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'' is a logic-based number placement puzzle. The objective is to fill a 9x9 grid so that each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9. The puzzle setter provides a partially completed grid.
The general way to solve the problem is when you insert a new number it should be unique in that particular region(blocks)and also in that particular row and column
Completed ''Sudoku'' puzzles are a type of Latin square, with an additional constraint on the contents of individual regions. Leonhard Euler is sometimes incorrectly cited as the source of the puzzle, based on his work with Latin squares.^[1]
The modern puzzle was invented by an American, Howard Garns, in 1979 and published by Dell Magazines under the name "'''Number Place'''".^[2] It became popular in Japan in 1986, after it was published by Nikoli and given the name Sudoku, meaning ''single number''. ^[3] It became an international hit in 2005.

Contents

Introduction

Strategies

Scanning

Marking up

Subscript notation

Dot notation

Analysis

Computer solutions

Difficulty ratings

Construction

Variants

Mathematics of Sudoku

History

Popularity in the media

Competitions

See also

References

External links

Introduction

The name ''Sudoku'' means "single digits"^{[4][5][6][7][8][9]}. The name is a trademark of puzzle publisher Nikoli Co. Ltd. in Japan History of Sudoku in our site Nikoli . The word sudoku derives from the Japanese phrase "Sūji wa dokushin ni kagiru," meaning "the numbers must be single," or "the numbers must occur only once." In Japanese, the word is pronounced []; in English, it is usually [] (BrE) [] (AmE) or [] (BrE) [] (AmE) (See IPA (International Phonetic Alphabet) or IPA chart for English for notation usage.) Other Japanese publishers refer to the puzzle as ''Number Place'', the original U.S. title, or as "Nanpure" for short.^[10] Some publishers spell the title as "Su Doku".
The numerals in ''Sudoku'' puzzles are used for convenience; arithmetic relationships between numerals are irrelevant. Any set of distinct symbols will do; letters, shapes, or colours may be used without altering the rules. In fact, ESPN published Sudoku puzzles substituting the positions on a baseball field for the numbers 1–9; and Viz magazine published a Doctor Who version of the game, using images of the television series' first nine leading actors in place of the numerals. Dell Magazines, the puzzle's originator, has been using numerals for ''Number Place'' in its magazines since they first published it in 1979.
The attraction of the puzzle is that the rules are simple, yet the line of reasoning required to solve the puzzle may be complex. The level of difficulty can be selected to suit the audience. The puzzles are often available free from published sources and may be custom-made using software.

Strategies

The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analyzing. The approach to analysis may vary according to the concepts and the representations on which it is based.

Scanning

Scanning is performed at the outset and throughout the solution. Scans need to be performed only once in between analyses. Scanning consists of two techniques as follows:

★ 'Cross-hatching:' the scanning of rows to identify which line in a region may contain a certain numeral by a process of elimination. The process is repeated with the columns. For fastest results, the numerals are scanned in order of their frequency, in sequential order. 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 numerals.' Counting based upon the last numeral discovered may speed up the search. It also can be the case, particularly in tougher puzzles, that the best way to ascertain the value of a cell is to count in reverse—that is, by scanning the cell's region, row, and column for values it ''cannot'' be, in order to see what remains.
Advanced solvers look for "contingencies" while scanning, narrowing a numeral's location within a row, column, or region to two or three cells. When those cells lie within the same row ''and'' region, they can be used for elimination during cross-hatching and counting. Puzzles solved by scanning alone without requiring the detection of contingencies are classified as "easy"; more difficult puzzles cannot be solved by basic scanning alone.

A method for marking likely numerals in a single cell by the placing of pencil dots. To reduce the number of dots used in each cell, the marking would only be done after as many numbers as possible have been added to the puzzle by scanning. Dots are erased as their corresponding numerals are eliminated as candidates.

Marking up

Scanning stops when no further numerals can be discovered, making it necessary to engage in logical analysis. One method to guide the analysis is to mark candidate numerals in the blank cells.

Subscript notation

In subscript notation, the candidate numerals are written in subscript in the cells. Because puzzles printed in a newspaper are too small to accommodate more than a few subscript digits of normal handwriting, solvers may create a larger copy of the puzzle. Using two colors, or mixing pencil and pen marks can be helpful.

Dot notation

The dot notation uses a pattern of dots in each square, where the dot position indicates a number from 1 to 9. The dot notation 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 easily erased.
An alternative technique is to mark the numerals that a cell ''cannot'' be. The cell starts empty and as more constraints become known, it slowly fills until only one mark is missing. 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.

