As Sudoku puzzle levels get harder you will find the simple scanning methods described above are not enough and more sophisticated solving techniques must be used. Hard puzzles require deeper logic analysis which is done with the aid of pencilmarks.
Sudoku pencilmarking is a systematic process writing small numbers inside the squares to denote which ones may fit in. After pencilmarking the puzzle, the solver must analyze the results, identify special number combinations and deduce which numbers should be placed where.
Here are some ways of using analyzing techniques:
1. Eliminating Squares using Naked Pairs in a Box
Eliminating Squares using Naked Pairs in a Box
In this example, the pencilmarks (in red) show that 4 and 9 can only be in Square c7 and Square c8. We don’t know which is which, but we do know that both squares are occupied.
In addition, Square a6 excludes 6 from being in the left column of Box 7. As a result the 6 can only be in Square b9.
Such cases where the same pair can only be placed in two boxes is called Disjoint Subsets, and if the Disjoint Subsets are easy to see then they are called Naked Pairs.
2. Eliminating Squares using Naked Pairs in Rows and Columns
Eliminating Squares using Naked Pairs in Rows and Columns
The previous solving technique is useful for deducing a number within a row or column instead of a box.
In this puzzle we see that 2 and 7 can only be in Square d9 and Square f9. Again we don’t know which is which, but we do know that both squares are occupied. The numbers which remain to be placed in Row 9 are 1, 6 and 8.
However, 6 can’t be placed in Square a9 or in Square i9, so the only possible place is Square c9.
3. Eliminating Squares using Hidden Pairs in Rows and Columns
Eliminating Squares using Hidden Pairs in Rows and Columns
Disjoint Subsets are not always obvious to see at first sight, in which case they are called Hidden Pairs.
If we take a very close look at the pencilmarks in Row 7 we can see that both 1 and 4 can only be in Box f7 and Box g7. This means that 1 and 4 are a Hidden Pair, and that Square f7 and Square g7 cannot contain any other number.
Using the scanning technique we see that 7 can only be in Square d7.
4. Eliminating Squares using X-Wing
Eliminating Squares using X-Wing
The X-Wing technique is used in rare situations which occur in some extremely difficult puzzles.
Scanning Column a we see that 4 can only be in Square a2 or Square a9. Similarly, 4 can only be in Square i2 or Square i9. Because of the X-Wing pattern where boxes are in the same row (or column), a new logic constraint occurs: it is obvious that in Row 2 the 4 can only be either in Square a2 or in Square i2, and it cannot be in any other square.
Therefore 4 is excluded from Square c2, and Square c2 must be 2.
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