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The attraction of the puzzle is that the rules are simple, yet the line of reasoning required to reach the solution may be complex A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. 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 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 (also known as a jigsaw variation); 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], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths. 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. There's no math involved, the grid has numbers, but nothing has to add up to anything else. 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. 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 strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. 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.

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 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. 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 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). A three-dimensional Sudoku puzzle was invented by Dion Church and published in the Daily Telegraph in May 2005. 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. The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in April 1984 as Suuji wa dokushin ni kagiru (????????), which can be translated as "the numbers must be single" or "the numbers must occur only once" (?? literally means "single; celibate; unmarried"). The puzzle was named by Kaji Maki (?? ??), the president of Nikoli. At a later date, the name was abbreviated to Sudoku (??, pronounced SUE-dough-coo; su = 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. 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 (also known as a jigsaw variation); 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], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths.

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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. 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. 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: Fill in the grid so that every row, every column, and every 3x3 box contains the digits 1 through 9. 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. [6] 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 [7] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku [8] 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. 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. 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. 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. Others like Magic Sudoku [5] adds some restrictions (diagonals from 1 to 9, and colors) to the standard sudoku to solve it with less numbers.

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. 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. 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). 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". 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. 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 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. 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.