(click to see larger version) An analysis in Sudoku, done in superscript notation, with all possible values for the squares written in. There are three squares which contain only three values: 4, 6, and 8. If 4, 6, or 8 were written in any square where they're red, it would be impossible to complete the squares where they're blue. Therefore, the numbers in red can be erased. This logic works with rows, columns, sections, and diagonals. (if applicable)

Analysis

The two main approaches to analysis are "candidate elimination" Goals of Sukoku-Grok and "what-if". Play Sudoku In "candidate elimination", progress is made by successively eliminating candidate numerals from cells to leave one choice. After each answer has been achieved, another scan may be performed—usually checking to see the effect of the contingencies. In general, if entering a particular numeral prevents completion of the other necessary placements, then the numeral in question can be eliminated as a candidate. One method works by identifying "matched cell groups". For instance, if precisely two cells within a scope (a particular row, column, or region) contain the same two candidate numerals (''p'',''q''), or if precisely three cells within a scope contain the same three candidate numerals (''p'',''q'',''r''), these cells are said to be matched. The placement of those candidate numerals anywhere else within that same scope would make a solution impossible; therefore, those candidate numerals can be deleted from all other cells in the scope.
In the "what-if" approach (also called "guess-and-check", "bifurcation", "backtracking" and "Ariadne's thread"), a cell with two candidate numerals is selected, and a guess is made. The steps are repeated until a duplication is found or a cell is left without a possible candidate, in which case the alternative candidate must be the solution. For each cell's candidate, the question is posed: 'will entering a particular numeral prevent completion of the other placements of that numeral?' If the answer is 'yes', then that candidate can be eliminated. If the "what-if" exercises for both candidates show that either one is possible, another pair should be tried. Alternatively, if the "what-if" exercises for both candidates imply an identical result, then that result must be true. The what-if approach requires a pencil and eraser or a good layout memory.
There are three kind of conflicts, which can appear during puzzle solving:
#basic conflicts - there are only N-1 different candidates in N cell in the area
#fish conflicts - when eliminating number from N rows/columns, it will disappear also from N+1 columns/rows.
#unique conflicts - this pattern means multiple solutions, all numbers in the pattern exist exactly two times in every area, row and column. If there is only one candidate in the cell, any virtual candidate can be added.
Encountering any of those would indicate that the puzzle is not uniquely solvable. So, if you encounter them as a consequence of "what-if", you use your eraser and go back to try untried alternatives.

Computer solutions

There are three general approaches taken in the creation of serious ''Sudoku''-solving programs: human solving methods, rapid-style methods, and pure brute-force algorithms. Human-style solvers will typically operate by maintaining a mark-up matrix, and search for contingencies, matched cells, and other elements that a human solver can utilize in order to determine and exclude cell values.
Many rapid-style solvers employ backtracking searches, with various pruning techniques also being used in order to help reduce the size of the search tree. The term rapid-style may be misleading: Most human-style solvers run considerably faster than a rapid-style solver, although the latter takes less time to write and is more easily adapted to larger grids. A purely brute-force algorithm is very simple and finds a solution to a puzzle essentially by "counting" upward until a string of 81 digits is constructed which satisfies the row, column, and box constraints of the puzzle.
Rapid solvers are preferred for trial-and-error puzzle-creation algorithms, which allow for testing large numbers of partial problems for validity in a short time; human-style solvers can be employed by hand-crafting puzzlesmiths for their ability to rate the challenge of a created puzzle and show the actual solving process their target audience can be expected to follow.
Although typical ''Sudoku'' puzzles (with 9×9 grid and 3×3 regions) can be solved quickly by computer,
the generalization to larger grids is known to be ''NP-complete''. Various optimisation methods
have been proposed for large grids.
Details of computer solutions may be found on the page on the Algorithmics of Sudoku.

Difficulty ratings

The difficulty of a puzzle is based on the relevance and the positioning of the given numbers rather than their quantity. Surprisingly, most of the time the number of givens does not reflect a puzzle's difficulty. Computer solvers can estimate the difficulty for a human to find the solution, based on the complexity of the solving techniques required. Some online versions offer several difficulty levels.
Most publications sort their ''Sudoku'' puzzles into four or five rating levels, although the actual cut-off points and the names of the levels themselves can vary widely. Typically, however, the titles are synonyms of "easy", "intermediate", "hard", and "challenging" (also known as "diabolical" or "evil"). An easy puzzle can be solved using only scanning; an intermediate puzzle may take markup to solve; a hard or challenging puzzle will usually take analysis.
Another approach is to rely on the experience of a group of human test solvers. Puzzles can be published with a median solving time rather than an algorithmically defined difficulty level.
Difficulty is a very complex topic, subject to much debate on the Sudoku forums, because it may depend on the concepts and visual representations one is ready to use.

