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very useful "almost" naked pair

 
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Sat Dec 08, 2007 5:12 pm    Post subject: very useful "almost" naked pair Reply with quote

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:

Code:

+--------------------------+--------------------------+--------------------------+
| 68      356     358      | 257     2456    47       | 1       23      9        |
| 2       156     9        | 3       568     158      | 57      4       567      |
| 4       1356    7        | 125     9       15       | 23      8       56       |
+--------------------------+--------------------------+--------------------------+
| 5       2346    2348     | 28      7       89       | 49      369     1        |
| 9       237     1        | 4       25      6        | 357     357     8        |
| 68      467     48       | 158     3       1589     | 4579    5679    2        |
+--------------------------+--------------------------+--------------------------+
| 37      259     25       | 6       1       37       | 8       259     4        |
| 17      8       45       | 79      45      2        | 6       179     3        |
| 137     2459    6        | 5789    458     347      | 2579    12579   57       |
+--------------------------+--------------------------+--------------------------+


If we take out the "2" in r3c4, then
a) r3c46 becomes a true naked pair and r2c6=8
b) we have a potential DP together with r6c46, which means r4c4=2 (UR type 3 I believe) and thus r1c4<>2

If, on the other hand, r3c4=2, then
a) r4c4=8, r46c6<>8 and r2c6=8
b) r1c4 cannot be 2.

So, no matter which way we look at it, we can set r2c6=8 and remove 2 from r1c4.

My questions to the experts, please:
- is this (r3c46) really a "almost naked pair", and/or what does the term usually mean?
- does this pattern or parts of the pattern have a name? It seems that only a few cells are involved and they all are somehow linked together. But I could not find a simple ALS-xz explanation for the eliminations made. There must be more to this "almost" situation than I can grasp at the moment ...


____

P.S. in the end (after some more coloring and an xyz wing), I had to use an xy-chain to finish the puzzle. A short one (97-73-34-49) and it removed 9 from cell(s) r9c4
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Sun Dec 09, 2007 2:14 am    Post subject: Reply with quote

AIC with ALS???

AIC with groups perhaps.
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sun Dec 09, 2007 2:22 am    Post subject: Reply with quote

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.
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Sun Dec 09, 2007 8:10 am    Post subject: Reply with quote

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


The interpretation of those group implications in terms of strong/weak links and (almost) locked sets was exactly what I was looking for.

Thanks!
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sun Dec 09, 2007 9:58 am    Post subject: Reply with quote

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.
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Sun Dec 09, 2007 11:26 am    Post subject: Reply with quote

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)
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sun Dec 09, 2007 11:17 pm    Post subject: Reply with quote

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

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.

Quote:
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

In principle, the "node" of an AIC can be anything for which a proper (pair of) link(s) can be constructed. So, don't limit yourself!

I read somewhere that an XY Wing can be used as an AIC node, so had been hoping to do so some day. Well, yesterday (8Dec07), I used an AIC with an XY Wing node to help solve the Daily Nightmare. There was a weak link to an "extra digit" in the potential XY Wing (so it was strongly linked to the XY Wing). The Wing in turned rendered some other candidates false (weak links) and off you go on your merry chain!

Quote:
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)

Of course, "(not a)=>b"and "a=>(not b)" are perfectly fine definitions of strong and weak links, respectively. However, I personally find the following definitions easier in practice (using your notating approach):
strong: not[(not a) and (not b)]
weak: not(a and b)

The first can be stated as "they cannot both be false" and the second as "they cannot both be true". The first is "strong" because something has to be true. The second is "weak" because maybe neither is true. Or at least that's how I think of it. (These work at least for bidirectional links. I understand that there are also unidirectional links. But... never mind!) Anyway, it seems that what matters is what is easiest for each person to remember and apply correctly.

I definitely do NOT think of a strong link as "a or b". That is a conjugate link, which is both strong and weak. There is much confusion about this. That arises, I believe, because many sudoku solvers remain in the house- and/or cell-constrained world of links. For links constrained in this way, the only possible type of strong link is the conjugate link. So, they learn it that way and start to think that is how it is. However, if and when one breaks free of the house/cell constraints, one needs to learn the more general case and let go of the special case. This is another of those life changing moments in sudoku evolution.
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Mon Dec 10, 2007 7:16 am    Post subject: Reply with quote

Asellus wrote:
Watch out Jean Arp! You've got competition!!


Laughing I'm flattered. The inspiration must have come from the prints by Joan Miro and Paul Klee in my office (no Hans Arp. Limited wallspace)...

Quote:

I definitely do NOT think of a strong link as "a or b". That is a conjugate link, which is both strong and weak. There is much confusion about this. That arises, I believe, because many sudoku solvers remain in the house- and/or cell-constrained world of links. For links constrained in this way, the only possible type of strong link is the conjugate link.


This seems to be of some major importance and I think I get the point. It is just a small matter, but on the chance of missing something I'd rather tidy up a small detail now than run around with some sort of half-knowledge, so please bear with me one more time:

When I use "or" in any implication, it is always in the sense of inclusive or, the "v" in (a v b), not in the sense of exclusive or, like (a|b).
I don't know how to produce the other (and, not) operators on my keyboard, much less sure how they will be displayed on screens all over the world, so I stick to the words. To be absoutely sure, what I am talking about is this:
Code:
a v b
T T T
T T F
F T T
F F F


which is exactly equivalent to (not a => b) and to not( (not a) and (not b) ).

But of course that is just elementary logic, so I assume the reason for making the point ("I definitely do NOT think of a strong link as 'a or b' ") is to avoid ambiguity:

I agree completely that one should not look at the strong link as either / or, and the statement "cannot both be false" avoids that trap very elegantly, and the ambiguous "one or the other" should be avoided.
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Mon Dec 10, 2007 11:00 am    Post subject: Reply with quote

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.)
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