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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 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. For most computer programmers, coding the search for cell values based on 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. 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] 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. 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. The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. The first world championship was held in Lucca, Italy from 10 to 12 March 2006 [20]; it was won by Jana Tylova, a 31-year-old accountant from the Czech Republic. The competition included variants; a full list can be found in the PDF here. The attraction of the puzzle is that the rules are simple, yet the line of reasoning required to reach the solution may be complex

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. 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: For most computer programmers, coding the search for cell values based on 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. 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. 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. 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. 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.

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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 general problem of solving Sudoku puzzles on n2 x n2 boards of n x n blocks is known to be NP-complete [9]. 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. The second notation uses a pattern of dots within each square, where the position of the dot represents a number from 1 to 9. Dot schemes differ and one method is illustrated here. 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 sharp pencil with an eraser end is recommended. 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. Here are some of the more notable single-instance variations: 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: The name Sudoku is the Japanese abbreviation of a longer phrase, "suuji wa dokushin ni kagiru (????????)," meaning "the digits must remain single"; it is a trademark of puzzle publisher Nikoli Co. Ltd in Japan. The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. 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.

Sudoku (Japanese) 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. You solve the puzzle with reasoning and logic. Even though most solving algorithms are able to solve puzzles in under a second, very fast solvers are preferred for trial-and-error puzzle-creation algorithms, which must be able to test large numbers of partial problems for validity in a short time. 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 (even a fairly naive implementation will typically not take a noticeable amount of time). 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. The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. 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. In the subscript notation the candidate numerals 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.