Construction

Building a ''Sudoku'' puzzle can be performed by predetermining the locations of the givens and assigning them values only as needed to make deductive progress. 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. 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.
Nikoli ''Sudoku'' are hand-constructed, with the author being credited; the givens are always found in a symmetrical pattern.^[11] Dell ''Number Place Challenger'' (see Variants below) puzzles also list authors. The ''Sudoku'' puzzles printed in most UK newspapers are apparently computer-generated but employ symmetrical givens; ''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''.

Variants

A nonomino ''Sudoku'' puzzle, sometimes also known as a Jigsaw Sudoku, for instance in the Sunday Telegraph

Solution numbers in red for above puzzle

Even though the 9×9 grid with 3×3 regions is by far the most common, 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. Larger grids are also possible. The ''Times'' offers a 12×12-grid ''Dodeka sudoku'' with 12 regions of 4×3 squares each. Dell regularly publishes 16×16 ''Number Place Challenger'' puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal). Nikoli offers 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.
Another variant is the combination of Sudoku with Kakuro on a 9 x 9 grid, and clues are given in terms
of cross sums. The clues can also be given by cryptic alphametics.

Hypersudoku puzzle.

Solution to Hypersudoku puzzle.

Many newspapers include the popular Hypersudoku. The layout is identical to a normal Sudoku, but with additional interior areas defined in which the numbers 1 to 9 must appear. The solving algorithm is therefore more complicated than for normal Sudoku puzzles, although there is the possibility of having access to more information to solve each of the shaded squares.
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'', ''The Age'' and ''The Sydney Morning Herald'' this form of puzzle is known as Samurai SuDoku. 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'' MathRec Sudoku devised by Steve Schaefer has an entire sentence embedded into the puzzle; the ''Super Wordoku'' Twodoku! 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 claims this as a feature designed to defeat solving programs. Cludoku is an alphabetical variation that is solved only by logic, after which the solver then has to find solutions to crossword-style clues found in the completed grid.
There is also a Sudoku version of the Rubik's Cube named Sudokube.
Finally, some of the more notable single-instance variations are:

★ 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.

★ The Sudoku_Solitaire concept was created by Leonard C. Russell for bluenitesystems. In it the numbers needed to complete the puzzle are played from a deck and discard pile, just like in the Solitaire card game.

Mathematics of Sudoku

:Main articles: Mathematics of Sudoku

A completed ''Sudoku'' grid is a special type of Latin square with the additional property of no repeated values in any 3×3 block. The relationship between the two theories is now completely known, after Denis Berthier has proven in his recent book, "The Hidden Logic of Sudoku"^[12], that a first order formula that does not mention blocks (also called boxes or regions) is valid for Sudoku if and only if it is valid for Latin Squares.
The number of classic 9×9 ''Sudoku'' solution grids was shown in 2005 by Bertram Felgenhauer and Frazer Jarvis to be 6,670,903,752,021,072,936,960 Sudoku enumeration problems : this is roughly 0.00012% the number of 9×9 Latin squares. Various other grid sizes have also been enumerated—see the main article for details. The number of ''essentially different'' solutions, when symmetries such as rotation, reflection and relabelling are taken into account, was shown by Ed Russell and Frazer Jarvis to be just 5,472,730,538 There are 5472730538 essentially different Sudoku grids ... and the Sudoku symmetry group .
The maximum number of givens provided while still not rendering a unique solution 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, プログラミングパズルに関心のある人は雑談しましょう Minimum Sudoku and 18 with the givens in rotationally symmetric cells.

History

Page from ''La France'' newspaper, July 6, 1895

Number puzzles first appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers from magic squares. ''Le Siècle'', a Paris-based daily, published a partially completed 9×9 magic square with 3×3 sub-squares in 1892. Supplément de l’article « Les ancêtres français du sudoku », , Christian, Boyer, Pour la Science, 2006 It was not a Sudoku because it contained double-digit numbers and required arithmetic rather than logic to solve, but it shared key characteristics: each row, column and sub-square added up to the same number.
Within three years ''Le Siècle's'' rival, ''La France'', refined the puzzle so that it was almost a modern Sudoku. It simplified the 9×9 magic square puzzle so that each row and column contained only the numbers 1–9, but did not mark the sub-squares. Although they are unmarked, each 3×3 sub-square does indeed comprise the numbers 1–9. However, the puzzle cannot be considered the first Sudoku because, under modern rules, it has two solutions. The puzzle setter ensured a unique solution by requiring 1–9 to appear in both diagonals.
These weekly puzzles were a feature of newspaper titles including ''L'Echo de Paris'' for about a decade but disappeared about the time of the First World War. Les fiendish French beat us to Su Doku Jack Malvern
According to Will Shortz, the modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Indiana, and first published in 1979 by Dell Magazines as ''Number Place'' (the earliest known examples of modern Sudoku). Garns's name was always present on the list of contributors in issues of ''Dell Pencil Puzzles and Word Games'' that included ''Number Place'', and was always absent from issues that did not. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon. It is unclear if Garns was familiar with any of the French newspapers listed above.
The puzzle was introduced in Japan by Nikoli in the paper ''Monthly Nikolist'' in April 1984 as , which can be translated as "the digits must be single" or "the digits are limited to one occurrence." At a later date, the name was abbreviated to ''Sudoku'' by , taking only the first kanji of compound words to form a shorter version. In 1986, Nikoli introduced two innovations: 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''.

