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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, 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). 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. When using marking, a couple of similar rules applied in a specified order can solve any sudoku puzzle, without performing any kind of backtracking. 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. Yoshimitsu Kanai published his computerized puzzle generator under the name Single Number for the Apple Macintosh [15] in 1995 in Japanese and English, for the Palm (PDA) [16] in 1996, and for the Mac OS-X [17] in 2005. This is a row, 9 cells wide. A filled-in row must have one of each digit. That means that each digit appears only once in the row. There are 9 rows in the grid, and the same applies to each of them. 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. This is a row, 9 cells wide. A filled-in row must have one of each digit. That means that each digit appears only once in the row. There are 9 rows in the grid, and the same applies to each of them. There's no math involved, the grid has numbers, but nothing has to add up to anything else. 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

Although for standard sudoku problems highly optimized and sophisticated backtracking programs are fastest, 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. 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. 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. 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. A three-dimensional sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. 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. 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 [18]). 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. [19] 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. Also found is the Circular sudoku, also known as Target sudoku, invented by Essex mathematician Peter Higgins. [3] [4] In this variant, all the numbers must appear in all the concentric rings as well as in all pairs of adjacent wedges. 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.

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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. Also found is the Circular sudoku, also known as Target sudoku, invented by Essex mathematician Peter Higgins. [3] [4] In this variant, all the numbers must appear in all the concentric rings as well as in all pairs of adjacent wedges. Cross-hatching: the scanning of rows (or columns) to identify which line in a particular region may contain a certain numeral by a process of elimination. This process is then repeated with the columns (or rows). For fastest results, the numerals are scanned in order of their frequency. It is important to perform this process systematically, checking all of the digits 1-9. This is a box, containing 9 cells in a 3x3 layout. A filled-in box must have one of each digit. That means that each digit appears only once in the box. There are 9 boxes in the grid, and the same applies to each of them. 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:or, or, and 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. 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. 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.

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 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. 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. When using marking, a couple of similar rules applied in a specified order can solve any sudoku puzzle, without performing any kind of backtracking. 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. Challenge Me, LLC will be hosting the first national face-to-face sudoku competition in the United States and the largest sudoku tournament to date. Scheduled to begin in Chicago, Illinois on June 10-11, 2006 participants from 32 regional tournaments will compete to become the champion of their region. The Regional Champions will then compete in a National Final to become the 2006 sudoku Champions winner and win a $50,000 grand prize (http://www.sudokuchampions.com). 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. You solve the puzzle with reasoning and logic. 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. 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:or, or, and