Strong Corner Marker for X-Wing AIC Deduction

Advanced methods and approaches for solving Sudoku puzzles

Postby daj95376 » Sun Oct 12, 2008 4:12 pm

aran wrote:Never could understand why notation which seeks to be succinct uses twice as many characters as necessary to specify a cell (r1c1 v a1).
Can you ?

This question was asked recently in the DailySudoku forum. One reply was a question as well.

How do you specify boxes:?:

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?p=13523&sid=14cfcfef0060882024cf38ebaeee7e86#13523
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Postby DonM » Sun Oct 12, 2008 4:43 pm

daj95376 wrote:
DonM wrote:But, unless I'm missing something ...

Myth Jellies' technique is involved to put it mildly -- especially since I never understood his first assertion.

Note the strong corner for 5's in r7c9, r1c9, and r7c6. Because of that strong corner, we can say that either the corner (5)r7c9 is true or the x-wing (5)r17/c69 is true.

What X-Wing ... what or for conjugate truths?


Subbing for Myth is a risky business, but as I see it:

With the typical X-wing the eliminations occur because one of the numbers in each leg of the X-wing is going to be true, we don't know which and it doesn't really matter. In 'strong corner', the premise is that if we were to eliminate the 'corner 5' (ie. (5)r7c9), since the opposing corner isn't present, we would be exposing (ie. we would know which numbers are true) what is really the pure X-wing: ie. both (5)r1c9 & (5)r7c6 are true or in other words, the X-wing is true. Thus, if that were the case, (5)r1c5, (5)r2c6 could be eliminated and thus they serve either individually or together as the weak links.
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Postby ronk » Sun Oct 12, 2008 5:00 pm

I agree with daj95376; an x-wing is not part of the deduction. All the cells of r1 and c6 could contain digit 5 candidates, and the AAIC would still hold.
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Postby DonM » Sun Oct 12, 2008 5:29 pm

ronk wrote:I agree with daj95376; an x-wing is not part of the deduction. All the cells of r1 and c6 could contain digit 5 candidates, and the AAIC would still hold.


If one thinks in term of what a core x-wing is, I think one can call it part of the deduction if one prefers that concept. Obviously, seeing it simply as what happens when you remove the corner digit from two adjoining conjugate pairs works also.
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Postby ronk » Sun Oct 12, 2008 5:49 pm

DonM wrote:
ronk wrote:I agree with daj95376; an x-wing is not part of the deduction. All the cells of r1 and c6 could contain digit 5 candidates, and the AAIC would still hold.

If one thinks in term of what a core x-wing is, I think one can call it part of the deduction if one prefers that concept.

Is that supposed to be a persuasive counter-argument? Indeed, is that even an argument?
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Postby DonM » Sun Oct 12, 2008 7:18 pm

ronk wrote:
DonM wrote:
ronk wrote:I agree with daj95376; an x-wing is not part of the deduction. All the cells of r1 and c6 could contain digit 5 candidates, and the AAIC would still hold.

If one thinks in term of what a core x-wing is, I think one can call it part of the deduction if one prefers that concept.

Is that supposed to be a persuasive counter-argument? Indeed, is that even an argument?


That sounds strangely familiar; have you been talking to my wife?:D
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Postby daj95376 » Sun Oct 12, 2008 9:29 pm

DonM wrote:In 'strong corner', the premise is that if we were to eliminate the 'corner 5' (ie. (5)r7c9), since the opposing corner isn't present, we would be exposing (ie. we would know which numbers are true) what is really the pure X-wing: ie. both (5)r1c9 & (5)r7c6 are true or in other words, the X-wing is true. Thus, if that were the case, (5)r1c5, (5)r2c6 could be eliminated and thus they serve either individually or together as the weak links.

All semantics aside, it sounds like a 'strong corner' boils down to either blue is true or else green is true for this PM. This pattern occurs all of the time and is nothing special by itself. In fact, my favorite variant is to also have a green cell in the box with the blue cell. I always force the green cells true and see if a contradiction appears quickly. Is that what's happening here?

Bottom Line: For Myth Jellies' PM, I reduced the effort to a simple forcing net based on a strong link in (5) -- (5)r1c9=r7c9 -- and a string link in (9). It was not a good PM to demonstrate the suitability of his technique.

BTW: I'm a big fan of Myth Jellies, and don't want this to appear as a personal attack on him or his contributions to Sudoku:!:
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Postby aran » Mon Oct 13, 2008 12:13 am

DonM
You have a point (and a point that has been made before), but I'm afraid it's too late to go back now- r#c# is pretty much the standard here and on Eureka. (Digressing: When it comes to compact, logical languages in general, I've often thought that we all would have done well to stick with Latin.)


