The advanced Sudoku Solutions include: X-Wing, XY-Wing, Swordfish patterns, and the Colors technique.



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These Sudoku solutions techniques are for the serious Sudoku addict!

These are very involved tactics and require extensive knowledge of Sudoku puzzle strategy.,

they are slightly different from strategies such as elimination, CRME, lone rangers, etc, in that they

do not follow the standard sub-grid, row, column recognition patterns involved with the afore-

mentioned solution types.

X-Wing

To solve Sudoku puzzles using this process, one must have recognition of number relationships

in the grid. To begin with the name X- wing refers to the top right corner and bottom left corner, or the bottom

left corner and top right corner, which form an X, hence x-wing.

THEORY:

The theory goes something like this:

(For one, we know that for any unique sudoku solution, the numbers 1-9 can only appear once in any

given row, column, or sub-grid.) So for sudoku solutions with this method proceed as follows:

If the number, say y, appears only twice in any given row, then we know it CAN only appear in

one of those two rows.

Further if y is also restricted to two columns (and no more than two columns) , and since y can

only appear once within each of the two rows, no column can have more than one y, and y will

appear only once in each of the columns contained within the rows, and any other candidates

in those columns can be eliminated.

(The converse of this theory also holds, that is interchange the words row and column above).

Perhaps an example will suffice. Consider the following Sudoku puzzle:

X-WING THEORY :

x

4

3

9

8

6

2

5

6

6

x

x

4

2

5

x

x

x

2

x

x

x

6

1

6

9

4

9

6

6

x

x

4

6

7

6

3

x

x

6

x

8

x

x

x

4

1

6

2

x

9

6

6

3

8

2

x

5

6

6

x

6

6

x

6

6

x

4

x

x

x

5

5

3

4

8

9

6

7

1

6



In this example we have highlighted all the possible values for our candidate (the number 6). Looking at the diagram we can observe the following:

  1. Only rows 1 and 9 meet the x-wing criteria ( that is 6's appear twice within the rows and the cells also share the same columns).

  2. The columns shared are 6 and 9, thus any other 6's in those columns can safely be eliminated. (Those highlighted in red)

Swordfish

Similar to an x-wing pattern, the swordfish theory proceeds as follows:.

Given a general puzzle with three rows that has candidate y, in each of the three rows: then y must be

restricted to the same three columns within those rows and:

  1. Candidate y must be assigned once in each row
  2. No included column (within the rows) can contain more than one y.
  3. Then y MUST be assigned exactly once (and only once ) in each of the three columns within these

three rows. (Therefore any other y's within those columns can be eliminated.)


Finally, as it was true for the x-wing, the converse is also true ( that is interchange the words, rows and columns) ,

and the theory holds. Again, here is the graphic example:

SWORDFISH:

1


5

5


8

5

3

5




9

6

1





4

5

8

1

5

6


9

5

5

4


3

5

5

6


2

5


6


5

1





8


5



7


6

5

5

3


4


1




1

7


6

3

5

1

5

3

6

5



2



In this example we have highlighted all the possible values for our candidate (the number 5). Looking at the diagram we can observe the following:


  1. Three rows (three, five & seven) have candidate 5 in no more than three cells (only two cells each in this example-highlighted in blue), and these cells all share the same three columns (three, four & seven). A "Swordfish" pattern is established.

  2. Other candidate 5's in these three columns (highlighted yellow) can be excluded safely.

    Note: As shown in this example, there does not have to be exactly 3 cells in each row (or column), there may be less. However, there must not be more that three candidates in the pattern defining rows.

XY-Wing

Extends the x-wing theory to include three cells as follows:

Given three cells:

  1. All cells must share two candidates.
  2. Within they share three candidates in the form ( xy, yz, xz).
  3. One cell ( the y 'root' with candidate xy), shares a group with the other two cells ( y 'branches'

with the other two candidates yz, xz).


Then any other cell which shares a group with both branch cells can exclude the z element that is common

to both branch cells.

Proof: If a root cell sharing a group with both branch cells has member 'z', then neither branch can be

assigned 'z'. Consequently one branch is assigned x and the other y, leaving the root without a

valid member.

Note: If all 3 cells in an xy wing share the same candidates (namely x,y,z), then this would reduce to a

simple triplet or naked triple.

