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Generalized XY-wing

 
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Mon Apr 07, 2008 10:11 pm    Post subject: Generalized XY-wing Reply with quote

Here is the generalized XY-wing:
Code:

+----------+----------+----------+
| .  SX .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| SY .  .  | .  .  .  | .  YZ .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+
| .  XZ .  | .  .  .  | .  -Z .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+

S, X, Y, and Z are any candidates. The only condition is that SX and SY are a strong link on S.

The logic is: One of SX and SY must be X or Y. One of XZ and YZ must be Z. We can eliminate Z in the cell marked -Z.

Sort of looks like a kite or turbot fish, but SX-XZ and SY-YZ do NOT have to be strong links in X and Y, respectively.

How about this:
Code:

+----------+----------+----------+
| .  SX .  | .  XZ .  | -Z -Z -Z |
| .  .  .  | .  .  .  | .  .  .  |
| SY .  .  | -Z -Z -Z | .  YZ .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+

Does this ever occur? Does it have another name?

Maybe this can be an S-wing, in Steve's honor! Cool

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=2455

Just theorizing,

Keith
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Asellus



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

PostPosted: Mon Apr 07, 2008 10:50 pm    Post subject: Reply with quote

As long as those cells are all bivalues, as seems to be the case, then there is no need for a strong link on S. It is just an XY Chain. Perhaps I'm missing something.
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Wed Apr 09, 2008 7:56 am    Post subject: Reply with quote

ZX...XS...SY...YZ

Z(X...X)(S...S)(Y...Y)Z

23...34...45...52

2(3...3)(4...4)(5...5)2

(2=3)-(3=4)-(4=5)-(5=2)... cells in ()... this is an AIC.

in all xy-chains, which this is, the first and last candidate has to be the same and the cells have to be bivalue. therefore creating a pincer and eliminating any other candidates they see.

this is also called a bidirectional y-cycle because you can make the same chain in both directions and the candidate not used in connecting the prior two cells must be used to connect the next two. thus called "y". and the cells must "see" each other.

so you would logically look for bivalue cells that have one number in common...( and can see each other ) (2,3) with (3,4) then (4,5) then (5,2) creating a pincer with candidate 2.

if 2 than not 3, 3 than not 4, 4 than not 5, 5 than not 2... then reverse...

if 2 than not 5, 5 not 4, 4 not 3, 3 not 2

so when seen in both directions, either end has to have a 2 thus eliminating any 2's the two ends see.

this works because the cell with two candidates is actually a strong link...cell (2,3) is 2=3 ... so am I correct that this is a formation of a AIC?? strong link for each cell, weak when connecting cells. srong-weak-strong-weak-etc. two directions?? I just want to be sure of that.

this is the same foundation that a w-wing is based on... and the xy-wing is a three cell xy-chain. its the same technique.

the misconceptions about what is and isn't a xy-chain might stem from the multitude of bivalue cells and the numerous geometric shapes they create around the puzzle board and possibly what kind of eliminations they do make, but all in all, it has its roots in one particular technique.

IMO, the hardest xy-chains to spot have 3 or more cells in the same row or column and don't interact in any specific order. the chain then overlaps but still works.


Last edited by storm_norm on Tue Apr 15, 2008 1:02 am; edited 1 time in total
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Asellus



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

PostPosted: Wed Apr 09, 2008 5:51 pm    Post subject: Reply with quote

storm_norm wrote:
so am I correct that this is a formation of a AIC??

Yes.
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Thu Apr 10, 2008 7:40 am    Post subject: Reply with quote

the three cell xy-chain or commonly known as the xy-wing

23...34...42

2(3...3)(4...4)(2
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Victor



Joined: 29 Sep 2005
Posts: 207
Location: NI

PostPosted: Thu May 01, 2008 10:26 am    Post subject: Reply with quote

Norm, you've written slightly carelessly - a syndrome that I'm personally very familiar with!
Quote:
so you would logically look for bivalue cells that have one number in common...( and can see each other ) (2,3) with (3,4) then (4,5) then (5,2) creating a pincer with candidate 2.

if 2 than not 3, 3 than not 4, 4 than not 5, 5 than not 2... then reverse...


You have to start off with If not 2 rather than If 2.
The logic runs:
if not 2, then 3 - because the 2 & 3 are strongly linked;
if 3 in the (2,3) cell, then not 3 in the (3,4) cell - because these 3s are weakly linked;
if not 3 in the (3,4) cell, then 4 in this cell;
..... then 2 in the last cell.
And it works in the opposite direction. And so the one possibility that we've eliminated is that both 2s are false.

To look at it in another way, it wouldn't do any good to say that if one end is true then the other isn't, because that doesn't address the possibility that both ends are true.
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