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Sudoku can be solved using stochastic (random-based) algorithms. [9] [10] An example of this method is to: Randomly assign numbers to the blank cells in the grid. Calculate the number of errors. "Shuffle" the inserted numbers until the number of mistakes is reduced to zero. A solution to the puzzle is then found.
Mathematical context. The general problem of solving Sudoku puzzles on n2 × n2 grids of n × n blocks is known to be NP-complete. [8] A puzzle can be expressed as a graph coloring problem. [9] The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring. The Sudoku graph has 81 vertices, one vertex for each cell.
Many Sudoku solving algorithms, such as brute force-backtracking and dancing links can solve most 9×9 puzzles efficiently, but combinatorial explosion occurs as n increases, creating practical limits to the properties of Sudokus that can be constructed, analyzed, and solved as n increases.
Taking Sudoku Seriously. Taking Sudoku Seriously: The math behind the world's most popular pencil puzzle is a book on the mathematics of Sudoku. It was written by Jason Rosenhouse and Laura Taalman, and published in 2011 by the Oxford University Press. The Basic Library List Committee of the Mathematical Association of America has suggested its ...
So, Sudoku is in NP (quickly checkable) but does not seem to be in P (quickly solvable). Thousands of other problems seem similarly fast to check but slow to solve. Researchers have shown that many of the problems in NP have the extra property that a fast solution to any one of them could be used to build a quick solution to any other problem ...
Backtracking is an important tool for solving constraint satisfaction problems, [2] such as crosswords, verbal arithmetic, Sudoku, and many other puzzles. It is often the most convenient technique for parsing, [3] for the knapsack problem and other combinatorial optimization problems.
The algorithm is still in NP--perhaps it is very fast for solving all proper 9x9 sudoku puzzles, but from a complexity theoretic point of view, this algorithm is still in the same category as the rest (still in NP, still in superpolynomial time on classical computers).
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