid framework can sometimes be achieved.

• One of the most common elimination tactics is "unmatched candidate deletion". Cells with identical sets of candidate numbers are said to be matched if the quantity of candidate numbers in each is equal to the number of cells containing them; essentially, these are perfectly coincident contingencies. For example, cells are said to be matched within a particular row, column, or region (scope) if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triplet of candidate numbers (p,q,r) and no others. The placement of these numbers anywhere else in the matching scope would make a solution for the matched cells impossible; thus, the candidate numbers (p,q,r) appearing in unmatched cells in the row, column or region scope can be deleted. This principle also works with candidate number subsets—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 in the scope can be deleted. The principle is true for all quantities of candidate numbers.

• A second related principle is also true — if the number of cells (in a row, column or region scope) where a set of candidate numbers only appear is equal to the quantity of candidate numbers, the cells and numbers are matched and only those numbers can appear in matched cells. Other candidates in the matched cells can be eliminated. For example, if (p,q) can only appear in 2 cells (within a specific row, column, region scope), other candidates in the 2 cells can be eliminated.

The first principle is based on cells where only matched numbers appear. The second is based on numbers that appear only in matched cells. The validity of either principle is demonstrated by posing the question 'Would entering the eliminated number prevent completion of the other necessary placements?' Advanced techniques carry these concepts further to include multiple rows, columns, and blocks. (See "X-wing" and "Swordfish" above.)

• In the what-if approach, a cell with only two candidate numbers is selected, and a guess is made. The steps above are repeated unless a duplication is found or a cell is left with no possible candidate, in which case the alternative candidate is the solution. In logical terms, this is known as reductio ad absurdum. Nishio is a limited form of this approach: for each candidate for a cell, the question is posed: will entering a particular number prevent completion of the other placements of that number? If the answer is yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as trial and error (and most published puzzles are built to ensure that it will never be necessary to resort to this tactic,) but it can arrive at solutions fairly rapidly.

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 numbers 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 minimises counting, marking up, and rubbing out.

Computer solutions

For computer programmers, coding the search for cell values based elimination, contingencies and multiple contingencies (required for harder Sudoku) is relatively straightforward. These programs emulate the human logic to solve a puzzle without resorting to guesses. Given the self-imposed constraints of most Sudoku publishers, this method generally succeeds.

It is also fairly simple to build a backtracking search. Typically this involves assigning a value (say, 1, or the nearest available number to 1) to the first available cell (say, the top left hand corner) and then moves on to assign the next available value (say, 2) to the next available cell. This continues until a conflict occurs, in which case the next alternative value is used for the last cell changed. If a cell cannot be filled, the program backs up one level (from that cell) and tries the next value at the higher level (hence the name backtracking). Although far from computationally efficient, this "brute force" method will find a solution, given sufficient computation time. A standard 9×9 puzzle can typically be "solved" in under two seconds using almost any programming language. An extremely difficult puzzle can take as much as 1 minute. A more efficient program could keep track of potential values for cells, eliminating impossible values until only one value remains for a cell, then filling that cell in and using that information for more eliminations, and so on until the puzzle is solved.

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.

A highly efficient way of solving such constraint problems is the Dancing Links Algorithm, by Donald Knuth. This method can be directly applied to solving Sudoku problems, counting all possible solutions for most puzzles in milliseconds. This is the method now preferred by many Sudoku programmers, mainly by virtue of its speed. A very fast solver is usually required for most trial-and-error puzzle-creation algorithms.

Difficulty ratings

Published puzzles often are ranked in terms of difficulty. Perhaps 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. It is based on the relevance and the positioning of the 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.

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; like 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 beside each puzzle; the givens are always found in a symmetrical pattern. Dell Number Place Challenger (see Variants below) puzzles also list author credits. 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 authoring credits. 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. 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 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; Daily SuDoku features new 4×4, 6×6, and simpler 9×9 grids every day as Daily SuDoku for Kids. [1] Even the 9×9 grid is not always standard, with Ebb regularly publishing some of those with nonomino regions; the 2005 U.S. Puzzle Championship had a Sudoku with parallelogram regions that wrapped around the outer border of the puzzle, as if the grid were toroidal. Larger grids are also possible, with Daily SuDoku's 12×12-grid Monster SuDoku [2], Dell regularly publishing 16×16 Number Place Challenger puzzles, 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 Daily Mail, which use 6×6 grids. The Daily Mail also features Super Sudoku X in its Weekend magazine: an 8×8 grid in which rows, columns, main diagonals, 2×4 blocks and 4×2 blocks contain each number once. Another dimension in use is digits with the same relative location within their respective regions; such puzzles are usually printed in colour, with each disjoint group sharing one colour for clarity.

