## Hodoku XY-Chains

As we explore Hodoku Chains and Loops (tech-chains.php) examples, here I contrast the premise starting, chain following school of Stephens/Hodoku, with the Sysudoku find-em-all technique of XY-rails. Also, a mathematically sensible order is suggested for remote pairs, as a Hodoku web of them is unraveled.

The XY-chain has a built-in alternate inference pattern. The slinks are within the bv cells, and any link between bv numbers will serve as an alternating wink. Not only is it easy to build, but the XY-chain advertizes its low hanging fruit with an ample field of bv.

If you don’t know what a bv is, it’s a bi-value cell, and you’re in the wrong place in the right blog. Scroll back and read the posts from the beginning. You’ll actually directed to reading them out of chronological order because, along the way, I discovered a very easy way to exploit the XY-chain and gather in every piece of that fruit. It’s the XY-railroad.

You may not have to follow the link, though, because I’m going to show the railroad for Hodoku’s second XY-chain example. The first was the XY-wing.

In the batting cage, you would score a double for the 3 removals, but not credit for a triple for the 9 removal as well. You get the extra base by simply extending the chain for two more bv, and detecting the subchains ending in 9.

In this case it doesn’t matter, because the 3 removals also take care of that illogical 9, but extending the XY-chain calls attention to the railroad.

Also in this case, the railroad is not very impressive, because of the confines of the batting cage. But in the real puzzle collections reviewed here, there are some doozies. In this example, you can see on the rails how the 136 bv group is isolated from the XY-chain group, but is solved along with it. That may be an artifact of the batting cage technology.

I’ll not bother to contrast the XY-rail as a human solving aid to the Hodoku chain notations that are now officially beneath our notice.

Despite his demonstrated skill as a computer programmer, Bernhard shows little mathematical insight in the organization of Hodoku Techniques. He overdefines the XY-chain to require the ending slinks, with matching terminals, what I call their toxic sets. He also says that an XY-chain is not an XY-chain if it has no victim. This even as he freely uses them without this requirement within AIC. I know you don’t believe he said that, but check it out. I’ll be right here when you get back.

The remote pair is one of the easiest advanced structures to recognize, because of the constellations of identical bv.

Bernhard also betrays his math instruction when he defines the XY-chain as a less restricted remote pair. It’s the other way around. A remote pair is a very restricted special case of XY-chain, with conjugate pairs at the ends of chains of length 4, 6, 8, . . . Hodoko places them as “any chain at least four cells long”.

Here is the Hodoku second remote pair example, translated to Sysudoku graphics. It shows a green one of length 4, a black one of length 4, a blue one of length 6, and a red one of length 8.

The chain ends are conjugate pairs. Conjugate pairs and connecting paths are easy to follow. The pairs work as if they, and the victim, are in the same unit.

Next week, we examine Hodoku X-chain views and examples, with the welcome aid of the Sysudoku X-panel.