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| | | |---|---| | [Math Explorers' Club](http://www.math.cornell.edu/~mec) | [Cornell University Department of Mathematics](http://www.math.cornell.edu/) | ### Contents [Home](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Home.html) [1\. Introduction to Sudoku](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Intro.html) [2\. Solving Strategy](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Solve.html) [3\. Counting Solutions](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Count.html) [4\. Solution Symmetries](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Symmetry.html) [5\. The 4×4 Case](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Four.html) [6\. Some More Interesting Facts](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/More.html) [7\. References](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/References.html) # The Math Behind Sudoku ## Solving Strategy When one hears that no math is required to solve Sudoku, what is really meant is that no arithmetic is required. The puzzle does not depend on the fact that the nine placeholders used are the digits from 1 to 9. Any nine symbols would serve just as well to create and solve the puzzles. In fact, mathematical thinking in the form of logical deduction is very useful in solving Sudokus. The most basic strategy to solve a Sudoku puzzle is to first write down, in each empty cell, all possible entries that will not contradict the One Rule with respect to the given cells. If a cell ends up having only one possible entry, it is a "forced" entry that you should fill in. Another way to proceed is to pick a number and a row, column, or block. Note all the cells in the row, column, or block in which the number can be placed without violating the One Rule. If the digit can only be placed in one cell in the neighborhood, you should fill that cell in. Once you've done this, the chosen number can be eliminated from being a possibility for any other cell in the neighborhood. These two strategies are usually not enough to completely fill in a Sudoku grid. You often need more complicated analysis methods to make progress, and sometimes you need to make a guess and proceed, backtracking if the guess results in a conflict. One more complicated strategy is to look at pairs or triples of cells within a row, column, or block. You might find that a pair of cells has only two options of entries, but don't know which goes where. What you can still gain from this observation is that those pair of numbers cannot occur anywhere else in the neighborhood. This will decrease the number of possibilities for the other cells in the neighborhood and help you get closer to a solution. Similarly, a triple of cells having only three possibilities of entries between them will eliminate these entries in all other cells in a neighborhood of this triple. If no entries are forced, try to pick a box with the fewest number of possibilities and pick one of them. Continue playing, using the strategies above and any other ones you discover. If you reach a contradiction (a repeated digit in a row, column, or block), you should retrace your steps and undo what you've done until you have no contradiction. **Exercise:** Here is a Sudoku puzzle for you to try:  You can find many more puzzles on the internet, in a whole range of difficulty levels. | | | | |---|---|---| | [Previous](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Intro.html) | [Top](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Solve.html) | [Next](https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Count.html) |

## Solving Strategy When one hears that no math is required to solve Sudoku, what is really meant is that no arithmetic is required. The puzzle does not depend on the fact that the nine placeholders used are the digits from 1 to 9. Any nine symbols would serve just as well to create and solve the puzzles. In fact, mathematical thinking in the form of logical deduction is very useful in solving Sudokus. The most basic strategy to solve a Sudoku puzzle is to first write down, in each empty cell, all possible entries that will not contradict the One Rule with respect to the given cells. If a cell ends up having only one possible entry, it is a "forced" entry that you should fill in. Another way to proceed is to pick a number and a row, column, or block. Note all the cells in the row, column, or block in which the number can be placed without violating the One Rule. If the digit can only be placed in one cell in the neighborhood, you should fill that cell in. Once you've done this, the chosen number can be eliminated from being a possibility for any other cell in the neighborhood. These two strategies are usually not enough to completely fill in a Sudoku grid. You often need more complicated analysis methods to make progress, and sometimes you need to make a guess and proceed, backtracking if the guess results in a conflict. One more complicated strategy is to look at pairs or triples of cells within a row, column, or block. You might find that a pair of cells has only two options of entries, but don't know which goes where. What you can still gain from this observation is that those pair of numbers cannot occur anywhere else in the neighborhood. This will decrease the number of possibilities for the other cells in the neighborhood and help you get closer to a solution. Similarly, a triple of cells having only three possibilities of entries between them will eliminate these entries in all other cells in a neighborhood of this triple. If no entries are forced, try to pick a box with the fewest number of possibilities and pick one of them. Continue playing, using the strategies above and any other ones you discover. If you reach a contradiction (a repeated digit in a row, column, or block), you should retrace your steps and undo what you've done until you have no contradiction. **Exercise:** Here is a Sudoku puzzle for you to try:  You can find many more puzzles on the internet, in a whole range of difficulty levels.