Encyclopedia
, also known as
Number Place or
Nanpure, is a logic-based placement
puzzle. The aim of the puzzle is to enter the digits 1 through 9 in each cell of a 9×9 grid made up of 3×3 subgrids so that each row, column, and region contains exactly one instance of each digit. A set of clues, or "givens", constrain the puzzle such that there is only one way to correctly fill in the remainder.
Completed sudoku puzzles are a type of Latin square, with the additional constraint on the contents of individual regions.
Leonhard Euler is sometimes cited as the source of the puzzle based on his work with Latin squares.
The modern puzzle
Sudoku was invented in
Indianapolis in 1979 by Howard Garns. Garns' puzzles appeared in Dell Magazines, which published them under the title "
Number Place". Sudoku became popular in
Japan in 1986, when puzzle publisher
Nikoli discovered the game in older Dell publications. The puzzles became an international hit in 2005.
Introduction
The name "
Sudoku" is the Japanese abbreviation of a longer phrase, , meaning "the digits must remain single". It is a trademark of puzzle publisher
Nikoli Co. Ltd. in
Japan. In
Japanese, the word is pronounced []; in
English, it is usually spoken with an Anglicised pronunciation, [] [] or [] [] 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.
Scanning
Scanning is performed at the outset and throughout the solution. Scans need 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.
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". 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 contain the same two candidate numerals , or if precisely three cells within a scope contain the same three candidate numerals , 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 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.
In the "what-if" approach , 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
Computers are, of course, capable of
exhaustively searching a
Sudoku puzzle for its solution. However, in general this sort of approach will only be achievable in reasonable time if the puzzle is relatively small in size , and/or the puzzle has a relatively high proportion of the cells already filled in correctly. Indeed, from a Computer Science perspective, Sudoku has been proven to belong in the class of
NP-Complete problems, implying that we cannot hope to find a polynomially bounded algorithm for all possible puzzles.
Given the above, 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. Note, however, that regardless of whether pruning methods are used or not, it is still the case that this sort of methodology might still take an unreasonable amount of time for many larger Sudoku problems. One alternative to this method is to employ stochastic-based optimisation methods instead such as Simulated Annealing, which has recently been shown to perform well on various instances of orders 3, 4 and 5.
Finally, another alternative uses finite domain constraint programming. A constraint program specifies the constraints of the puzzle ; a finite-domain solver applies the constraints successively to narrow down the solution space until a solution is found. Backtracking or stochastic optimisation techniques 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. Dell
Number Place Challenger 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
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 , 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 is a British [i] newspaper [i], currently a tabloid [i], first publish ...
, 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 Sudoku. In
The Times is a national newspaper [i] published daily in the United Kingdom [i] since 1785, and unde ...
and
The Sydney Morning Herald is a major Australian [i] broadsheet [i] newspaper [i] published ...
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 is the name of two North American weekly magazine [i]s about television [i] programming, on ...
, 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 was founded in 1855 [i], and is one of only two remaining daily British [i] ...
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.
A new variant without any given numbers for start comes up this summer 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 from one through nine for each region. Therefore, each cell's border line inside every region is notched in the meaning of < – or vice versa.
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 : 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 . 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
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 9x9 magic square with 3x3 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 9x9 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 3x3 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.
The modern Sudoku was designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor, and first published in 1979. Sadly, he died in 1989 before getting a chance to see his creation as a worldwide phenomenon. 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 , which can be translated as "the numbers must be single" or "the numbers must occur only once." The puzzle was named by , 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" . 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. Knowing that British newspapers have a long history of publishing
crosswords and other puzzles, he promoted
Sudoku to
The Times is a national newspaper [i] published daily in the United Kingdom [i] since 1785, and unde ...
in Britain, which launched it on 12 November 2004 .
It was rapidly introduced by
The Daily Telegraph, the
Daily Mail and
The Independent is a British [i] compact [i] newspaper [i] published by Tony O'Reilly [i] ...
.
By April and May 2005 the puzzle became a national phenomenon and was introduced to several other national British newspapers including
The Guardian is a British [i] newspaper [i] owned by the Guardian Media Group [i]. ...
,
The Sun , and
The Daily Mirror is a British [i] left-wing tabloid [i] daily newspaper [i]. ...
.
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 . 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 is the BBC [i]'s weekly television [i] and radio [i] programme listings magazine [i]. ...
featured a weekly Super Sudoku.
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 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.
Sudoku is now also a very popular part of the games available for mobile phones. Gameloft, Guardian Unlimited and Playcom are some of the distributors.
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..
See also
References
External link
– An active listing of
Sudoku links.
