About Sudoku
Sudoku 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.
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.
The name Sudoku means «single digits». The word sudoku derives from the Japanese phrase «Suji wa dokushin ni kagiru», meaning «the numbers must be single», or «the numbers must occur only once».
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.
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.
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.
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», 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 9x9 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 9x9 Latin squares. Various other grid sizes have also been enumerated. 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.
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, and 18 with the givens in rotationally symmetric cells.
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