Alea jacta est...or...castigat ridendo mores ?
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Postby aran » Mon Oct 13, 2008 12:08 pm

Simpler to think of it like this ?
Middle corner false =>ends true.
Ends true => cross-hatch with them.
Cross-hatch = most basic technique of all
=> no new name required....
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Postby DonM » Mon Oct 13, 2008 1:12 pm

When you read Myth's explanation, it seems that part of the potential is: 'the extra weak links in the opposite cornered box can often combine to form a useful strong link in that box.' Overall, the logic of the components of this pattern certainly doesn't appear all that complicated and it can be described in different ways, but having solved the puzzle where it came from, I can say that I wouldn't have thought of using this combination the way Myth did so I'll be looking more closely for it in the future.

Still, maybe Myth will happen by to give his take on all this.
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Postby aran » Mon Oct 13, 2008 1:26 pm

DonM
When you read Myth's explanation, it seems that part of the potential is: 'the extra weak links in the opposite cornered box can often combine to form a useful strong link in that box.'

That just means cross-hatching...
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Postby eleven » Fri Oct 17, 2008 11:12 am

After looking for it for some time now, i can say that this pattern is definitely rare and we have still to wait, until a puzzle is found, which can be cracked best with it.

All what i found up to now, i already posted here: http://forum.enjoysudoku.com/viewtopic.php?p=62403#p62403

So - though it is rather quickly checked, if a grid has it, i will stop doing it now. But maybe one of those eloquent Eureka people can prove me wrong with good samples:)
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Postby Allan Barker » Fri Oct 17, 2008 7:46 pm

Myth,

Congratulations, the strong corners are both simple and elegant. I often see similar things in set logic but lack the knowledge and skill to know what to do with them outside of sets. I call them strong triplets (3-way) because of the 'exit path'. These occur quite frequently, I think we just don't look for them. I have seen a few cases where they are the only reasonable elimination possible.

Here are a couple more from the same grid, both of which eliminate candidate 3r7c7. There is a shorter chain that eliminates the same candidate. These are just examples. I have noted the corner with the letter 'A'. The AIC parts are not quite the same and here there is no X-wing.

PS. Why don't you make a "weak corner" where the exit path is a strong bilocation set? I think these are potentially even more deadly.

In example 1, the bifurcation caused by the strong corner comes together in a 3 candidate set 2C8, but that is OK.

Code: Select all
  +-----------------------------------------------------------+
  | 7     6     1     | 9     (25)  4     | 8     35(2) 23(5) |
  | 9     2     3     | 178   157   1568  | 16    15    4     |
  | 5     4     8     | 12    3     126   | 1269  7     1269  |
  +-----------------------------------------------------------+
  | 8     3     4     | 6     12    7     | 5     9     12    |
  | 16    5     9     | 128   4     128   |126(3) 1(23) 7     |

  | 16    7     2     | 5     9     3     | 4     8     16    |
  +-----------------------------------------------------------+
  | 23    8     7     | 123   6     125   |12-3(9) 4    1(59) |
  | 4     9     6     | 1237  1257  125   | 123   13(25)8     |
  | 23    1     5     | 4     8     9     | 7     6     23    |
  +-----------------------------------------------------------+

set
5C9: 5r7c9A======================5r1c9
       |  A                        |
5B9: 5r7c9A=5r8c8                  |
       |      |                    |
1N5:   |      |           2r1c5==5r1c5
       |      |             |
2C8:   |    2r8c8==2r5c8==2r1c8
       |             |
3R5:   |           3r5c8=============3r5c7
       |                               |
9R7: 9r7c9=============================|==9r7c7
                                       |   |
                                        \ /
                                         X=3r7c7
= strong link
| weak link
A label for candidate in two strong links (strong corner)


Here is another from the same grid that eliminates 3r7c7. This is not exactly the same because the bifurcation caused by the strong corner comes together in a 3 candidate set 2C8, but that is OK.

Example 2. I have no idea how to express this as AICs but it but weak set 3r7 is always occupied by 3r7c1 or 3r7c4, thus elimining any other candidates such as 3r7c7.

