## Double XY-Wing Triangle

Advanced methods and approaches for solving Sudoku puzzles

### Double XY-Wing Triangle

Double XY-Wing Triangle

This is the second time I have run into this solving pattern so I thought I would pass it on. In the example below the first XY-wing has a pivot at r7c9 and 7 pincers at r7c3 and r8c8. This eliminates 7 from r8c1 which becomes the pivot for the second XY-wing with 7 pincers at r7c3 and r8c7. The resultant triangle is indicated by '. Since r8c78 are peers, at least one of these must be not 7 and its Z-conjugate r7c3 must be 7.

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` |-----------------+-----------------+-----------------| |  467   8   467  |   3    5   67   |   2    9    1   | |   9    5    1   |  468  46    2   |   3   68    7   | |   3   67    2   | 1678   9  1678  |   4   68    5   | |-----------------+-----------------+-----------------| |   8   36    5   |  16    2   136  |  79   47   49   | |   1    2    9   |  478  47   478  |   6    5    3   | |  467  367  467  |   5   36    9   |   1    2    8   | |-----------------+-----------------+-----------------| |   2    9  '67   |  467   1    5   |   8    3   46   | | *567   1    3   |   9    8   467  | '57  '47    2   | |  567   4    8   |   2   367  367  |  579   1   69   | |-----------------+-----------------+-----------------| `
Bud

Posts: 56
Joined: 24 August 2008

Bud starting from your initial cell we could also find this chain
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`(6=4)r7c9-(4)r7c4=(4-6)r8c6=(6)r8c1 => r7c3<>6,r8c6<>7`
Glyn

Posts: 357
Joined: 26 April 2007

An interesting overlap in logic.

The XY-Wing [r7c9]/[r7c3]+[r8c8] not only produced an elimination, but it also identifies [r7c3]=7=[r8c8].

The XY-Wing [r8c1]/[r7c3]+[r8c7] doesn't produce an elimination, but it identifies [r7c3]=7=[r8c7].

Putting these links together results in a short forcing network:

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`          [r8c7]         /      \[r7c3]=7=        contradiction!  =>  [r7c3]=7         \      /          [r8c8]`

I don't know how often this will be of use, but it may have possibilities given the number of XY-Wings present in so many puzzles.
daj95376
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Off topic: The strong link r8c78 =7= r9c7 along with the potential deadly pattern r89c17<57> eliminates 7 from r89c1.

On topic: Bud's observation is a different take on transport, but uses the same principle. I like it. re'born

Posts: 551
Joined: 31 May 2007

Very... very nice to see you come back Adam...!
I'm sorry if something's wrong... I was dru... so much... HCMC's vodka... ttt
ttt

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Location: vietnam

Off topic :
hp36r9c56=>-6r9c9=>6r7c9 : <6>r7c3
~hp36r9c56=>7r9c5/6=>np46r7c49 : <6>r7c3
aran

Posts: 334
Joined: 02 March 2007

ttt wrote:Very... very nice to see you come back Adam...!
I'm sorry if something's wrong... I was dru... so much... HCMC's vodka... ttt

LoL Thanks for the kind words, ttt. But I'm not really back...at least, don't let my wife know I'm here.

aran wrote:Off topic :
hp36r9c56=>-6r9c9=>6r7c9 : <6>r7c3
~hp36r9c56=>7r9c5/6=>np46r7c49 : <6>r7c3

If I understand your notation, aran, this is a corresponding AIC?
(6)r7c9 = (6)r9c9 - (6=37)r9c56 - (7=46)r7c49, => r7c3<>6.
re'born

Posts: 551
Joined: 31 May 2007

re'born wrote:
aran wrote:Off topic :
hp36r9c56=>-6r9c9=>6r7c9 : <6>r7c3
~hp36r9c56=>7r9c5/6=>np46r7c49 : <6>r7c3

If I understand your notation, aran, this is a corresponding AIC?
(6)r7c9 = (6)r9c9 - (6=37)r9c56 - (7=46)r7c49, => r7c3<>6.

Nicer written as a nice loop r7c3 -6- r7c9 =6= r9c9 -6- r9c56 -7- r7c49 -6- r7c3 ==> r7c3<>6
ronk
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Location: Southeastern USA

re'born
If I understand your notation, aran, this is a corresponding AIC?
(6)r7c9 = (6)r9c9 - (6=37)r9c56 - (7=46)r7c49, => r7c3<>6

As written, the hidden pair approach considered the (in this case) 2 element strong inference set (hidden pair 36, not hidden pair=>7).
So two AICs which converged on r7c3 <6>.
It could be written as a single AIC loop starting from the elimination cell which would not however reflect the actual logic
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`7r7c3=7r7c4-(7=36)r9c56-6r9c9=6r7c9-(6=7)r7c3`

Would you agree that your own AIC is in fact a DIC sending two streams from r7c9 (the ~6 stream and the 4 stream) and therefore the step (7=46)r7c49 isn't "strictly" true insofar as there is not a pair46 but a 4 and a 6? Capillary fissure.
aran

