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Sudoku

2007 Schools Wikipedia Selection. Related subjects: Recreation

   A sudoku puzzle
   Enlarge
   A sudoku puzzle

   Sudoku (数独, sūdoku^ ?), also known as Number Place or Nanpure, is a
   logic-based placement puzzle. The objective is to fill the grid so that
   every column, every row and every 3×3 box contains the digits 1 to 9.
   The puzzle setter provides a partially completed grid so that there is
   only one solution.

   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.

   The modern puzzle was invented by an American, Howard Garns, in 1979
   and published by Dell Magazines under the name "Number Place". It
   became popular in Japan in 1986, when it was published by Nikoli and
   given the name Sudoku. It became an international hit in 2005.

Introduction

   The name "Sudoku" is the Japanese abbreviation of a longer phrase,
   "Sūji wa dokushin ni kagiru" (数字は独身に限る, "Sūji wa dokushin ni kagiru"^
   ?), meaning "the digits must occur only once". It is a trademark of
   puzzle publisher Nikoli Co. Ltd. in Japan. In Japanese, the word is
   pronounced [sɯːdokɯ]; in English, it is usually spoken with an
   Anglicised pronunciation, [səˈdəʊkuː] (BrE) [səˈdoʊkuː] (AmE) or
   [ˈsuːdəʊku] (BrE) [ˈsuːdoʊku] (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. Some non-Japanese 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. 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.

Solution methods

   The strategy for solving a puzzle may be regarded as comprising a
   combination of three processes: scanning, marking up, and analyzing.
   The top right region must contain a 5. By hatching across and up from
   5s elsewhere, the solver can eliminate all the empty cells in the
   region which cannot contain a 5. This leaves only one possibility
   (shaded green).
   Enlarge
   The top right region must contain a 5. By hatching across and up from
   5s elsewhere, the solver can eliminate all the empty cells in the
   region which cannot contain a 5. This leaves only one possibility
   (shaded green).

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:
     * 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, from high to low.
       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.
   Enlarge
   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. There are
   two popular notations: subscripts and dots.
     * In the 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.

     * The second 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 up" the numerals that a cell
   cannot be. A cell will start empty and as more constraints become
   known, it will slowly fill 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.

Analysis

   The two main approaches to analysis are "candidate elimination" and
   "what-if". 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 unless 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. 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:
    1. basic conflicts - there are only N-1 different candidates in N cell
       in the area
    2. fish conflicts - when eliminating number from N rows/columns, it
       will disappear also from N+1 columns/rows.
    3. unique conflicts - this pattern means multiple solutions, all
       numbers in the pattern exist exactly two times in every area, row
       and column. If there are 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 two general approaches taken in the creation of serious
   Sudoku-solving programs: human solving methods and rapid-style methods.
   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.

   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 vanilla 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.

Difficulty ratings

   The difficulty of a puzzle is based on the relevance and the
   positioning of the given numbers rather than their quantity.
   Surprisingly, the number of givens does not always 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". 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.

Construction

   Building a Sudoku puzzle can be performed by pre-determining 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. 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
   Enlarge
   A nonomino Sudoku puzzle, sometimes also known as a Jigsaw Sudoku, for
   instance in the Sunday Telegraph

   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, with Daily SuDoku's
   16×16-grid Monster SuDoku , 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. Even larger sizes, for
   instance 100×100, have been claimed.

   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.

   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 devised by Steve Schaefer
   has an entire sentence embedded into the puzzle; the Super Wordoku 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.

   Here are some of the more notable single-instance variations:
     * 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 online journal Speculative Grammarian has published a trio of
       linguistics-themed Sudoku-like puzzles called LingDoku, which
       require the solver to solve for two or more variables at once,
       including a simple 3×3 puzzle,, a slightly more complicated 4x4
       puzzle, and a Samurai-style puzzle with 3 overlapping 3×3 grids.

   Greater-Than or Comparison Sudoku (sample)
   Enlarge
   Greater-Than or Comparison Sudoku (sample)

   A new variant without any given numbers for start comes up this summer
   (2006) from India by Yoogi Games, named Greater-Than or Comparison
   Sudoku. All rules of a standard Sudoku are the same, but you have to
   find out by comparisons the sequence of the digits (or letters if you
   want, but the symbols used must be in ascending order) from one through
   nine for each region. Therefore, each cell's border line inside every
   region is notched in the meaning of < (less than) – or vice versa. A 3D
   variant of Sudoku is Sudoku-Ball™,. Sudoku-Ball contains 14 networked
   sudokus that follow the idea of the 2D samurai puzzle but than
   converted into 3D. The physical game is not yet available on the market
   but is due to release early 2007.

   An example of a Hexudoku. Enlarge
   An example of a Hexudoku.

   A Hexudoku is a variant of the game Sudoku, which uses a grid of 16x16
   squares instead of 9x9. The name originates from the contraction of the
   greek-, and latin prefixes hexa-deca, which means 16, and the word
   sudoku.

   Similar to sudoku, the goal of the game is to complete a not-completely
   filled grid, such that each row, each column, and each of the 16 marked
   4x4 squares contains each number between 1 and 16 exactly once. With
   256 squares, the hexudoku has more than 3 times as many squares as the
   regular sudoku.

   Instead of the numbers 1 to 16, sometimes the hexadecimal digits 0-9,
   a-f are used, or a variant starting at 1: 1-9, a-g. Also a letter-only
   variant is sometimes used.

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 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
   (sequence A107739 in OEIS) : 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 (sequence A109741 in OEIS). Both results have been
   confirmed by independent authors.

   The maximum number of givens 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, and 18
   with the givens in rotationally symmetric cells.

History

   Page from La France newspaper, July 6, 1895
   Enlarge
   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. 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.

   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, and first published in 1979 by Dell
   Magazines as Number Place (the earliest known examples of modern
   Sudoku), because Garns' 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.
   Sadly, he died in 1989 before getting a chance to see his creation as a
   worldwide phenomenon.

   The puzzle was introduced in Japan by Nikoli in the paper Monthly
   Nikolist in April 1984 as Suuji wa dokushin ni kagiru (数字は独身に限る, Suuji
   wa dokushin ni kagiru^ ?), which can be translated as "the numbers must
   be single" or "the numbers must occur only once." At a later date, the
   name was abbreviated to Sudoku by Maki Kaji (鍜治 真起, Kaji Maki^ ?),
   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.

Competitions

     * The first world championship was held in Lucca, Italy from 10 to 12
       March 2006; it was won by Jana Tylová, a 31-year-old accountant
       from the Czech Republic. The competition included numerous
       variants.
     * The second world championship will be held in Prague from March 28
       to April 1, 2007;

Related games

     * Minesweeper (computer game)
     * Paint by numbers

   Retrieved from " http://en.wikipedia.org/wiki/Sudoku"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
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