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"Intersection of four groups" generalization of &q

 
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Sun Oct 16, 2005 5:38 am    Post subject: "Intersection of four groups" generalization of &q Reply with quote

Subject: the technique called "Intersection of four groups"

I was working a lot on the technique called "X-wing":
The X-wing that has the same number at intersection of 2 rows and 2 columns (the name cames from the airplains flying in this formation) and having this same number ONLY 2 times in the rows or in the columns - permits us to eliminate the rest of this number from the 2 columns respective 2 rows (if there are any additional such ones).

I think that one of the "generalization" of this X-wing is the technique called "intersectioon of four groups". It is the combination of 4 intersection rows, columns, 3x3 blocks.

* If we have 2 rows and 2 columns - then we speak about the "X-wing".

The other cases are:

* 2 rows, 1 column & 1 block

* 1 row, 2 columns & 1 block

* 2 rows, 2 blocks

* 2 columns, 2 blocks

Let's take an example:

X-X
6=6======
X-X
=== the sign "=" means no "6"
==6
6==
X-X
X-X
X-X

We have here number 6:
- in row 2 exact twice
- in column 1 exact twice
- in column 3 exact twice
- in 3x3 block (4,1) - (6,3) exact twice
We have the same number 6 at the intersection of four groups:
one row, 2 columns and 1 3X3 block.

We can quickly deduce that from all places marked with "X" the "6" can be eliminated!

For the rest of the cases, I let you draw your own diagrams.

see u,
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Sun Oct 16, 2005 11:45 am    Post subject: Generalized "X-Wing" Reply with quote

Good observation, someone. Excellent insight!

In your list of cases, what happened to one row, one column, & two blocks? Oh -- I guess that one is not possible, is it. dcb
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Sun Oct 16, 2005 7:37 pm    Post subject: Reply with quote

Yup, only 3 of them can "intersect", so this would be a X-wing with a broken wing. It can't fly ;-)

P.S. you know the XY-wing. Now I found out that there is also a "XYZ-wing". For example:

* XYZ * ... YZ ...
= = =
XZ = =

From this pattern we can exclude "Z" from positions marked "*"
(of course that left all numbers must be in a 3x3 block).

see u,
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fordmodelt



Joined: 07 Nov 2008
Posts: 2
Location: Melbourne, Australia

PostPosted: Fri Nov 07, 2008 5:27 pm    Post subject: Reply with quote

I'm having a lot of trouble with this Intersection of 4 Groups concept. In the example given by Someone, don't the restrictions "In column 1 exact twice" and "In column 3 exact twice" just knock out all the Xs in your example anyway? The 2 row 2 column description of Intersection of 4 Groups seems to me to basically X-wing, which is fine, but I'm trying to find visual examples of the other Intersection of 4 Groups options: 2 rows, 1 square, and 1 column; 2 rows and 2 squares; 1 row, 1 square, and 2 columns; 2 columns and 2 squares. I've been stuck at the same level in Sudoku for months and it's driving me nuts. Shocked I just can't crack the toughest level puzzles unless there happens to be a x-wing in there. Desperately looking for other techniques to help me take the next step...
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nataraj



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

PostPosted: Fri Nov 07, 2008 7:41 pm    Post subject: Reply with quote

Hello, fordmodelt, welcome to this forum!

So it's your (first) "birthday" on the forum (not your 100th like the model T's this year), have fun!

I would not look at those "intersections of .." a lot, that idea seems to have been a very dead end ...

If you'd like to go beyond x-wings, you could either look at a technique called coloring or multi-coloring (with some very common examples being the skyscraper and the kite. Both are somewhat similar to x-wings but the two "lines" are not of the same length and sometimes not even parallel), or you could look at swordfish and jellyfish (same as x-wing, but with three and four rows/columns instead of two), or you leave the fishy business for a while and look at wings instead, especially the xy-wing.

You'll find excellent explanations of these techniques in
sudopedia, Ruud's sudoCue.net or Andrew Stuart's strategy pages, among others.

The "Daily" section of this forum always has lots of discussions about the more difficult puzzles, and people are usually very helpful. On this site, your next step definitely needs to be the xy-wing. To add one more to the list of references: look here for an explanation of the xy-wing

Enjoy!
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keith



Joined: 19 Sep 2005
Posts: 3165
Location: near Detroit, Michigan, USA

PostPosted: Fri Nov 07, 2008 10:51 pm    Post subject: Reply with quote

Interesting! Where to go next?

I agree with Nataraj. If your only "advanced" move is X-wing, next you should look at XY-wing. Since it only involves cells that have two candidates, it is relatively easy to spot.

Next, look at XYZ-wing. They are relatively rare, but with X-, XY-, and XYZ-wings in your arsenal, you will be able to solve this site's "Very Hard" puzzles.

As a side trip, take a look at Unique Rectangles. They are sort of like X-wings, but the logic is different. (Please do not start another religious argument as to whether uniqueness assumptions are valid!)

Now, the most important thing you will ever learn about Sudoku: Two strong links. http://www.sudoku.com/boards/viewtopic.php?t=3326

An X-wing is two strong links. So is a kite, a skyscraper, and a turbot fish. (Don't worry about the names.)

After that, it's the home stretch. Look on this site for explanations of W-wings and M-wings.

Another side trip is coloring, multi-coloring, and Medusa coloring. You'll easily understand these, once you get the two strong links into your head.

Keith


Last edited by keith on Sat Nov 08, 2008 2:22 am; edited 1 time in total
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fordmodelt



Joined: 07 Nov 2008
Posts: 2
Location: Melbourne, Australia

PostPosted: Sat Nov 08, 2008 12:24 am    Post subject: Reply with quote

Thankyou nataraj and keith - I'll follow your suggestions and see how I go. I've got this far using my own logic processes, so it came as a bit of a surprise to find that what I was doing was the same as what everyone did anyway! However my own logic has hit a dead end - though I had started playing around with colouring techniques of my own without much success. Thanks for your help!
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