4/2/2024 0 Comments Hard sudoku with solution4 Details of enumerating distinct grids (9×9).3.1.5 Sum number place ("Killer Sudoku").3.1.3 Sudoku with additional constraints.2.2.1 Sudoku with additional constraints.2.1.2.5 Number of essentially equivalent grids.2.1.2.3 Fixed points and Burnside's lemma.2.1.2.1 Validity preserving transformations. ![]() 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
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