In particular, an N× N square where N is prime can only be tiled with irregular N-ominoes. It seems clear (already from enumeration arguments), that not all Sudokus can be generated this way.Ī Sudoku whose regions are not (necessarily) square or rectangular is known as a Jigsaw Sudoku. Namely, one has to take subgroups and quotient groups into account: There are significantly fewer Sudoku grids than Latin squares because Sudoku imposes the additional regional constraint.Īs in the case of Latin squares the (addition- or) multiplication tables ( Cayley tables) of finite groups can be used to construct Sudokus and related tables of numbers. 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 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) or different constraint types ( Samunamupure). 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. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004. analyzing the properties of completed puzzles.analyzing the properties of completed grids.The analysis of Sudoku falls into two main areas: 2.1.2.2 Fixed points and Burnside's lemma.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. A puzzle with a unique solution must have at least 17 clues, and there is a solvable puzzle with at most 21 clues for every solved grid. There are 26 types of symmetry, but they can only be found in about 0.005% of all filled grids. The main results are that for the classical Sudoku the number of filled grids is 6,670,903,752,021,072,936,960 ( 6.67 ×10 21), which reduces to 5,472,730,538 essentially different groups under the validity preserving transformations. Sudoku puzzles can be studied mathematically 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. A 24-clue automorphic Sudoku with translational symmetry

Elimination math calculator