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Marty R. · Posted: Tue May 09, 2006 12:48 am Post subject: Simple coloring

This was the April 27 "Nightmare." With about 75% of the cells solved, I did a coloring chain on "1", starting from r7c2. As you can see, the chain ended up with two "+" in r7. I didn't think this was possible and I double- and triple-checked the chain to make sure that it was composed only of strong links.

I thought this meant that my solution thus far contained an error and that I would end up with duplicates. But just for laughs, I figured with the signs reversed, there would be two "-" in r7, which meant that r7c3 had to be the cell containing the "1", and that finished off the puzzle with the correct solution.

So I either made an error in the chain, despite triple-checking, or I learned something new. Which was it? Could that be a vaild chain?

keith · Posted: Tue May 09, 2006 1:44 am Post subject: Of Course!

Marty,

Of course! If you have an x-y (any other notation) chain of strong links, and you find that "x" is a contradiction, then "y" must be the solution.

In your example, "+" is a contradiction, "-" must be the cells that have the value.

The reasons this is so powerful:

Chains of strong links (conjugate pairs) are easy to find.

They can lead to solution of the whole chain, not just elimination of a possibility at the ends.

They are the foundation of many other colorfully named techniques.

Best wishes,

Keith

David Bryant · Joined: 29 Jul 2005 Posts: 559 Location: Denver, Colorado

Marty R. · Posted: Tue May 09, 2006 3:58 pm Post subject:

Thanks Keith and David. I was obviously under a bad assumption. Having read so much about how these chains are reversible from true-false to false-true, it didn't seem that an invalid chain could exist. Live and learn!

TKiel · Joined: 22 Feb 2006 Posts: 292 Location: Kalamazoo, MI

This has always been my favorite kind of colouring chain for two reasons: Instead of just an exclusion one makes an assignment and that placement happens to a cell in the chain, not one outside it. Maybe I like it because it doesn't seem to occur that often.

I don't remember exactly how I found out about this possible result of colouring but I don't think it was in any of the websites that explain techniques. Do any of them use a chain like this in their examples?