Of course it may be easier to visualize, refer to the diagram below:

XY-WING THEORY:

457

27

6

2345

25 {B}

2345

1

9

8

59

1

29

258

2589

7

3

6

4

49

8

3

14

6

149

2

5

7

2

9

7

18

4

18

6

3

5

3

5

4

9

7

6

8

2

1

1

6

8

25

3

25

4

7

9

78

4

5

2378

28{A}

238

9

1

6

789

27

29

6

1

58{C}

57

4

3

6

3

1

457

59

459

57

8

2


In the diagram we have labeled cell A (root cell) containing {2,5 ,8}.

Also we have the two branch cells as B which contains subset {2, 5} and C which contains subset {5,8}.

Therefore by the xy theory, we can safely eliminate 5 from those cells within the group contained by A,B, and C. ( This cell is highlighted in red ).

Colors Technique

An interesting little technique, basically used to try to narrow candidates only in two cells within a given

group ( say sub-grid, row, or column). the two cells would have a conjugate, or opposing relationship, that is,

when one is true , the other is false. The idea of the colors technique in essence then is to assign colors to

these variables , or states.

For any given sudoku there may be any number of these 'conjugate pairs' present at any given time.

Some may even link together forming a chain of alternate true-false cell states, and these chains may expose

candidates which can then be excluded safely.

Note: Whenever two cells within a group ( sub-grid, row or column), have the same color, this would indicate

the color must be the 'false' color, since this is impossible, or an illegal state.


Again we provide a visual example to clarify the discussion.I

COLORS TECHNIQUE



1

4

6

9

2

8

3

7

5

5

9

9

6

1

7

8

4

2

8

7

2

4

3

5

9

1

6

7

2

1

3

5

9

6

8

4

9

6

8

2

7

4

5

9

1

4

9

5

1

8

6


9

9

2

5


8

9

1

4

6


9

8

4

7

6

2

1

5

9

6

1

9

5

4

3


9

8





In this diagram we choose to start with row2 column2, and assign it color green , it's conjugates at row2 column3 and row6 column2 are then colored blue.


We proceed around the grid in a like manner.

With this method we see that the blue color has more than one member in a sub-grid, therefore the blue

nine's must be false and we can now safely add the green colored nines.


BRUTE FORCE ALGORITHMS:

There are approximately 67,000, 000, 000, 000, 000, 000, 000( YES, that is 6.7 x 10 to the 21st power)

Sudoku puzzle solutions (or game combinations) possible.

Brute-Force algorithms are basically computer programs that will solve Sudoku puzzles.

A good program might be a practical way to solve Sudoku puzzles, (so long as the puzzle is a valid one,

that is one of the 6.7 x 10^ 21 grids).

Basically it is a numbers crunching game,

A brute force algorithm visits the empty cells in some order, filling in digits sequentially from the

available choices, or backtracking (removing failed choices) when stymied. For our purposes

assume a algorithm order of left to right, top to bottom. (The algorithm could, however,

visit the empty cells in any order)

Briefly, a brute force program(or a person doing it manually) would solve a puzzle by placing the

digit "1" in the first cell and checking if it is allowed to be there. If there are no violations

(checking row, column, and box constraints) then the algorithm advances to the next cell, and places

a "1" in that cell. When checking for violations, it is discovered that the "1" is not allowed, so the value is

advanced to a "2". If a cell is discovered where none of the 9 digits is allowed, then the algorithm leaves

that cell blank and moves back to the previous cell. The value in that cell is then incremented by one.

The algorithm is repeated until the allowed value in the 81st cell is discovered. The construction of 81

numbers is parsed to form the 9 x 9 solution matrix.

Most Sudoku puzzles will be solved in just a few seconds with this method, but there are exceptions.

SUMMARY ADVANCED SUDOKU SOLUTION TECHNIQUES:

These are the main Advanced Sudoku Solution techniques.

Although we have only mentioned the four main ones, there are many many more ;

however we do not believe these techniques are effective as the basic techniques,

including simple Elimination, CRME, Twins, Triplets and Brute force.

All of those techniques should give you enough muscle to solve a majority of Sudoku puzzles

with ease. To review those techniques simply click here.





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