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Earl · Joined: 30 May 2007 Posts: 677 Location: Victoria, KS

After a few moves, x-wing, xyz-wing, I was stymied. So I guessed.
Any logical solution?

Earl

storm_norm · Joined: 18 Oct 2007 Posts: 1741

nataraj · Posted: Mon Dec 03, 2007 7:59 pm Post subject:

8 can be removed from r4c7 and from r8c9 by multi-coloring aka x-cycles
I'll write down the chains (one has to visualize the weak links in rows 8 and 9 in the diagram). The two chains overlap, oly the end is different.

-r2c7=r1c8-r1c2=r9c2-r9c4=r8c4-r8c9=r4c9-
-r4c9=r4c7-r2c7=r1c8-r1c2=r9c2-r9c4=r8c4-

nataraj · Posted: Mon Dec 03, 2007 8:54 pm Post subject:

A better picture than the one I posted a while ago is probably this one:

Some of the lines are solid, those connections are "strong" links (max 2 "8"s in a row or column or box). Some others are drawn as "weak" links (they share a house, but there might be other roommates. maybe not. or the roommates have gone for a beer ...). It helps to remember that every strong link can be viewed as a weak link but not vice versa (not every weakling can be viewed as strong!). E.g., in box 1 there are only two "8"s, but i painted the connection as "weak". Why? Because we are looking for a certain pattern here ...

When you look at the picture there is a certain regularity to the pattern of strong and weak links:

strong-weak-strong-weak-strong-...

But at one point (r4c9) the pattern is broken. There are TWO strong links coming out of that cell.

This is what is called a discontinuity. And in this case (both are strong) it means that r4c9=8.

But wait ....

didn't I say just a minute ago you could view all strong links as weak links? Let's try that...

amazing. Now the discontinuity has moved to another cell: r4c7. AND ...

now there are two WEAK links coming out of the "odd" cell. That means r4c7 CANNOT be 8 (and that is totally, I mean absolutely, really consistent with the conclusion we drew earlier, namely that r4c9=8. right?)

In the same way, we can color the link between r4c9 and r8c9 "weak". That's possible, too. Just don't color all of them "weak" because remember we basically need alternating sequences of strong and weak links EXCEPT FOR ONE SINGLE discontinuity.

Again, we have a cell with two weak links coming out of it, which means r8c9<>8. And that again is totally, absolutely consistent ...
(Personally I never look for those double strong discontinuities. The double weak are much more frequent. And they do the job just as well)

And that's all folks. Nataraj's world of solid and dotted lines aka The Big X-Cycle Show.

(Guess I got carried away)

Asellus · Posted: Tue Dec 04, 2007 1:12 am Post subject:

nataraj's single-digit AIC, or X-Chain, is a nice example of the technique.

I have found another quite interesting route to the solution from storm_norm's grid in a single step involving <4>s. Notice the otherwise useless {459} XYZ-Wing at R5C25|R6C6. Also notice the otherwise useless Finned X-Wing in R35.

If we put these two useless things together, they become useful. Notice that if the Fin (R5C3) is true, then the only XYZ-Wing Cell remaining for <4> is R6C6, which then must be <4>. Then, a glance at Box 2 shows that if R6C6 is <4>, then R8C4 must be <4> and R8C5 cannot be <4>.

However, if the Fin is false, then the X-Wing is true and R8C5 cannot be <4>. Either way, the <4> in R8C5 is toast and the puzzle is solved.