Completed puzzles are always a type of Latin square with an additional constraint on the contents of individual regions. For example, the same single integer may not appear twice in the same row, column or in any of the nine 3×3 subregions of the 9x9 playing board.
Sudoku puzzle (black) solved (in red)
French newspapers featured variations of the puzzles in the 19th century, and the puzzle has appeared since 1979 in puzzle books under the name
The results in the following text refer to classic Sudoku, disregarding jigsaw, hyper and others.
A completed Sudoku grid is a special type of Latin square with the additional property of no repeated values in any of the 9 blocks of contiguous 3×3 cells. The relationship between the two theories is now completely known, after it was proven that a first-order formula that does not mention blocks (also called boxes or regions) is valid for Sudoku if and only if it is valid for Latin Squares (this property is trivially true for the axioms and it can be extended to any formula).
The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960 (sequence A107739 in OEIS), or approximately 7021666999999999999♠6.67×1021. This is roughly 6994120000000000000♠1.2×10−6 times the number of 9×9 Latin squares. Various other grid sizes have also been enumerated. The number of essentially different solutions, when symmetries such as rotation, reflection, permutation and relabelling are taken into account, was shown to be just 5,472,730,538 (sequence A109741 in OEIS).