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, after it was published by Nikoli and given the name Sudoku, meaning single number. It became an international hit in 2005.
Introduction
The name Sudoku means "single digits". The name is a trademark of puzzle publisher Nikoli Co. Ltd. in Japan. In Japanese, the word is pronounced [sɯːdokɯ]; in English, it is usually (AmE) 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 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.
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 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.
The partially filled sub-square determines that 3,5, and 6 must go in the top row. These create a contingency for the far right hatched cell based on the complete row across. It must be a 4.
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.
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" 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 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.[14] 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 puzzleEven 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,[15] 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.
Solution to Hypersudoku puzzle.Many news papers and include the popular Hypersudoku now also playable online. 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[17] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku[18] 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.
Websudoku
No comments:
Post a Comment