Identical Pair Forcing Pattern

Advanced methods and approaches for solving Sudoku puzzles

Identical Pair Forcing Pattern

Postby Bud » Tue Sep 02, 2008 8:48 pm

Here is a simple xy pattern I have encountered in some of the puzzles I have worked. I have added it to my pattern techniques toolbox. The first time it was a puzzle breaking technique, so I thought I should share it with other members. The pattern consists of two cells with identical xy pairs which are not in the same house but are in the same box row (or column}. The not x (-x) and not y (-y) cells are necessary for the pattern to work. Because of the pattern. xy cell r3c3 forces xy cell r2c9 to be y. The logic is as simple as the pattern. r3c3 = y implies that r2c9 = y since there are no other y candidates in box 3 except in row 3. r3c3 = x implies that x is in row 2 of box 2 since there are no x candidates in row 1 of box 2. Therefore x cannot be in row 2 of box 3 and r2c9 = y.

Code: Select all
 

  .     .    .  | -x  -x   -x |  -y  -y  -y
  .     .    .  |  .   .   .  |  -y  -y   xy
  .     .    xy |  .  ,    .  |    .    .    .
-------------------------------------------
Last edited by Bud on Wed Sep 24, 2008 9:20 am, edited 1 time in total.
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Postby daj95376 » Tue Sep 02, 2008 11:24 pm

Code: Select all
 |---------------+---------------+---------------|
 |   .   .   .   |  -x  -x  -x   |  -y  -y  -y   |
 |   .   .   .   |   .   .   .   |  -y  -y   xy  |
 |   .   .   xy  |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
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Postby udosuk » Wed Sep 03, 2008 12:39 am

Code: Select all
+---------------+---------------+---------------+
|   .   .   .   |  -x  -x  -x   |  -y  -y  -y   |
|   #   #   #   |   *   *   *   |  -y  -y   xy  |
|   .   .   xy  |   *   *   *   |   #   #   #   |
+---------------+---------------+---------------+

Another way to look at your pattern is:

Grouped W-wing (aka Grouped Y-wing in some regions of the world)

The * cells (r23c456) contain all the x's in b2.

If r2c456 contain x, r2c9=y.
If r3c456 contain x, r3c3=y.

Therefore, all # cells (r2c123+r3c789), seeing r2c9+r3c3, can't be y.

Then r2c9 becomes a hidden single of y in b3.

Can someone write the nice-loop notation of this move for me?:?:
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Postby Myth Jellies » Wed Sep 03, 2008 2:53 am

(y=x)r2c9 - (x)r2c456 = (x)r3c456 - (x=y)r3c3 => r2c123, r3c789 <> y
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Postby denis_berthier » Wed Sep 03, 2008 3:03 am

udosuk wrote:
Code: Select all
+---------------+---------------+---------------+
|   .   .   .   |  -x  -x  -x   |  -y  -y  -y   |
|   #   #   #   |   *   *   *   |  -y  -y   xy  |
|   .   .   xy  |   *   *   *   |   #   #   #   |
+---------------+---------------+---------------+

Another way to look at your pattern is:

Grouped W-wing (aka Grouped Y-wing in some regions of the world)

The * cells (r23c456) contain all the x's in b2.

If r2c456 contain x, r2c9=y.
If r3c456 contain x, r3c3=y.

Therefore, all # cells (r2c123+r3c789), seeing r2c9+r3c3, can't be y.

Then r2c9 becomes a hidden single of y in b3.

Can someone write the nice-loop notation of this move for me?:?:


First, notice that, as it is presented, this is a complex pattern, as it requires looking at 10 cells. It should therefore be applied only after elementary rules.

This pattern is also incompletely specified, because:
- if x is absent, as a candidate, from block 2 row 3, then an elementary interaction rule entails r2c9 <> x, then singles and elementary constraints propagation rules entail r2c9=y and r3b3 <> y

- if x is absent, as a candidate, from block 2 row 2, then an elementary interaction rule entails r3c3 <> x, then singles and elementary constraints propagation rules entail r3c3= y, r3b3 <> y

These 2 cases are therefore subsumed by singles, elementary constraints propagation rules and elementary interaction rule.


What remains is the following pattern (x present, as a candidate, in both rows 2 and 3 of block 2):

Code: Select all
+---------------+---------------+---------------+
|   .   .   .   |  -x  -x  -x   |  -y  -y  -y   |
|   #   #   #   | x in segment  |  -y  -y   xy  |
|   .   .   xy  | x in segment  |   #   #   #   |
+---------------+---------------+---------------+


where "x in segment" means that x appears, as a candidate, in the segment.

