page_title: Mathematics of Sudoku
section_title: Automorphic Sudokus
hierarchical_section_title: Mathematics of Sudoku / Puzzles / Automorphic Sudokus
caption_attribution_description: English: The 9x9 Sudoku solution grid with most non-trivial automorphisms. This image is the same as "Sudoku Most Canonical grid.svg" by Dobrichev but has been trimmed to remove the transparent border to match the style of other Sudoku images.
context_page_description: Sudoku puzzles can be studied mathematically to attempt to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of givens possible?" or "In what way can Sudokus exhibit symmetry?" through the use of combinatorics and group theory.
  For the classical Sudoku the number of filled Sudoku grids is 6,670,903,752,021,072,936,960, which reduces to 5,472,730,538 essentially different groups under the validity preserving transformations. There are 26 types of symmetry, but they can only be found in about 0,005% of all filled grids. 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. The largest possible minimal puzzle found so far has 40 clues.
  Similar results are known for variants and smaller grids. No results are known for Sudokus larger than the classical 9x9 grid, although there are estimates which are believed to be fairly accurate.
context_section_description: A Sudoku grid is automorphic if it can be transformed in a way that leads back to the original grid, when that same transformation would not otherwise lead back to the original grid.  One example of a grid which is automorphic would be a grid which can be rotated 180 degrees resulting in a new grid where the new cell values are a permutation of the original grid. Automorphic Sudokus are Sudoku puzzles which solve to an automorphic grid. Two examples of automorphic Sudokus, and an automorphic grid are shown below.
  In the first two examples, notice that if the Sudoku is rotated 180 degrees, and the clues relabeled with the permutation (123456789) -> (987654321), it returns to the same Sudoku. Expressed another way, these Sudokus have the property that every 180-degree rotational pair of clues (a, b) follows the rule (a) + (b) = 10.
  Since these Sudokus are automorphic, so too their solutions grids are automorphic.  Furthermore, every cell which is solved has a symmetrical partner which is solved with the same technique (and the pair would take the form a + b = 10). Notice that in the second example, the Sudoku also exhibits translational (or repetition) symmetry; clues are clustered in groups, with the clues in each group ordered sequentially (i.e., n, n+1, n+2, and n+3).
  The third image is the Most Canonical solution grid. This grid has 648 automorphisms and contributes to all ~6.67×10²¹ solution grids by factor of 1/648 compared to any non-automorphic grid.
  In these examples the automorphisms are easy to identify, but in general automorphism is not always obvious. The table at right shows the number of the essentially different Sudoku solution grids for all existing automorphisms.
caption: The "Most Canonical" solution grid (648 automorphisms)
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