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Backtracking is a depth-first search (in contrast to a breadth-first search), because it will completely explore one branch to a possible solution before moving to another branch. Although it has been established that approximately 5.96 x 10 26 final grids exist, a brute force algorithm can be a practical method to solve Sudoku puzzles.
Traditionally, as with regular sudoku puzzles, the grid layout is symmetrical around a diagonal, horizontal or vertical axis, or a quarter or half turn about the centre. This is a matter of aesthetics, though, rather than obligatory: many Japanese puzzle-makers will make small deviations from perfect symmetry for the sake of improving the puzzle.
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.
The general problem of solving Sudoku puzzles on n 2 ×n 2 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.
He has also written puzzles for events including the World Sudoku Championship, U.S. Puzzle Championship, the MIT Mystery Hunt, Gen Con, and the Microsoft Puzzle Picnic. [4] In early 2012, Snyder founded his publishing company Grandmaster Puzzles. On April 9, 2012, he began selling his first title from the newly formed company, The Art of Sudoku.
The book also includes discussions on the nature of mathematics and the use of computers in mathematics. [4] After an introductory chapter on Sudoku and its deductive puzzle-solving techniques [1] (also touching on Euler tours and Hamiltonian cycles), [5] the book has eight more chapters and an epilogue.
It is particularly useful for efficiently implementing backtracking algorithms, such as Knuth's Algorithm X for the exact cover problem. [1] Algorithm X is a recursive , nondeterministic , depth-first , backtracking algorithm that finds all solutions to the exact cover 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.
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