SUDOKU

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. However, original puzzles printed in a newspaper usually are too small to accommodate more than a few digits of normal handwriting. Thus, solvers often 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".

Candidate elimination

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. One method works by identifying "matched cells". 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 these numerals anywhere else within that same scope would make a solution impossible; thus, the candidate numerals (p,q,r) scope can be deleted. When all else fails, ask 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.

The "What-If" Approach

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.

Computer solutions

A computer program is capable of exhaustively searching a Sudoku puzzle for solutions, thereby determining whether it is valid or not, with great ease relative to a human attempting the same. There are two general approaches taken in the creation of serious Sudoku-solving programs: Human solving method and rapid-style method.

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 still employ backtracking searches, but with various shortcuts and optimizations to reduce the width of the search tree. Another alternative uses finite domain constraint programming. A constraint program specifies the constraints of the puzzle (the fact that every number in each row, each column, and each 3×3 region must be unique, and the provided "givens"); a finite-domain solver applies the constraints successively to narrow down the solution space until a solution is found. Backtracking may be applied when alternate values cannot be excluded.

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.

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". 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.^[1] 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

An extra-regions Sudoku puzzle (Source: NRC Handelsblad)

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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 [1], 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.

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 and The Sydney Morning Herald this form of puzzle is known as Samurai SuDoku. [2] 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 [3] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku [4] 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.

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 number of linguistics-themed Sudoku-like puzzles called LingDoku, which require the solver to solve for two variables at once, including a simple 3x3 puzzle, and a slightly more complicated 4x4 puzzle.

Greater-Than or Comparison Sudoku (sample)

A new variant without any given numbers for start comes up this summer (2006) from India byYoogi 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.

Mathematics of Sudoku

Main article: 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 [5] (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 [6] (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 [7] [8], and 18 with the givens in rotationally symmetric cells.

History

Page from Le Siècle newspaper, November 19, 1892

Page from La France newspaper, July 6, 1895

Le Siècle, a French daily, produced a 9x9 grid with 3x3 sub-squares as early as 1892, but used double-digit numbers rather than the familiar 1-9 [9]. In 1895, another French daily, La France, created a puzzle that used the numbers 1-9 but did not mark the 3x3 sub-squares. 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.[10]

The modern Sudoku was designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor, and first published in 1979.^[2] Garns added a third dimension to the traditional Roman practice of Latin Squares and presented the creation as a puzzle, providing a partially-completed grid and requiring the solver to fill in the rest. The puzzle was first published in New York by the specialist puzzle publisher Dell Magazines in its magazine Dell Pencil Puzzles and Word Games, under the title Number Place.

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." The puzzle was named by Maki Kaji (鍜治 真起, Kaji Maki^?), the president of Nikoli. At a later date, the name was abbreviated to Sudoku, taking only the first kanji of compound words to form a shorter version. In 1986, Nikoli introduced two innovations: the number of givens 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 6 years he developed a computer program to produce puzzles quickly.^[3] 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 immense surge in popularity of Sudoku in British newspapers and internationally has led to it being dubbed in the world media in 2005 the "fastest growing puzzle in the world" and the "Rubik's cube of the 21st century".

By April and May 2005 the puzzle had become popular in these publications and it was rapidly introduced to several other national British newspapers including The Independent, The Guardian, The Sun (where it was labelled Sun Doku), and The Daily Mirror. As the name Sudoku became well-known in Britain, the Daily Mail adopted it in place of its earlier name "Codenumber". Newspapers competed to promote their Sudoku puzzles, with The Times and the Daily Mail each claiming to have been the first to feature Sudoku.

The rapid rise of Sudoku from relative obscurity in Britain 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 [11]). 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. 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.

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

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. 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.

Dr. House was clearly seen working on a Web Sudoku puzzle on his office computer in one scene of the December 13 2005 episode of House, M. D.; Sudoku is now banned on the studio set due to the cast constantly playing it.

In the The Da Vinci Code parody from the 2006 MTV Movie Awards, Jessica Alba finds a body on the ground with a Sudoku puzzle on his stomach. Alba begins to take her pen out and tries to think quickly.

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 are also Sudoku video games, such as Sudoku XP for PCs, Go! Sudoku for the PSP, "Dr. Sudoku" for the GBA and two sudoku games for the DS: Sudoku Mania and Sudoku Gridmaster. Sudoku puzzles are also featured in Brain Age and Big Brain Academy. In addition, various LCD handhelds exist, using either a number pad and directional keys or a touch screen.

Competitions

• The first world championship was held in Lucca, Italy from 10 to 12 March 2006 [12]; it was won by Jana Tylová, a 31-year-old accountant from the Czech Republic. The competition included numerous variants.^[4].

See also

• List of newspapers featuring Sudoku
• List of Nikoli puzzle types
• List of Sudoku terms and jargon
• Latin square

References

1. ^ Rules and history of Sudoku from Nikoli.
2. ^ Garns, H. "Number Place." Dell Pencil Puzzles & Word Games. No. 16, May p. 6, 1979.
3. ^ Wayne Gould's sudoku.com website.
4. ^ World Sudoku Championship 2006 Instructions Booklet.

External links

• Sudoku at the Open Directory Project – An active listing of Sudoku links.