Popularity in the media

In 1997, retired Hong Kong judge Wayne Gould, 59, a New Zealander, saw a partly completed puzzle in a Japanese bookshop. Over six years he developed a computer program to produce puzzles quickly.^[13] 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 rapid rise of ''Sudoku'' in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as when ''The Guardian's ''G2'' section advertised itself as the first newspaper supplement with a ''Sudoku'' grid on every page G2, home of the discerning Sudoku addict ). 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. On 2 August, the BBC's programme guide ''Radio Times'' featured a weekly Super Sudoku.
Even the Greeks owe the publication of their first Sudoku magazine to British influence. It was at Heathrow airport in the summer of 2005 that a Greek computer magazine publisher first laid eyes on a British Sudoku magazine and - realising the opportunity - proceeded to purchase the necessary software and quickly launch the first local Sudoku magazine that became an instant success.
In the United States, the first newspaper to publish a Sudoku puzzle by Wayne Gould was The Conway Daily Sun (New Hampshire), in 2004.^[14]

The world's first live TV ''Sudoku'' show, 1 July 2005, Sky One.

The world's first live TV ''Sudoku'' show, ''Sudoku Live'', was a puzzle contest first 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. The audience at home was in a separate interactive competition.
Later in 2005, the BBC launched SUDO-Q, a game show that combines Sudoku with general knowledge. However, it uses only 4x4 and 6x6 puzzles.
Sudoku software is now also very popular on PCs, websites, and mobile phones. It has also been released on portable video game handhelds such as the Nintendo DS, PlayStation Portable, and the Game Boy Advance. One of the most popular video games which features Sudoku is ''. Critically and commercially well received, it generated particular praise for its Sudoku implementation,[1][2][3] and sold more than 8 million copies worldwide.[4]

Competitions

★ The first world championship was held in Lucca, Italy from 10 to 12 March 2006; Sudoku title for Czech accountant it was won by Jana Tylová, a 31-year-old accountant from the Czech Republic. The competition included numerous variants.^[15]

★ The second world championship was held in Prague from March 28 to April 1, 2007. The individual competition was won by Thomas Snyder of the United States, with Japan winning the team contest. Report on the 8th General Assembly of the World Puzzle Federation official WSC 2007 website

★ Hosted by renowned puzzle master Will Shortz, ''The Philadelphia Inquirer Sudoku National Championship'' is the first U.S. Sudoku Championship ever. This unprecedented event will be held October 19 – 21, 2007 in Philadelphia, PA. Prizes ranging from $100 to $10,000 will be awarded for a variety of categories. Solvers can choose to compete at the Beginner, Intermediate or Advanced level. The winner of the Advanced level will win $10,000, plus a spot on the US National Sudoku Team in which they will compete in the 2008 World Sudoku Championship in India. ''The Philadelphia Inquirer Sudoku National Championship''

See also

★ Mathematics of Sudoku

★ List of Sudoku terms and jargon

★ Killer sudoku

★ Sudokube (3D variant)

★ Latin square

★ Logic puzzle

★ List of Nikoli puzzle types

★ Kakuro

★ Nonogram (aka Paint by numbers, O'ekaki)

References

1. On magic squares Leonhard Euler
2. Sudoku Variations
3.
4. History of Sudoku: Roots and Development of Sudoku
5. The History of Sudoku Gil Galanti
6. Sudoku FAQ
7. 数独
8. 数独研究所
9. 脳◎ 数字パズル
10. Ed Pegg Jr.'s Math Games: Sudoku Variations
11. Rules and history of Sudoku from Nikoli
12. The Hidden Logic of Sudoku Denis Berthier
13. Wayne Gould's sudoku.com website
14. ''New York Times'' corrections column, April 2, 2007, p. A2
15. World Sudoku Championship 2006 Instructions Booklet

External links

★ – An active listing of ''Sudoku'' links.

★ Father of Sudoku puzzles next move BBC