# THE LOGIC OF SUDOKU PDF

Denis Berthier. The Hidden Logic of Sudoku (second edition) - Conclusion. . action rules (that may be put to practical use for solving puzzles). The Hidden Logic of Sudoku (second edition) - Introduction. .. But, as we shall also see, there are lots of puzzles that need rules of a. is the underlying mathematics and logic behind some solution techniques?). All of In the 9 × 9 sudoku puzzles there are 27 such virtual lines.

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Dell was known as a specialist when it game to puzzles of logic and ability. Dell published Sudoku as “Number. Place” in its Math Puzzles and Logic Problems. For many who are new to Sudoku puzzles, finding a solution can be A Sudoku puzzle is defined as a logic-based, number-placement puzzle. Sudoku is a (sometimes addictive) puzzle presented on a square grid that is logical and mathematical approaches to solving sudoku puzzles.

This example illustrates the number 4 as the unique candidate for the cell marked in red. The example shows that the number 7 can only be inserted in the red cells of the middle row.

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Thus you can remove 7 as a possible candidate from the rest of the row. In the middle and the middle-left blocks, the number 8 must be placed in one of the red cells. This means, we can eliminate 8 from the upper and lower rows in the middle-right column.

Naked Subset The example shows that row number 1 and row number 5 both have a cell in the same column containing only the candidate numbers 4 and 7. These two numbers appear as candidates in all of the other open cells in that column too, but since they are the only two candidates in rows 1 and 5, these two numbers cannot appear anywhere else in the row, thus you can remove them.

In the example, the two candidate pairs circled in red, are the sole candidates. Since 4 and 7 must be placed in either of these two cells, all of the pairs circled in blue, can remove those numbers as candidates.

In this puzzle, this means 1 becomes sole candidate in the second row; 2 becomes sole candidate in row 6; and thus, 6 is sole candidate for row number 4. You can also use this technique if you have more than two candidates.

For example, let us say the pairs circled in red were instead triple candidates of the numbers 1, 4, 7. This would mean those three numbers would have to be placed in either rows 1, 2 or 5.

We could remove these three numbers as candidates in any of the remaining cells in the column. Hidden subset This is similar to Naked subset, but it affects the cells holding the candidates.

## Mini Sudoku

In this example, we see that the numbers 5, 6, 7 can only be placed in cells 5 or 6 in the first column marked in a red circle , and that the number 5 can only be inserted in cell number 8. Since 6 and 7 must be placed in one of the cells with a red circle, it follows that the number 5 has to be placed in cell number 8, and thus we can remove any other candidates for this cell; in this case, 2 and 3.

X-Wing This method can work when you look at cells comprising a rectangle, such as the cells marked in red. In this example, let's say that the red and blue cells all have the number 5 as candidate numbers. Now, imagine if the red cells are the only cells in column 2 and 8 in which you can put 5. In this case you obviously need to put a 5 in two of the red cells, and you also know they cannot both be in the same row. Well, now, this means you can eliminate 5 as the candidate for all the blue cells.

## Introduction to Sudoku

This is because in the top row, either the first or the second red cell must have a 5, and the same can be said about the lower row. Taking a careful look at square b4 we can see that 3, 4, 7 and 8 are already used in the same box, 1 and 6 are used in the same row, and 5 and 9 are used in the same column. Eliminating all the above numbers leaves 2 as the single candidate for square b4.

Eliminating numbers from rows, columns and boxes: There are more complex ways to find numbers by using the process of elimination. In this example the 1 in square c8 implies that either square e7 or square e9 must contain 1. Whichever the case may be, the 1 of column e is in box 8 and it is therefore not possible to have 1 in the centre column of box 2.

So the only square left for 1 in box 2 is square d2. Searching for missing numbers in rows and columns: This method can be particularly useful when rows and columns are close to completion.

Seven of the nine squares contain the numbers 1, 2, 3, 4, 5, 8 and 9, which means that 6 and 7 are missing.

However, 6 cannot be in square h6 because there is already 6 in that column. Therefore the 6 must be in square b6. Analyzing techniques 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.

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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. This leaves square e1 as the only possible place into which 9 can fit in. Scanning in two directions: The same technique can be expanded by using information from perpendicular rows and columns.

## Sudoku rules

In this example, row 1 and row 2 contain 1s, which leaves two empty squares in the bottom of box 3. However, square g4 also contains 1, so no additional 1 is allowed in column g. This means that square i3 is the only place left for 1. Searching for Single Candidates: Often only one number can be in a square because the remaining eight are already used in the relevant row, column and box.

Taking a careful look at square b4 we can see that 3, 4, 7 and 8 are already used in the same box, 1 and 6 are used in the same row, and 5 and 9 are used in the same column.

Eliminating all the above numbers leaves 2 as the single candidate for square b4. Eliminating numbers from rows, columns and boxes: There are more complex ways to find numbers by using the process of elimination. In this example the 1 in square c8 implies that either square e7 or square e9 must contain 1. Whichever the case may be, the 1 of column e is in box 8 and it is therefore not possible to have 1 in the centre column of box 2.

So the only square left for 1 in box 2 is square d2. Searching for missing numbers in rows and columns: This method can be particularly useful when rows and columns are close to completion.

## Sudoku Solver

Seven of the nine squares contain the numbers 1, 2, 3, 4, 5, 8 and 9, which means that 6 and 7 are missing.When you removed the 1s, you now expose a naked single 3 in R3,C9. The grid is also divided into nine 3x3 sub-grids named boxes which are marked box 1 through box 9.

It is all down to permutations. Searching for missing numbers in rows and columns: This method can be particularly useful when rows and columns are close to completion.

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. Soon after, Sudoku-fever swept England.