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PostPosted: Sun Apr 19, 2009 6:50 pm 
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When I solve (vanilla) sudokus requiring unknown advanced techniques, I sometimes do not spot the intended advanced technique(s) for solving the puzzle.
When I get stuck, I resort to XY-Chains, since they normally will allow me to solve the puzzle, unless it requires ultra advanced techniques like XY-X Chain, XYT Chain, Hidden Unique Rectangle, BUG, ALS, Sue de Coq, etc.

Here is how I try to find XY-Chains.

First the definition of an XY-Chain from sudopedia:
  • An XY-Chain is a chain of cells, where all cells in the chain have only 2 candidates, i.e. are bivalue cells.
  • Because the chain is entirely made up by bivalue cells, the link between the 2 candidates in each cell is a strong link, which allows the use of weak links between the cells.
  • The shortest XY-Chain is an XY-Wing with only 3 cells.

When searching for an XY-Chain, I first pick a cell C where it would be nice to eliminate the value X.
For the value X in cell C to be a target for an XY-Chain it is an absolute requirement that cell C sees two bivalue cells, which both have X as a candidate.
If C cannot see two bivalue cells, which both have X as a candidate, its value X cannot be eliminated by an XY-Chain!

Assume two bivalue cells, which both see cell C and have X as a candidate are found.
I mark (colour) one of these two cells as the "head" of the chain. Let's call this cell H, which has the two values X and Y.
The other cell is the "tail" of the chain. Let's call this cell T, which has the two values X and Z. (Y and Z can be identical or different)
When colouring an XY-Chain, I use pink for C, green for H, blue for T and lilac for the other cells in the chain.

Before continuing, let us recall how the XY-Chain must function for X to be eliminated from C.
  1. Assume H=X -> C<>X
  2. Assume H=Y => find a chain of bivalue cells between H and T, which makes T=X -> C<>X

For an XY-Chain the same will be true for the opposite direction, from tail to head, as opposed to an XY-X Chain.
  1. Assume T=X -> C<>X
  2. Assume T=Z => the found chain of bivalue cells between T and H is such that H=X -> C<>X

The tricky part is of course item 2. Sometimes this chain comes together pretty naturally, and sometimes it takes a lot of sweat.

Here an example of a puzzle, which I originally solved using an XY-Chain before spotting the intended advanced technique, which is an XY-Wing, which I assumed all along, but did not spot before after finding an XY-Chain and while looking for a shorter/better XY-Chain.

Image
090000050002107400130508097970000016000000000680000045710902064006301700040000020

The nice thing about this puzzle is that it can be solved with only Naked Singles plus a single advanced technique, for instance an XY-Wing or an XY-Chain.
And there are numerous different XY-Chains that will solve the puzzle, hence it is ideal for practicing spotting XY-Chains.

Here two examples of the XY-Chains that can be constructed to solve this puzzle.
The first example is a "neat chain" and the second example demonstrates how a strong link between two cells in the chain must be used to proceed.

EXAMPLE 1:
XY-Chain on 8 with 6 cells:
(8=6)r5c4 <-> (6=4)r1c4 <-> (4=3)r1c6 <-> (3=9)r6c6 <-> (9=2)r6c7 <-> (2=8)r4c7
(slightly modified Eureka notation)
-> r4c5 <> 8
Image

The chain detailed:
=> Assume r5c4=6 -> r1c4=4
-> r1c6=3
-> r6c6=9
-> r6c7=2
-> r4c7=8
-> r4c5 <> 8
=> Assume r5c4=8 -> r4c5 <> 8

Here the values of all the bivalue cells in the XY-Chain are such, that when a cell takes one of its two possible values, this value is also one of the the two values of the next cell in the chain, which then automatically can be omitted from this next cell, and so on from head to tail and vice versa.
Sometimes it is not possible to construct such a neat chain, and a strong link between two cells in the chain must be used to proceed.

EXAMPLE 2:
XY-Chain on 8 with 7 cells:
(8=6)r5c4 <-> (6=4)r1c4 <-> (4=3)r1c6 <-> (3=9)r6c6 <-> (9=5)r5c6 <-> (8=9)r5c7 <-> (2=8)r4c7
(slightly modified Eureka notation)
-> r4c5 <> 8
Image

The chain detailed:
=> Assume r5c4=6 -> r1c4=4
-> r1c6=3
-> r6c6=9
-> r5c6=5
-> r5c7=9
(There is a strong link on 9 between the cells r5c6 and r5c7, so if r5c6 <> 9, then r5c7 = 9, and vice versa.)
-> r4c7=8
(There is a strong link on 8 between the cells r5c7 and r4c7, so if r5c7 <> 8, then r4c7 = 8, and vice versa.)
-> r4c5 <> 8
=> Assume r5c4=8 -> r4c5 <> 8



The XY-Wing that can be used to solve the puzzle:
Hidden Text:
Image

The XY-Wing viewed as an XY-Chain:
Hidden Text:
Image

An XY-Chain with 4 cells, which solves the puzzle:
Hidden Text:
Image

An XY-Chain, which eliminates the same value from two cells and solves the puzzle:
Hidden Text:
Image

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