Other kinds of extra restrictions can be mathematical 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 commonly seen. 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. [3] 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, which use letters rather than numbers, have also emerged; of course, there is no functional difference in the puzzle unless the letters spell something. Recent variants have just that, often in the form of a word reading along a main diagonal once solved; determining the word in advance can be viewed as a solving aid. The Code Doku [4] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku [5] 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 unique variations:

• A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005.
• The 2005 U.S. Puzzle Championship includes a variant called Digital Number Place: rather than givens, most cells contain a partial given—a segment of a number, with the numbers drawn as if part of a seven-segment display.
• Wei-Hwa Huang created a meta-Sudoku, where the object is to finish drawing the 5×5 grid's pentomino-region borders so as to leave a uniquely solvable puzzle with no identically-shaped regions.

Mathematics of Sudoku

The general problem of solving Sudoku puzzles on n² x n² boards of n x n blocks is known to be NP-complete [6]. This gives some indication of why Sudoku is difficult to solve, although on boards of finite size the problem is finite and can be solved by a deterministic finite automaton that knows the entire game tree.

Solving Sudoku puzzles 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 <math>(x,\, y)</math>, where x and y are integers between 1 and 9. In this case, two distinct vertices labelled by <math>(x,\, y)</math> and <math>(x',\, y')</math> are joined by an edge if and only if:

• X = X'
• <math> y = y'\,</math> or,
• <math> \lceil x/3 \rceil = \lceil x'/3 \rceil</math> and <math> \lceil y/3 \rceil = \lceil y'/3 \rceil</math>

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 Latin square. There are significantly fewer valid Sudoku solution grids than Latin squares because Sudoku imposes the additional regional constraint. Nonetheless, the number of valid Sudoku solution grids for the standard 9×9 grid was calculated by Bertram Felgenhauer in 2005 to be 6,670,903,752,021,072,936,960 [7] (sequence A107739 in OEIS). This number is equal to 9! × 72² × 2⁷ × 27,704,267,971, the last factor of which is prime. 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 [8] (sequence A109741 in OEIS). 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 [9] [10], and 18 with the givens in rotationally symmetric cells.

For more results and conjectures, see the Mathematics of Sudoku page.

History

The puzzle was designed by Howard Garns, a 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 Nikoli in the paper Monthly Nikolist in April 1984 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" (独身 literally means "single; celibate; unmarried"). The puzzle was named by Kaji Maki (鍜治 真起), the president of Nikoli. At a later date, the name was abbreviated to Sudoku (数独, pronounced SUE-dough-coo; sū = number, doku = single); it is a common practice in Japanese to take only the first kanji of compound words to form a shorter version. In 1986, Nikoli introduced two innovations which guaranteed the popularity of the puzzle: 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. 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 (the English translation of 'sЕ«doku') for the Apple Macintosh [11] in 1995 in Japanese and English, and in 1996 for the Palm (PDA) [12].

Bringing the process full-circle, Dell Magazines, which publishes the original Number Place puzzle, now also publishes two Sudoku magazines: Original Sudoku and Extreme Sudoku. Additionally, Kappa reprints Nikoli Sudoku in GAMES Magazine under the name Squared Away; the New York Post, USA Today, The Boston Globe, Washington Post, 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 Rubik's Cube, another logic puzzle 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 Zealander, was enticed by seeing a partly completed puzzle in a Japanese bookshop. He went on to develop a computer program to produce puzzles quickly; this took over six years. 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 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 variously as "the Rubik's cube of the 21st century" or the "fastest growing puzzle in the world".

There is no doubt that it was not until The 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 of the other national 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, and The Guardian claiming (though perhaps ironically) that its hand-made puzzles, licensed from Nikoli, offered a superior experience (complete with "almost imperceptible witticisms") to the computer-generated grids found in other papers.

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 [13]). 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 142). On 2nd August 2005 the BBC's programme guide Radio Times started to feature a weekly Super Sudoku.

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.

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 26th.

See also

• List of newspapers featuring Sudoku
• List of Nikoli puzzle types
• Mathematics of Sudoku

References

• Rules and history from the Nikoli website
• Keys to Solution at Puzzle Japan
• Sudoku.com Website of Wayne Gould, populariser of Sudoku. Also includes forum which discusses solution techniques and mathematics of Sudoku.
• Let's Make Sudoku! – Explains step-by-step how to make Sudoku puzzles by hand.
• Sudoku Variations article at MAA Online; also includes the history of the puzzle's invention
• Complexity and Completeness of Finding Another Solution and its Application to Puzzles Mathematical reference proving NP-completeness.
• Frazer Jarvis's Sudoku page Contains programs, data, an article with Bertram Felgenhauer detailing the enumeration of Sudoku grids, and the results of Ed Russell.
• A step-by-step guide to Sudoku by Michael Mepham.
• Commentary on the sudden popularity of Sudoku in Britain:
  • The puzzling popularity of Su Doku (BBC News, 22 April 2005)
  • So you thought Sudoku came from the Land of the Rising Sun… (The Observer, 15 May 2005)
  • Do you sudoku? (The Economist, 19 May 2005)

External links

• Sudoku at the Open Directory Project – An active listing of Sudoku links.
• Sudoku Programmers Forum

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