This is a grouped-nrc-chain of length 3: {y x}r3c3 - x{r3c4/5/6 r2c456} - {x y}r2c9
which admits any of yr2c1/2/3 and any of yr3c7/8/9 as a target.

This is also a basic grouped-AIC or NL.

here c4/5/6 means "any of c4, c5 or c6" and simirlarly for c7/8/9.

Conclusion: this complex pattern is subsumed by much simpler rules.
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Postby denis_berthier » Wed Sep 03, 2008 3:10 am

I've been writing my post in parallel with Myth.
Of course we get the same result.
Just one thing: Myth's NL is valid only if x appears as a candidate in the 2 mentioned segments.
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Postby Myth Jellies » Thu Sep 04, 2008 3:49 am

Sorry, my above post only showed an AIC reliant on just the two xy-bivalues and the lack of an x-candidate in r1c456.

In your setup, Bud, the elimination of y's from r3c789 then forces r2c9=y. That extra deduction is somewhat incidental. The removal of the y's is not at all dependent upon the missing y-candidates in box 3.

It is a deduction based on a maximum of three strong inference set (SIS) subpatterns, thus making it roughly equivalent in complexity to an xy-wing or a W-wing.

Denis wrote:Just one thing: Myth's NL is valid only if x appears as a candidate in the 2 mentioned segments.


Denis's statement is incorrect. The strong inference & AIC/NL will still hold even if x does not have a viable candidate in one of the mentioned segments (one of the segments will still be true). If only one segment contains candidate x then you will have a more obvious locked candidates deduction though, and the rest of the AIC will become rather unnecessary.

You can use this marker of two equivalent bivalue cells (something that is very easy to see) to find various similar patterns. For example if the xy-bivalues are in r1c1 and r9c9, then you can look for an empty rectangle (missing candidates) for one of the candidates in either r23c78 or r78c23 to complete the pattern and eliminate the other candidate from r1c9 and r9c1.

And you don't have to limit yourself to bivalued cells either. Certain other ALS's will also work.
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Postby denis_berthier » Thu Sep 04, 2008 3:57 am

Myth Jellies wrote:
Denis wrote:Just one thing: Myth's NL is valid only if x appears as a candidate in the 2 mentioned segments.


Denis's statement is incorrect. The strong inference & AIC/NL will still hold even if x does not have a viable candidate in one of the mentioned segments (one of the segments will still be true). If only one segment contains candidate x then you will have a more obvious locked candidates deduction though, and the rest of the AIC will become rather unnecessary.

That's exactly what I said in a more correct way in the previous post. If there is no x in any of the 2 segments, there's no AIC at all.
But you've certainly not read it before jumping to the conclusion that my statement was incorrect.
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Postby daj95376 » Thu Sep 04, 2008 10:37 am

Given Myth Jellies' AIC for udosuk's grouped W-Wing, it appears to me that Bud's diagram can be reduced to:

Code: Select all
 (y=x)r2c9 - (x)r2c456 = (x)r3c456 - (x=y)r3c3 => r2c123, r3c789 <> y
 |---------------+---------------+---------------|
 |   .   .   .   |  -x  -x  -x   |   .   .   .   |
 |   ~   ~   ~   |   .   .   .   |   .   .   xy  |
 |   .   .   xy  |   .   .   .   |   ~   ~   ~   |
 |---------------+---------------+---------------|

The location -- or lack of -- for y cells is incidental.
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Postby denis_berthier » Thu Sep 04, 2008 11:14 am

daj95376 wrote:Given Myth Jellies' AIC for udosuk's grouped W-Wing, it appears to me that Bud's diagram can be reduced to:

Code: Select all
 (y=x)r2c9 - (x)r2c456 = (x)r3c456 - (x=y)r3c3 => r2c123, r3c789 <> y
 |---------------+---------------+---------------|
 |   .   .   .   |  -x  -x  -x   |   .   .   .   |
 |   ~   ~   ~   |   .   .   .   |   .   .   xy  |
 |   .   .   xy  |   .   .   .   |   ~   ~   ~   |
 |---------------+---------------+---------------|

The location -- or lack of -- for y cells is incidental.


