dailysudoku.com

[nataraj]

In today's nightmare http://www.sudocue.net/daily.php, after removing 3 from r4c7 by this short coloring chain: -r4c3=r1c3-r1c8=r3c7- I came to this position, which has a remarkable "almost" naked pair in box 2:

[storm_norm]

AIC with ALS???

AIC with groups perhaps.

[Asellus]

Hi nataraj,

I'll take a stab at answering your questions.

First... yes, r3c46 is an "almost naked pair," which is just a form of ALS. There is a strong link between the single digit and the pair since they can't both be false. This looks like (2)r3c4=({15})r3C46 and can be exploited in chains.

You are exploiting this ALS in a couple of nice ways, but neither of them is the "xz" method of exploiting two related ALSs.

Regarding the potential UR, with the {15} locked pair in r3 and the strongly linked <1> in r6c46, we know that neither of r6c46 can be <5> and vice versa. In other words, the DP creates a weak link between the {15} locked pair in r3 and the two <5>s in r6 since they can't both be true. This looks like ({15})r3c46-(5)r6c46 and, again, can be exploited in chains. (Note that the 2-cell reference on the right side of the link refers to the grouped candidates. Also, I think this might be Type 4 though I'm not sure.)

It turns out that DPs (of all sorts) often create useful strong or weak links such as this that are powerful in chains. I've only recently begun to appreciate and to try to exploit this.

One more detail: those two <5>s in r6c46 are part of a 3-cell {1589} ALS and either one of the <5>s is true or the {189} locked set is true. They can't both be false so are strongly linked. This looks like (5)r6c46=({189})r4c6|r6c46. (The "|" character means "and" and is used to unite cell or candidate groups in Eureka notation.)

I believe we now have enough to write out your four AICs in a comprehensible manner:


First, for that <8> in r2c6:

For the "false <2>"...
(2)r3c4=({15})r3c46-({15}=8)r2c6

[Note that last strong link. Think of it as a "bivalue" of a {15} pair and an <8>.]

For the "true <2>"...
(2)r3c4-(2=8)r4c4-(8)r46c6=(8)r2c6


Now, for that <2> in r1c4:

For the "true <2>", it is just a weak link...
(2)r3c4-(2)r1c4

And saving the best for last, for the "false <2>"...
(2)r3c4=({15})r3c46-(5)r6c46=({189})r4c6|r6c46-(8=2)r4c4-(2)r1c4

(That's one nice lookin' chain!)

I hope that all makes sense. By the way, it might be a good idea to post questions such as this in the "Techniques" forum.

[nataraj]

Asellus,

thank you very much, indeed, for your explanation! I've already digested the first part, but need a break (and will try to do some visuals) before getting to the second part, putting it all together:

[Asellus]

nataraj,

You're most welcome! I thought it might be what you were looking for.

About those ALS links, I should make a clarification. In all the instances in which you used them above, you used them as strong links for inference purposes so that's how I presented them. However, they are actually conjugate links. It's also true that both parts can't be true. That means that they are also weakly linked. E.g., (2)r3c4-({15})r3c46 is also valid.

[nataraj]

I think I got the AICs together now. Hardest part was to get it through my head that those candidates 1,5 in r2c6 and also the 1,8,9 in box 5 could be used as a group. I had hever tried that (only looked at groups of whole cells) before. I am sure this new way of thinking will take a lot more practice, but now that I was able to paint the chains it has become much easier.

So now this is how we arrived at r2c6=8: two strong links at the discontinuity.


and this is how to eliminate 2 from r1c4 (two weak links):



... and I still find it hard to think of strong link as (a or b) rather than (not a)=>b, and of weak link as not(a and b) rather than (a=>not b)

[Asellus]

Watch out Jean Arp! You've got competition!!

Very nice and colorful illustrations... something like that goes through my head when constructing AICs. It is also nice how you show the complete loops. For the record, the customary Eureka notation would also be written as complete loops:

(8)r2c6=(8)r46c6-(2=8)r4c4-(2)r3c4=({15})r3c46-({15}=8)r2c6; r2c6=8

and

(2)r1c4-(2)r3c4=({15})r3c46-(5)r6c46=({189})r4c6|r6c46-(8=2)r4c4-(2)r1c4; r1c4<>2

Usually, the "2 weak link" discontinuity is used to eliminate candidates. However, the converse is just as valid as you've shown: using a "2 strong link" discontinuity to place candidates.

[nataraj]

[Asellus]

Inclusive is good. I just assumed "or" was XOR. We're on the same page.

I guess I was conflating Hans Arp and Joan Miro. Easy to do, I suppose. (Me, I was in Schiele heaven when I visited Vienna.)