Mathematics of Sudoku

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. 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. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004. For classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 (6.671e21), which reduces to 5,472,730,538 essentially different solutions under the validity preserving transformations. There are 26 possible types of symmetry, but they can only be found in about 0.005% of all filled grids. An ordinary puzzle with a unique solution must have at least 17 clues. There is a solvable puzzle with at most 21 clues for every solved grid. The largest minimal puzzle found so far has 40 clues in the 81 cells. Similar results are known for variants and smaller grids. 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. Ordinary Sudokus (proper puzzles) have a unique solution. A minimal Sudoku is a Sudoku from which no clue can be removed leaving it a proper Sudoku. Different minimal Sudokus can have a different number of clues. This section discusses the minimum number of givens for proper puzzles. Many Sudokus have been found with 17 clues, although finding them is not a trivial task. 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 that the minimum number of clues in any proper Sudoku is 17. 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.