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In your newspaper or book, you will see a Sudoku puzzle, a grid, usually of 9 rows and 9 columns, giving 81 cells in a 9x9 grid:
Within a grid, cells are grouped into boxes. In a 9x9 grid, they are grouped into 9 boxes, each of 3 rows and 3 columns. Thus there are 9 cells in a box. And there are 9 boxes in a grid.
It can be useful to focus in turn on each horizontal band, which is a set of three boxes adjacent horizontally:
After that, it can be useful to focus in turn on each vertical stack, which is a set of three boxes adjacent vertically:
Between a quarter and a third of the cells contain a given initial value. The rest are empty and your challenge is to discover every unknown value.
This is a game of logic. You can be especially brave by writing your solution in PEN. Sudoku is not like a crossword puzzle, where more than one answer might be correct, at least temporarily. A delight of sudoku is that every proper puzzle has a unique solution, and only one value is correct for each cell.
Your goal is to fill in the grid so every cell in a row has a different symbol. Also every cell in a column must have a different symbol. And every cell in a box must have a different symbol. For a 9x9 grid, the symbols are the numbers from 1 to 9.
Scanning is the best way to begin. You can win a lot of success by scanning, which you do by using these two techniques: cross-hatching and counting.
This is especially useful in the opening game.
Initially, this technique is more manageable if applied to each band in turn and then each stack.
Start with the top band, which is the top set of 3 boxes. Do you see any digit that occurs in two of the 3x3 boxes? If you do, notice that the digit is shown both in:
That will give you one to three candidate cells. If there is only one candidate cell, then you have discovered a singleton: enter the digit into the empty cell. For example
The only possible value that can go at the (6th column of 2nd row) is a 4.
If there is more than one candidate cell, check if the digit of interest appears in one or two of the vertical columns: this is the 'Cross-hatching' part. If that leaves you with only one candidate cell, you have discovered a singleton: enter the digit into the empty cell.
Continue for any other digits that appear twice in the top band, to discover all the top-band singletons.
Do the same thing for the middle band.
And finally for the bottom band.
Now you are ready to do the same thing by scanning the stacks. Begin with the left-most stack, which is the set of 3 boxes on the left of the grid. Do you see any digit that occurs in two of those 3x3 boxes? If you do, notice that the digit is shown in
Continue in a similar way to the way you scanned for the top band.
Do the same thing for the middle stack.
And finally for the right-most stack.
Of course, by the time you have done all this, you have entered values in a lot of cells. So repeat the cross-hatchin until you can find nothing more.
For each of the most-filled rows, columns, and boxes, in turn identify which digits (of 1-9) are missing.
See more at middle-game counting.
In the basic rules, we saw that the straight-forward application of counting can be useful in the end-game of the whole grid or of a particular column, row, or box.
It can also be useful in letting you identify (say) a pair of values that must be in two cells. Although you are not able to fill in those values (either could go in either cell), this information can reduce the number of possible values for cells of an intersecting box, row, or column.
A subset of the above middle-game counting, whereby one determines certain values that cannot go in a cell, though their exact location elsewhere (in an associated box, row, or column) is still being worked out. See Teach Yourself Sudoku.
See Teach Yourself Sudoku.
This highly advanced technique is used when cells at the corners of any-size rectangle have pairs of candidates, each with a common candidate number. While you don't know where the candidate number goes, you do know that it must be in diagonally opposite corners.
This clever technique lets you eliminate any other instances of this candidate from the rest of the non-corner sides of the rectangle.
See Teach Yourself Sudoku for an example that happens to use the numeral 6.
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