dailysudoku.com

[Johan]

After staring quite some time at the grid, i used this step to do some eliminations on <6>. It's a kind of an xy-chain with an interruption in it.
Starting with <2> in R6C4 I must end with <6> in the upper right corner of Box 6 for having two <6>-pincer cells, that results in elimination of both <6>'s in R6C7 and R4C5.

R6C4=2 => R6C1=8 and now the chain is interrupted, but the <8> in R6C1 forces <2> in R4C1, because of the naked [15] pair, so i can proceed the xy-chain ending with <6> in R4C9.
Looking at the sudopedia site, i didn't find any feedback on this, perhaps it's just a forcing chain?

[TKiel]

Johan,

I'm no expert but I don't believe that is an XY-chain, so it must be some other type of chain.

Do you know what a W-wing is? Because there is one that performs the exact same eliminations that your mystery chains does. Bi-value cells <26> that are not peers connected by a strong link on one of the values <2>. Eliminates all 6's in cells that see both of the bi-value cells.

[Steve R]

Yes it’s a forcing chain. This particular type is called an als chain (or sometimes just an xy chain). In an xy chain proper the nodes are cells with two candidates but, when you think about it, the logic is the same if, instead, a node consists of an almost locked set: a set of n cells in a subhouse with n + 1 candidates.

The chain itself can be written in several ways. Here you could take the als as r367c1 with candidates (1258). Written as a nice chain it becomes:

-6- r6c4 -2- r367c1 -5- r4c1 -2- r4c9 -6-

The chain turns into a nice loop if you add {r4c5, r6c7} at both ends.

Alternatively you could take the als as r37c1 with candidates (158), giving

-6- r6c4 -2- r6c1 -8- r37c1 -5- r4c1 -2- r4c9 -6-

Sudopedia, verbose as ever, calls it an als xy chain

Steve

[Johan]

Tracy,

Because my knowledge of W-wing is limited I must thank you for the useful info you gave me, it took me quite some time to grasp this matter.


Steve,

Thanks for defining this chain
Like Tracy mentioned the W-wing had exactly the same eliminations as the ALS xy-chain, so they must share a common factor, maybe the strong links on <6>?


Johan

[Steve R]

Well, the w- wing and the als chain serve the same purpose here but they are very different in chain terms.

An als chain consists entirely of weak links – more precisely weakly inferential links. These are indicated by a minus sign. So, if a and b are cells or alss

a -2- b

stands for “2 is a candidate for a and b but can be entered in at most one of them.” This arises when a and b share a house.

The w-wing is also a nice chain but it contains just one strong – more precisely strongly inferential – link. Strongly inferential links are written with an equals sign. Thus

a =2= b

stands for “2 is a candidate for a and b and it must be entered into one of them.” A link of this sort typically arises when a and b are of conjugates with respect to 2. Note, however, that conjugacy gives rise to two quite distinct links: a -2- b (weakly inferential) and a =2= b (strongly inferential).

The w-wing involved here, written as a nice chain, would be

-6- r4c9 -2- r5c7 =2= r5c5 -2- r6c4 – 6-

The meaning of the chain is exactly what the w-wing says: one of r5c5 and r5c7 must contain 2 so one of r4c9 and r6c4 must contain 6.

If this seems like gibberish, I suggest you work a little with als chains to begin with. By the way, the crucial characteristic of nice chains is that they are double implication: they are to be read from right to left as well as left to right.

Steve