page_title: Mathematics of Sudoku
section_title: Maximum number of givens
hierarchical_section_title: Mathematics of Sudoku / Puzzles / Maximum number of givens
caption_attribution_description: English: The first discovered minimal 9x9 Sudoku puzzle with 40 givens. Original image uploaded by Dobrichev (13 April 2017). This image is the same except the white border is removed, so that it matches 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: The most clues for a minimal Sudoku is believed to be 40, of which only two are known. If any clue is removed from either of these Sudokus, the puzzle would have more than one solution (and thus not be a proper Sudoku). In the work to find these Sudokus, other high-clue puzzles were catalogued, including more than 6,500,000,000 minimal puzzles with 36 clues. About 2600 minimal Sudokus with 39 clues were also found.
  If dropping the requirement for the uniqueness of the solution, 41-clue minimal pseudo-puzzles are known to exist, but they can be completed to more than one solution grid. Removal of any clue increases the number of the completions and from this perspective none of the 41 clues is redundant. With slightly more than half the grid filled with givens (41 of 81 cells), the uniqueness of the solution constraint still dominates over the minimality constraint.
  As for the most clues possible in a Sudoku while still not rendering a unique solution, it is four short of a full grid (77). If two instances of two numbers each are missing and the cells they are to occupy are the corners of an orthogonal rectangle, and exactly two of these cells are within one region, there are two ways the last digits can be added (two solutions).
caption: A minimal Sudoku with 40 clues.[74]
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