How to spot XY-Chains

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How to spot XY-Chains

Postby coolowl » Fri Dec 07, 2012 4:44 pm

First, pardon the simplicity of the information I seek but I tried running a search on this but got nowhere. This forum's search engine was unsuccessful as XY is too short, chain produced like a million results and xy-chain produced nothing. :(

My question is how do you search for XY-Chains? Whatever I'm doing is not working. I'm jumping all over the place, wasting time and getting unproductive results. There must be a better way. I would love to hear ideas or techniques of how people are really using XY-chains. Are you looking for a common "pivot cell" first, two bi-value cells in adjacent rows, randomly picking a cell? What?

BTW, I've looked through several how-to sites and found some nice ones that explain how to solve the XY-chain, but that's not what I'm looking for. I know how the chain works. I know how to move from one cell to another. I'm just looking for ways to spot them quicker rather than just randomly selecting a cell and seeing where it leads. There must be some logic to using this technique other than, "Oh, here's a bi-value cell. Let's try this one." You know what I mean?

Thanks for any help you can provide.
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Re: How to spot XY-Chains

Postby JasonLion » Fri Dec 07, 2012 7:01 pm

Start by scanning the entire board looking for all of the possible links in an XY-Chain. Each "link" will consist of a pair of bi-value cells that share a house and a digit. Much of the time there won't be enough links available to make anything out of them, but at a certain stage of the game bi-value cells will often become common and XY-Chains become useful. You should quickly become able to tell at a glance if exploring XY-Chains is going to be a waste of time or not.

Assuming you find enough links to provide some hope, look around for possible connections between links, cells that participate in two or more links. Unless there are a huge number of possible links, you will quickly narrow things down to only a few connected links that might profitably participate in a chain. Then scan around for possible valid chains. Again, there aren't usually all that many possibilities, since they must start and end on the same digit, and there aren't usually all that many potential chains to look at. Having found a valid chain, check to see if it causes any eliminations. This global check for connected links is usually quick and gives you a quick sense of what potential there is for chains in the board positions. Basically you are trying to "see" the net of all possible chains. With practice this becomes a very valuable technique, though it can be quite tedious when you are just starting out.

Only very rarely are there so many candidate chains that it takes a significant amount of time to check through all of them. When that happens, it is sometimes worth breaking things down by the digit that might be eliminated, and use that to narrow down the candidate list to something manageable. Using more complex chain types will give you more potential chains to explore. The more possible chains there are, the more value there is in exploring the chains to see if they will make any eliminations, but the more work you need to do to scan them for eliminations.

Advanced players sometimes start with a pencil mark that they want to eliminate, often picked through intuition, and then check to see if there is a chain that can eliminate it. That sense of where the elimination might be is very valuable, and only develops with lots and lots of practice. Other times they scan through the net of potential chains, watching for ones that look "promising". Again, this sense of promising only comes with lots and lots of practice.
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Re: How to spot XY-Chains

Postby Marty R. » Fri Dec 07, 2012 7:28 pm

I've been in similar discussions in the past. An XY-Chain is a neat-looking solution. Unfortunately, at least for me, finding one has too much of a trial-and-error feel to it. Thus I find it more satisfying to find a pattern-based solution where you know what eliminations can be made without having to go on a fishing expedition. That said, I use XY-Chains because too many of the puzzles that I see require a chain for the obligatory one-step solution. I say obligatory because everyone else posts one-steppers so I try to keep up with the Joneses.
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Re: How to spot XY-Chains

Postby David P Bird » Sat Dec 08, 2012 10:19 am

Equivalence Marking uses tick marks to show which candidates must be true or false together to reveal contradictions.

7....6.411...547...4.1.....23....1...1..4..5...5....94.....3.8...891...752.4....9 Vanhegan Extreme for Dec 7 2012.

Code: Select all
 | 7     5     23    | 238   2389  6     | 2389  4     1     |
 | 1     8'9"  2369' | 238   5     4     | 7     236   238   |
 | 3"68" 4     236   | 1     23789 2789  | 23589 236   2358  |
 | 2     3     4     | 568   689   589   | 1     7     68    |
 | 689'  1     67"9" | 23678 4     278   | 2368  5     2368  |
 | 68    7'8"  5     | 23678 23678 1     | 2368  9     4     |
 | 4'9"  7"9'  1     | 2567  267   3     | 2456  8     256   |
 | 3'4"  6     8     | 9     1     25    | 2345  23    7     |
 | 5     2     3"7'  | 4     678   78    | 36    1     9     |
Here the tick marks were made to represent the alternate solutions for column 2. Where the candidates are either bilocal in a house or part of a bivalue in a cell the marking is extended (in this example the marking has been restricted to stack 1).

This results in contradictions in r3c1 and r5c3 where the double tick option would require these cells to hold two truths. Hence all the single tick candidates are true, which collapses the puzzle.

To notate such deductions as AICs, start with a weak link between two contradictory candidates and work back though the marks alternating the weak and strong inferences until an elimination can be made in one of the starting cells:

(8=9)r2c2 - (9)r2c3 = (9-7)r5c3 = (7)r6c2 => r6c2 <> 8

(9)r7c2 = (9-8)r2c2 = (8-3)r3c1 = (3)r8c1 – (3=7)r9c3 => r7c2 <> 7

This method can be extended to make other types of eliminations: Further Information
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Re: How to spot XY-Chains

Postby Cubbie » Sun Dec 09, 2012 2:54 am

To simplify this question, since an xy-wing is actually the shortest form of an xy-chain.
Similar to a swordfish is an extension of an x-wing.
Why not first ask yourself "How to spot an xy-wing".
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