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The Logical Solution to Sudoku
Since we can't have more than one number in a column, row, or region then we can use the numbers that are visible--the givens--to help us determine where the missing numbers should go. For columns and rows you can mentally (or physically) draw a line vertically or horizontally where the numbers are to help you see what spaces have been eliminated for a given number. For example, Figure 3 shows 3 red lines have been drawn over the column and rows that contain sevens. When we do that it exposes the fact that there is only one position left in the upper left region that could contain the 7 that should go in that region. When the Sudoku puzzle is solved correctly, there will have to be a seven in every region. We can see that there is only one grid space in the upper left region where that seven can be.
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Figure 3
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Let's review the logic. Since we know that there can only be one of each number in a column, row, and region, then we know that there is only one place for the seven to go in the upper left region. This is because the sevens that have been given to us when we start the puzzle help us to eliminate all the other possible open grid spaces in the upper left region.
There are often situations when trying to solve a Sudoku game in which you can use this method only to find out there is more than one option within a grid. For example, look at figure 4. The red lines again are drawn over the column and rows that contain a seven. This time, they reveal that there are two possible places that the seven can be in within the middle left region (highlighted in yellow).

Figure 4
However, notice that if you first use logic to figure out where the seven goes in the upper left region, then you will be able to figure out where the seven goes in the middle left region using the same logic (see Figure 5). When the seven is added to the upper left region, it removes one of the grids as a possible location for the seven because, once again, there can only be one of any number within a column, row, and region.

Figure 5
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