page_title: Mathematics of Sudoku
section_title: Sudoku with additional constraints
hierarchical_section_title: Mathematics of Sudoku / Puzzles / Minimum number of givens / Sudoku with additional constraints
caption_attribution_description: English: Ocean's Sudoku Disjoint Groups with 11 clues, created by Ocean, August 31, 2007. This image is the same as "Oceans SudokuDG11 Puzzle.svg" uploaded by Ocean but the border has been trimmed to match the style of other Sudoku images. At the time of publishing of the original image, this puzzle (plus a dozen others found and published at the same time) were probably the first known Sudoku-DG puzzles with only 11 clues.
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: Additional constraints (here, on 3×3 Sudokus) lead to a smaller minimum number of clues.
  Disjoint Groups: some 12-clue puzzles have been demonstrated by Glenn Fowler. Later, 11-clue puzzles were also found. It is not known if this is the best possible.
  Hypercube: various 8-clue puzzles have been demonstrated by Guenter Stertenbrink. This is the best possible.
  Magic Sudoku: a 7-clue example has been provided by Guenter Stertenbrink. It is not known if this is the best possible.
  Anti-Knight Sudoku: a 9-clue example has been provided by Reddit user u/wand125. It is suspected that this is the best possible.
  Sudoku X: a list of 7193 12-clue puzzles has been collected by Ruud van der Werf. It is not known if this is the best possible.
  NRC Sudoku: an 11-clue example has been provided by Andries Brouwer. It is not known if this is the best possible.
  2-Quasi-Magic Sudoku: a 4-clue example has been provided by Tony Forbes. It is suspected that this is the best possible.
caption: Disjoint Groups: 11 clues
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