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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. There is no doubt that 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. 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. 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. 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. 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. This is a column, 9 cells tall. A filled-in column must have one of each digit. That means that each digit appears only once in the column. There are 9 columns in the grid, and the same applies to each of them. 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".

Here are some of the more notable single-instance variations: 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. 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 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. 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. 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. 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".

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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. 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. Sudoku is recommended by some teachers as an exercise in logical reasoning. 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. 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. 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 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 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 [12] [13], and 18 with the givens in rotationally symmetric cells.

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". You solve the puzzle with reasoning and logic. Sudoku is recommended by some teachers as an exercise in logical reasoning. 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. 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. The world's first live TV Sudoku show, 1 July 2005, Sky One.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. 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. 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.