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Sudoku (数独 sūdoku, digit-single) (/suːˈdoʊkuː/, /-ˈdɒk-/, /sə-/, originally called Number Place) is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 subgrids that compose the grid (also called "boxes", "blocks", or "regions") contains all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a single solution.
Completed games are always a type of Latin square with an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice in the same row, column, or any of the nine 3×3 subregions of the 9x9 playing board.
Mathematics of Sudoku
The general problem of solving Sudoku puzzles on n2×n2 grids of n×n blocks is known to be NP-complete. Many computer algorithms, such as backtracking and dancing links can solve most 9×9 puzzles efficiently, but combinatorial explosion occurs as n increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved as n increases. A Sudoku puzzle can be expressed as a graph coloring problem. The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring.
The fewest number of clues possible for a proper Sudoku is 17 (proven January 2012, and confirmed September 2013). Over 49,000 Sudokus with 17 clues have been found, many by Japanese enthusiasts. Sudokus with 18 clues and rotational symmetry have been found, and there is at least one Sudoku that has 18 clues, exhibits two-way diagonal symmetry and is automorphic. The maximum number of clues that can be provided while still not rendering a unique solution is four short of a full grid (77); if two instances of two numbers each are missing from cells that occupy the corners of an orthogonal rectangle, and exactly two of these cells are within one region, the numbers can be assigned two ways. Since this applies to Latin squares in general, most variants of Sudoku have the same maximum.
The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960 (sequence A107739 in the OEIS), or around 6.67×1021. This is roughly 1.2×10−6 times the number of 9×9 Latin squares. Various other grid sizes have also been enumerated—see the main article for details. The number of essentially different solutions, when symmetries such as rotation, reflection, permutation, and relabelling are taken into account, was shown to be just 5,472,730,538 (sequence A109741 in the OEIS).
Unlike the number of complete Sudoku grids, the number of minimal 9×9 Sudoku puzzles is not precisely known. (A minimal puzzle is one in which no clue can be deleted without losing uniqueness of the solution.) However, statistical techniques combined with a puzzle generator show that about (with 0.065% relative error) 3.10 × 1037 minimal puzzles and 2.55 × 1025 nonessentially equivalent minimal puzzles exist.