What strategies do you all use when solving Sudoku puzzles algorithmically? What are your thoughts on this problem?

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Enumerate the 81 cells of the Sudoku board with numbers from 81. Then the graph associated with a Sudoku puzzle consists of 81 vertices (one for each cell of the board), together with edges as follows: two vertices are connected by an edge if the cells that they correspond to are in the same column, row or 3x3 box. We have thus represented the Sudoku grid as a graph. With 81 vertices and several hundred edges it would be a big graph if one wanted to draw it, so let us just think about it without attempting to produce a graphical representation. What about the numbers in the cells of the Sudoku puzzle, how do we represent those? Assign each number from 1 to 9 a color. Now color the vertices corresponding to cell that contains a given number in the color of the number.

It is now easy to see that completing a Sudoku puzzle without vioating the Sudoku condition is equivalent to coloring the vertices of the corresponding graph while ensuring that no two adjacent vertices have the same color.

It is now easy to see that completing a Sudoku puzzle without vioating the Sudoku condition is equivalent to coloring the vertices of the corresponding graph while ensuring that no two adjacent vertices have the same color.