#Sudoku puzzles 9x9 generator

In this case, two distinct vertices labeled by ( x, y) and ( x′, y′) are joined by an edge if and only if: The vertices are labeled with ordered pairs ( x, y), where x and y are integers between 1 and 9. The Sudoku graph has 81 vertices, one vertex for each cell. The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring. Ī puzzle can be expressed as a graph coloring problem. The general problem of solving Sudoku puzzles on n 2× n 2 grids of n× n blocks is known to be NP-complete. Solving Sudokus from the viewpoint of a player has been explored in Denis Berthier's book 'The Hidden Logic of Sudoku' (2007) which considers strategies such as 'hidden xy-chains'. See Glossary of Sudoku for other terminology. A minimal puzzle is a proper puzzle from which no clue can be removed without introducing additional solutions. A puzzle is a partially completed grid, and the initial values are givens or clues. A band is a part of the grid that encapsulates 3 rows and 3 boxes, and a stack is a part of the grid that encapsulates 3 columns and 3 boxes. Other variants include those with irregularly-shaped regions or with additional constraints ( hypercube). A rectangular Sudoku uses rectangular regions of row-column dimension R× C. Unless noted, discussion in this article assumes classic Sudoku, i.e. There are many Sudoku variants, partially characterized by size ( N), and the shape of their regions. same arrangement where all instances of one digit is switched with another digit). 2.55 × 10 25 minimal puzzles that are not pseudo-equivalent (i.e.

#Sudoku puzzles 9x9 generator

However, statistical techniques combined with a generator ( 'Unbiased Statistics of a CSP – A Controlled-Bias Generator'), show that there are approximately (with 0.065% relative error): The number of minimal Sudokus (Sudokus in which no clue can be deleted without losing uniqueness of the solution) is not precisely known. 8×8(2×4) Sudoku: The fewest clues is 14.6×6(2×3) Sudoku: The fewest clues is 8.A Sudoku with 24 clues, dihedral symmetry (a 90° rotational symmetry, which also includes a symmetry on both orthogonal axis, 180° rotational symmetry, and diagonal symmetry) is known to exist, but it is not known if this number of clues is minimal for this class of Sudoku. The fewest clues in a Sudoku with two-way diagonal symmetry (a 180° rotational symmetry) is believed to be 18, and in at least one case such a Sudoku also exhibits automorphism. A paper by Gary McGuire, Bastian Tugemann, and Gilles Civario, released on 1 January 2012, explains how it was proved through an exhaustive computer search based on hitting set enumeration that the minimum number of clues in any proper Sudoku is 17. Many Sudokus have been found with 17 clues, although finding them is not a trivial task. Ordinary Sudoku A Sudoku with 17 cluesĪ Sudoku with 17 clues and diagonal symmetry Ī Sudoku with 18 clues and orthogonal symmetry Īn automorphic Sudoku with 24 clues and complete geometric symmetry Ī Sudoku with 19 clues and two-way orthogonal symmetry This section discusses the minimum number of givens for proper puzzles. Different minimal Sudokus can have a different number of clues. A minimal Sudoku is a Sudoku from which no clue can be removed leaving it a proper Sudoku. Ordinary Sudokus ( proper puzzles) have a unique solution. No exact results are known for Sudokus larger than the classical 9×9 grid, although there are estimates which are believed to be fairly accurate. Similar results are known for variants and smaller grids. The largest minimal puzzle found so far has 40 clues in the 81 cells. There is a solvable puzzle with at most 21 clues for every solved grid. An ordinary puzzle with a unique solution must have at least 17 clues. There are 26 possible types of symmetry, but they can only be found in about 0.005% of all filled grids. įor classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 ( 6.671 ×10 21), which reduces to 5,472,730,538 essentially different solutions under the validity preserving transformations. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004. The analysis of Sudoku is generally divided between analyzing the properties of unsolved puzzles (such as the minimum possible number of given clues) and analyzing the properties of solved puzzles. Mathematics can be used to study Sudoku puzzles to answer questions such as 'How many filled Sudoku grids are there?', 'What is the minimal number of clues in a valid puzzle?' and 'In what ways can Sudoku grids be symmetric?' through the use of combinatorics and group theory. For solving and generating algorithms, see Sudoku solving algorithms.Ī 24-clue automorphic Sudoku with translational symmetry This article is about the mathematical analysis of Sudoku puzzles.