Code: Select all
  +-----------------------------------------------------------+
  | 7     6     1     | 9     25    4     | 8     235   2(35) |
  | 9     2     3     | 178   157   1568  | 16    15    4     |
  | 5     4     8     | 12    3     126   | 1269  7     1269  |
  +-----------------------------------------------------------+
  | 8     3     4     | 6     12    7     | 5     9     12    |
  | 16    5     9     | 128   4     128   | 1236  123   7     |
  | 16    7     2     | 5     9     3     | 4     8     16    |
  +-----------------------------------------------------------+
  | (23)  8     7     | (123) 6     (125) | 12-39 4     19(5) |
  | 4     9     6     | 1237  1257  125   | 123   1235  8     |
  | 2(3)  1     5     | 4     8     9     | 7     6     2(3)  |
  +-----------------------------------------------------------+

       X=3r7c7
       |
3C1: 3r7c1A=============================3r9c1 
       |  A                               |   
7N1: 3r7c1A=2r7c1                         |   
       |      |                           |   
7N4: 3r7c4==2r7c4==1r7c4                  |   
       .      |      |                    |   
7N6:   .    2r7c6==1r7c6==5r7c6           |   
       .                    |             |   
5C9:   .                  5r7c9==5r1c9    |   
       .                           |      |   
3C9:   .                         3r1c9==3r9c9 
       ^weak set 3r7

= strong link
| weak link
A label for candidate in two strong links (strong corner)
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Postby DonM » Fri Oct 17, 2008 8:19 pm

Hmmm. So some of us think there is something of real interest here and others think it's just ho-hum time. (Nice post Allan B!).

I still think the particularly useful idea is the exit strong link Myth used. IMO, it is one thing to see the logic behind the 2 weak-link exit cells as simple cross-hatching which it is, but it's another thing to use those exit cells as a group to create a strong link with the remaining cell in the box as part of a chain. Hindsight is 20/20 with these things. If I was seeing a lot of solvers making innovative use of patterns to make strong links (eg. such as the way ttt regularly does), maybe I'd think differently. In fact, fwiw, I'm surprised at how little full end-stage (ie. say from the ssts point on) manual solving of difficult puzzles using relatively standardized nice loops or AICs (ala Carcul- bless him for still making an occasional appearance) is going on at all here these days... compared to high-end theory, which is good but doesn't help solve our everyday difficult puzzles. Not making any big point- just an observation. (and that could be wrong:) )

In other words, there a few manual solvers out there who have the ability to 'see' strong links where the rest of us miss them even though in retrospect the underlying logic is relatively simple. I see Myth's message as simply to be on the lookout for this overall pattern- it may give you another strong link to work with that you would have otherwise missed.

Incidentally, I said earlier that I had a good example of this pattern in another puzzle. Turns out it wasn't as good as I thought, but I hope to have another one before too long.
Last edited by DonM on Sat Oct 18, 2008 4:32 am, edited 3 times in total.
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Postby David P Bird » Fri Oct 17, 2008 11:33 pm

Here's a chain using what could be termed weak corners that I found a while back.

Code: Select all
 |  1     2     3    |  4     5     6    |  7     8     9    |
-|-------------------|-------------------|-------------------|
 | 3578  _6__  457   | 3489  378   2     | 35    1     3579  |
 | 137   2347  1247  | 49    5     169_  | 236_  3679  8     |
 | 13578 23578 9     | 138   1378  168_  | 4     3567  23567 |
-|-------------------|-------------------|-------------------|
 | 1689  89    3     | 7     29    5     | 1268  68__  4     |
 | 15789 45789 1457  | _6__  29    3     | 1258  578   1257  |
 | 2     57    567_  | 18    18    4     | 9     3567  357   |
-|-------------------|-------------------|-------------------|
 | 3569  2359  8     | 1239  13    19    | 7     4     13569 |
 | 4     23579 257   | 1238  _6__  1789  | 1358  3589  1359  |
 | 3679  1     67__  | 5     4     789   | 368_  2     369__ |
-|-------------------|-------------------|-------------------|


(7)r8c6 = (7)r9c6 - (7=6)r9c3 - (6#>0)r6c3,r7c1 = (6#2)r6c8,r7c9 - (6)r2c8,r3c89 = (6)r2c7 - (6)r2c6 = (6-8)r3c6 = (78)AHT:r89c6 => r8c6 <> 19

(6)r9c3 and (6)r2c8,r3c89 are both weak corners linking to nodes split between different houses. The logic is slightly confusing as different cases have to be used for the split nodes they connect with; the first is whether they are both false or not, and the second whether they are both true or not. This treatment makes the chain bidirectional.

There is a strong overlap here with a braiding/travelling pairs as we are interested in which diagonal direction (6) repeats in the stacks and tiers of boxes.

I believe Myth's strong corners aren't particularly new but simply haven't been named before. As Myth has already said chained UR deductions often use them and they can be embedded in finned fish patterns where I once suggested calling them "fish eggs" http://www.sudoku.org.uk/SudokuThread.asp?fid=4&sid=10008&p1=2&p2=11
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