Where do you see any AIC if there is not at least one x candidate in each of the 2 segments r2b2 and r3b2 ?
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Postby daj95376 » Thu Sep 04, 2008 11:51 am

denis_berthier wrote:Where do you see any AIC if there is not at least one x candidate in each of the 2 segments r2b2 and r3b2 ?

Bud specifically indicated there weren't any x candidates in [r1c456]. In the absence of other eliminations for x candidates in b2, then Myth Jellies AIC is accurate (IMO) for the pattern presented.

Obviously, if another mini-row in b2 was void of x candidates, then Locked Candidates 1 would apply and it would make the discussion of this pattern useless in the first place!
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Postby denis_berthier » Thu Sep 04, 2008 11:56 am

daj95376 wrote:Obviously, if another mini-row in b2 was void of x candidates, then Locked Candidates 1 would apply and it would make the discussion of this pattern useless in the first place!

That's what I said in my first post. But when one describes a pattern, all the conditions should be written. Otherwise, we waste our time with such nonsensical discussions. The presence of an x in each of the 2 segments is compulsory if you want to have an AIC.

In any case, the initial pattern is not a new pattern at all.
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Postby ronk » Thu Sep 04, 2008 12:26 pm

denis_berthier wrote:But when one describes a pattern, all the conditions should be written. Otherwise, we waste our time with such nonsensical discussions.

I too think the pattern presented in the opening post was clear enough ... and that you are being argumentative.
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Postby denis_berthier » Thu Sep 04, 2008 12:41 pm

ronk wrote:
denis_berthier wrote:But when one describes a pattern, all the conditions should be written. Otherwise, we waste our time with such nonsensical discussions.

I too think the pattern presented in the opening post was clear enough ... and that you are being argumentative.

Then you don't have a clear notion of clearness.
The initial pattern is a mixture of different patterns and it can therefore not be clear enough - except if you're making the apology of vagueness.

The proof? It was presented as a new pattern.

No more time to waste with such Eureka-style remarks, with no technical content.
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Postby Myth Jellies » Fri Sep 05, 2008 1:27 am

Denis wrote:But when one describes a pattern, all the conditions should be written....The presence of an x in each of the 2 segments is compulsory if you want to have an AIC.


Once again, Denis, your statement is blatantly incorrect. Once you know that r1c456 does not contain x then you know that r23c456 must contain x. Therefore ANY division of those six cells containing x must form a valid strong inference set and that includes (x)r2c456=(x)r3c456.

The possible existence of a "simpler" pattern within those six cells has no effect whatsoever on the validity of this SIS or the validity of the AIC in question. One is not required to spot all "simpler" patterns prior to spotting a more complex one.

The beauty of this thread is that it presents an infrequently used way of looking at and discovering things. In specific, we are finding a SIS and a potential AIC deduction by looking for the holes where things aren't rather than concentrating on where things are. Some people may be up to this challenge and some may, for whatever reason, pooh-pooh it.

I've always been one to concentrate on the hinge of exsiting candidates vs. the empty rectangle of candidate holes; but I think I will put a quick search for appropriately placed candidate holes between two matching bivalue cells in my basic repertoire. Heck, with a marker as easy to spot as this, I might even consider investigating a fin/kraken arm in those holes.
Code: Select all
Example of non-chute case.
 |---------------+---------------+---------------|
 |   xy  .   .   |   .   .   .   |   .   .  -y   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .  ~x  ~x   |   .   .   .   |   .   .   .   |
 |   .  ~x  ~x   |   .   .   .   |   .   .   .   |
 |  -y   .   .   |   .   .   .   |   .   .   xy  |
 |---------------+---------------+---------------|
(y=x)r1c1 - (x)r789c1 = (x)r9c123 - (x=y)r9c9 => r1c9, r9c1 <> y

a Kraken/AAIC case (note strong link for z between r7c3 and r9c1)
 |---------------+---------------+---------------|
 |   xy  .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |  ~z  ~xz  .   |   .   .   .   |   .   .   .   |
 |  ~z  ~xz ~xz  |   .   .   .   |   .   .   .   |
 |  -y  ~z  ~z   |   .   .   .   |   .   .   xy  |
 |---------------+---------------+---------------|

(y=x)r1c1 - (x)r789c1
             ||
            (x)r9c123 - (x=y)r9c9
             ||
            (x-z)r7c3 = (z)r9c1  ==> r9c1 <> y

Last edited by Myth Jellies on Thu Sep 04, 2008 10:09 pm, edited 1 time in total.
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