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Sudoku solving algorithms. A typical Sudoku puzzle. A standard Sudoku contains 81 cells, in a 9×9 grid, and has 9 boxes, each box being the intersection of the first, middle, or last 3 rows, and the first, middle, or last 3 columns. Each cell may contain a number from one to nine, and each number can only occur once in each row, column, and box.
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.
The general problem of solving Sudoku puzzles on n 2 ×n 2 grids of n×n blocks is known to be NP-complete. [29] 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 ...
Main articles: Sudoku, Mathematics of Sudoku, Sudoku solving algorithms. The problem in Sudoku is to assign numbers (or digits, values, symbols) to cells (or squares) in a grid so as to satisfy certain constraints. In the standard 9×9 Sudoku variant, there are four kinds of constraints:
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 problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [ 3]: ND22, ND23. Vehicle routing problem.
Optimal solutions for the Rubik's Cube. Optimal solutions for the Rubik's Cube are solutions that are the shortest in some sense. There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second is to count the number of outer-layer twists, called "face turns".
Dancing Links. In computer science, dancing links ( DLX) is a technique for adding and deleting a node from a circular doubly linked list. 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 ...