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Sudoku solving techniques

Advanced techniques
Before starting on the advanced techniques, make sure you have looked at the basic techniques first.

2.1 - X-Wing
This solving method considers four interlinked cells. The technique takes two rows and finds four empty cells perpendicular to form a rectangle. The cells must be related to each other using a common value that is held as a candidate for each of the empty cells, there must only ever be two empty cells on each of the two rows that house that certain common value. Either top-left and bottom-right or top-right and bottom-left of this pattern will hold the true candidates, any other empty cells in the columns connecting the X-Wing can have the common value withdrawn from their candidate list.


example A
Because our three selected red cells all have1 as their number we can create an X-wing. The shadowed columns have all their candidate number 1's removed.



2.2 - Swordfish
Very similar to the X-Wing method, the swordfish uses three rows that have up to nine empty cells that house the same candidate number, two empty cells from each row are selected and they must connect with another selected empty cell from a different row through its column. This time we get two rectangle patterns in our puzzle. We can then eliminate the candidate number from any other empty cells along the selected rows or columns excluding our ‘swordfish’ cells. The final true positions for the candidate number will be in a diagonal formation using our ‘swordfish’ cells.


example B
Our red cells all have the same candidate number 1, the shadowed squares are allowed to have the number 1 removed from their candidate lists, this is one of the trickiest concepts to grasp.


2.3 - XY-Wing
An XY-Wing makes good use of combining empty cells that follow a certain pattern. When you have three empty cells along a row and a column you get an L shape .The two outside cells hold a candidate number that is the same as one of the centre cell candidate numbers. The two cells also hold one other matching candidate number each. When you follow the two outside cells along either their column or row you will find another cell that will force the pattern into a square. If this empty cell holds the matching candidate number then it can be removed from that cells candidate list. The XY-Wing technique also has other patterns that work in the same kind of way.


example C
The L shape here is branched out from our cell that holds the 4 and the 5. If either 4 ot 5 is the correct number for that cell we can follow the L-shape to its other points, where the 4 or the 5 from those cells would be removed, leaving an 8. This tells us that the shadowed cell must have candidate number 8 removed.


2.4 - Colouring
For the colouring technique we are only interested in two cells that share a row, column or block and hold an identical candidate number.
This means that one of the cells must hold the true candidate number, for this to work we must colour each of the two cells different colours, one of the coloured cells could also have the same relationship with another cell along another row, column or block, we can then colour the new cell with the opposite colour of the previous cell, so we get a kind of chain forming of opposite colours. When we have two opposite coloured cells in relation with a common cell (forming an L shape) we know that one of those cells must be the true candidate number, This then allows us to remove the candidate number from the new cell.


example D
We got to this point by first finding cell A and B, who hold the same candidate number. From here we found cell C, which links nicely with cell B and then finally cell D which also has candidate 1. Now either D or A because of their colours MUST be a real 1, forcing the shadowed cell to remove its candidate 1.



2.5 - Forcing Chains
To start a chain you must have a candidate link between empty cells, a link in a chain can be part of a row, column or block. One cell must only have two candidates and one candidate must always be the same as the next link in the chain. We take the first link in our chain and systematically go through pretending that one of our candidate numbers is the true one, this will move through each link in the chain whittling down the candidates to one possible true answer (like trial and error). When we reach a dead end, we start over and we do exactly the same thing again with the second candidate number following the path until it reaches another dead end (should be on the same cell as the last guess). Now if the two values that were found in our final cell were matching, we know that our cell has to be that value.

example E
We form our chain by starting with cell A and creating a connection with cell B through one of its candidate numbers, we then continue our chain and find cell C which holds a 3 which links with cell B, we continue on the same root for cell D and E. We then start from Cell A and fill in all the values if 2 was indeed the correct answer, we also in turn fill in all the correct answers for all the cells if number 2 was the correct answer. We find that our shadowed square always returns the same value 2. Meaning that whichever route you take that shadwed cell E will always be a 2.


2.6 - Nishio (Limited Trial and Error)
This is a technique that some Sudoku solvers find conterversial. For each candidate you ask the question: Will placing this number here stop me from completing the other placements of this number? If yes then the candidate can be eliminated.




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