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.
2.3 - XY-Wing
|| 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.
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.
2.4 - Colouring
|| 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.
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.
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.
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.