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Joined: 02 March 2007

aran wrote:re'born
If I understand your notation, aran, this is a corresponding AIC?
(6)r7c9 = (6)r9c9 - (6=37)r9c56 - (7=46)r7c49, => r7c3<>6

As written, the hidden pair approach considered the (in this case) 2 element strong inference set (hidden pair 36, not hidden pair=>7).
So two AICs which converged on r7c3 <6>.
It could be written as a single AIC loop starting from the elimination cell which would not however reflect the actual logic
Code: Select all
`7r7c3=7r7c4-(7=36)r9c56-6r9c9=6r7c9-(6=7)r7c3`

aran wrote:Would you agree that your own AIC is in fact a DIC sending two streams from r7c9 (the ~6 stream and the 4 stream) and therefore the step (7=46)r7c49 isn't "strictly" true insofar as there is not a pair46 but a 4 and a 6? Capillary fissure.

This part I definitely don't understand. Would you explain what you mean by "not a pair46 but a 4 and a 6"? I don't see the difference. As for it being a DIC, I can only answer hypothetically since I don't know the formal definition of a double implication chain. It does not appear to me that my AIC uses two streams, but I've been wrong about these things a lot more often than I've been right (ask ronk ).
re'born

Posts: 551
Joined: 31 May 2007

re'born
aran wrote:

Would you agree that your own AIC is in fact a DIC sending two streams from r7c9 (the ~6 stream and the 4 stream) and therefore the step (7=46)r7c49 isn't "strictly" true insofar as there is not a pair46 but a 4 and a 6? Capillary fissure.

This part I definitely don't understand. Would you explain what you mean by "not a pair46 but a 4 and a 6"?

Your chain starts (6)r7c9 =. That is : r7c9 is not 6. So you can't subsequently in the same logic say it is 46. On the other hand you can subsequently say it's 4 which is what starts your second inference stream which combined with the first stream inference (not 7 r7c4) places 6 at r7c4.
aran

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Joined: 02 March 2007

aran wrote:Your chain starts (6)r7c9 =. That is : r7c9 is not 6. So you can't subsequently in the same logic say it is 46.

I see absolutely nothing wrong with the derived strong inference ... (6)r7c9 = (46)r7c49. All it says is ... if r7c9 is not digit 6, then r7c49 contains the digit pair 46.

For the AIC notation ... mea culpa, mea culpa. ronk
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Ronk
aran wrote:
Your chain starts (6)r7c9 =. That is : r7c9 is not 6. So you can't subsequently in the same logic say it is 46.

I see absolutely nothing wrong with the derived strong inference ... (6)r7c9 = (46)r7c49. All it says is ... if r7c9 is not digit 6, then r7c49 contains the digit pair 46.

If you say r7c49 is a digit pair 46, then (to my way of thinking) that reserves the possibility that the 46's may split 4r7c4 and 6r7c9, which is excluded by virtue of (6)r7c9=.
So it's a white lie aran

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Joined: 02 March 2007

aran wrote:Ronk
aran wrote:
Your chain starts (6)r7c9 =. That is : r7c9 is not 6. So you can't subsequently in the same logic say it is 46.

I see absolutely nothing wrong with the derived strong inference ... (6)r7c9 = (46)r7c49. All it says is ... if r7c9 is not digit 6, then r7c49 contains the digit pair 46.

If you say r7c49 is a digit pair 46, then (to my way of thinking) that reserves the possibility that the 46's may split 4r7c4 and 6r7c9, which is excluded by virtue of (6)r7c9=.
So it's a white lie As I see it, an AIC (or Nice Loop) is not to be read as a novel, but a collection of short stories That is, when I write (6)r7c9 = (6)r9c9 - (6=37)r9c56 - (7=46)r7c49,
I mean it as a collection of 10 independent statements.
1. If r7c9<>6, then r9c9=6.
2. If r9c9<>6, then r7c9<>6.
3. If r9c9=6, then r9c56<>6
4. If r9c56=6, then r9c9<>6
...
The derived strong inference is then the Cliff Notes (okay, the metaphor is breaking down here) and is comprised of two statements.
1. If r7c9<>6, then r7c49=.
2. If r7c49<>, then r7c9<>6.
The first one can be used to say that if r7c9<>6, then r7c9=4, but this isn't the point of the AIC and it is a not a white lie to omit a conclusion that is unnecessary.
re'born

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Joined: 31 May 2007

Re'born
I agree with what you write about independent statements etc and the
collection of short stories
metaphor is absolutely fine.
What you then go on to say is this I believe :
if it is a true statement : 46r7c49 not a pair =>r7c9 is 6
ie ~46r7c49=>6r7c9
then the logical converse must hold :
~6r7c9=>~~46r7c49.
There can be no disagreement so far.
The double "not" above is crucial because in "language" we might say that "not not a pair" means a pair...but does it ?
A pair here means 4,6 or 6,4 in r7c49 (and both remain possible)
Examine : not not (4,6 or 6,4).
not (4,6 or 6,4)=>neither.
not neither : is satisfied by the existence of either, without implying that both remain possible.
Consequently I say "not" to :
it is a not a white lie

And not not a white lie is a white lie aran

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