![]() In our prior work on a Sudoku solver we used a polynomial time approximation to Nishio based on graph matching algorithms, that always makes correct deductions but that, by ignoring the block structure of a Sudoku puzzle, is not always able to solve puzzles that can be solved by Nishio such as the one in Figure 1.įor instance, consider the Sudoku puzzle in Figure 1. Nishio deductions on a 9 × 9 board are within the ability of skilled humans to solve mentally, often without written notes, but it is not obvious how to implement them in a computer simulation of human reasoning. This technique is often called Nishio, after Tetsuya Nishio, although the same word may also apply to unrestricted trial-and-error solution. One of the more complex deduction rules commonly used by human Sudoku puzzlers considers the possible positions still open to a single digit d, and eliminates possible placements whenever they cannot be part of a consistent placement of d in every row, column, and block of the puzzle, regardless of how the other digits are placed. Sudoku puzzles have also been used as a test case for many varied computer problem-solving techniques including constraint programming, reduction to satisfiability, harmony search, metaheuristics, genetic algorithms, particle swarms, belief propagation, projection onto the Birkhoff polytope, Gröbner bases, mixed integer programming, and compressed sensing. Sudoku puzzles may be reduced to instances of exact satisfiability (satisfiability of CNF formulae by an assignment that makes exactly one term in each clause true) with O ( n 2 ) variables and clauses or to instances of exact set cover with O ( n 2 ) sets and elements, again achieving a time bound exponential in n 2. ![]() A coloring algorithm of Lawler, specialized to these graphs (using the fact that any color class in a valid coloring may be represented as a permutation) can solve any Sudoku puzzle in worst case time O ( 2 n 2 ⋅ n ! ⋅ n O ( 1 ) ). ![]() Worst case bounds on the time to solve Sudoku problems may be found by reducing Sudoku to graph list coloring on a graph with n 2 vertices, one for each cell, with a color for each number that may be placed within a cell and with edges connecting pairs of cells that belong to the same row, column, or block. In practice, however, it is not difficult for computers to solve 9 × 9 Sudoku puzzles using backtracking together with simple deductive rules to determine the consequences of each choice and prune the search tree whenever an inconsistency is discovered. This variability allows us to apply the tools of computational complexity to Sudoku, and it has been shown that, when generalized to n × n grids, solving Sudoku is NP-hard. In summary, I would like to be able to selectively turn off the "easier" stratagies just as can be done with the tougher ones.Common variants of Sudoku shrink the puzzle to a 6 × 6 grid with 2 × 3 blocks, or expand it to a 16 × 16 grid with 4 × 4 blocks more generally, Sudoku puzzles may use any a b × a b grid with blocks of size a × b. Because these can't be turned off like the more difficult stratagies, I can't force the Solver around it to see if one of the other stratagies that I normally use, would allow the puzzle to be solved without using the naked double or triple. In some cases however, the Solver finds a hidden double or triple. In most cases, I find that I simply overlooked something silly. When I get stumped on a Sudoku, I import it into your Solver, uncheck stratagies I don't usually look for, and step through it. I personally find that Pointing Pairs, Box/Line Reduction, X-Wing, and Unique Rectangles, are much easier to spot than Hidden Triples. How do you determine what the difficulty level should be for a given stratagy? I have written several times but I must repeat "This is a great site".
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