![]() The puzzle is then completed by assigning an integer between 1 and 9 to each vertex, in such a way that vertices that are joined by an edge do not have the same integer assigned to them.Ī Sudoku solution grid is also a Latin square. 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 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. For n=3 (classical Sudoku), however, this result is of little relevance: algorithms such as Dancing Links can solve puzzles in fractions of a second.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. Horizontally adjacent rows are a band, and vertically adjacent columns are a stack. Other variants include those with irregularly-shaped regions or with additional constraints (hypercube) or different constraint types (Samunamupure).Ī puzzle is a partially completed grid, and the initial values are givens or clues. 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. Analysis has largely focused on enumerating solutions, with results first appearing in 2004. Also studied are computer algorithms to solve Sudokus, and to develop (or search for) new Sudokus. The analysis of Sudoku falls into two main areas: analyzing the properties of (1) completed grids and (2) puzzles.
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