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ERs--beyond the basics

 
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Marty R.



Joined: 12 Feb 2006
Posts: 5063
Location: Rochester, NY, USA

PostPosted: Sat Sep 20, 2008 9:23 pm    Post subject: ERs--beyond the basics Reply with quote

I know the very basics of an ER (and make frequent use of them): eliminate a candidate that sees both the ERI and the second part of the strong link. Is there more that can be done with these that falls short of rocket science? Wink
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Sun Sep 21, 2008 1:17 am    Post subject: Reply with quote

You can use them to look for 2-string kites at the same time, if the vertex of the ER does NOT have the candidate.

X is the candidate, cells marked "-" are not X, @ is the vertex, which has not X as a candidate, # cannot be X.

Code:
+-----------+-----------+-----------+
|  -  X  -  |  .  .  .  |  .  .  .  |
|  X  @  X  |  -  -  -  |  -  X   - |
|  -  X  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  X  .  |  .  .  .  |  .  #  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Sun Sep 21, 2008 3:58 am    Post subject: Reply with quote

Marty,

I think of ER's (Empty Rectangles) as kites with "grouped" coloring. Let me try to explain.

Suppose you have this:
Code:
Diagram 1
+-------+-------+-------+
| . * . | . . . | . . . |
| * . . | . . . | . @ . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . * . | . . . | . * . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
where there are strong links between the cells marked * in B1 and R5, not necessarily in C2. This is a kite. It eliminates the candidate in cell @. The logic is:
R2C1 is true. @ is not true. Or:
R2C1 is not true, R1C2 is true, R5C2 is not true, R5C8 is true, @ is not true.
Either way, @ is not true.

(Actually, the best explanation of a kite / skyscraper was given by Tracy of Kalamazoo. In this case, it would be: One of the cells in C2 is not true. So, at least one of R2C1 and R5C8 is true. @ is not true.)

Now, the ER comes in when you observe that "true" in B1 only needs to be in R2 or C2:
Code:
Diagram 2
+-------+-------+-------+
| . * . | . . . | . . . |
| * * * | . . . | . @ . |
| . * . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . * . | . . . | . * . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
The logic is:
Any cell in B1R2 is true. @ is not true. Or:
Any cell in B1C2 true, R5C2 is not true, R5C8 is true, @ is not true.
Either way, @ is not true.

This type of logic, where cells in a line within a block act as a single cell, is called (I believe) "grouped coloring".

I do not recall any explanation of ER's that talks about eliminating any candidate other than in the cell I have called @.

So far as grouped coloring is concerned, you need to ask for the "ravel tint" when you next get eyeglasses. Very Happy

Keith
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Sun Sep 21, 2008 4:11 am    Post subject: Reply with quote

wapati wrote:
You can use them to look for 2-string kites at the same time, if the vertex of the ER does NOT have the candidate.

X is the candidate, cells marked "-" are not X, @ is the vertex, which has not X as a candidate, # cannot be X.

Code:
+-----------+-----------+-----------+
|  -  X  -  |  .  .  .  |  .  .  .  |
|  X  @  X  |  -  -  -  |  -  X   - |
|  -  X  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  X  .  |  .  .  .  |  .  #  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
Wapati,

I do not understand. (First, what you are calling the "vertex" cell is usually called the "hinge" cell.)

Suppose, in your diagram, R1C2 is X. I do not see how # must then not be X.

I do not see anything special about whether the hinge cell does, or does not, contain the candidate.

Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5063
Location: Rochester, NY, USA

PostPosted: Sun Sep 21, 2008 5:01 am    Post subject: Reply with quote

Keith,

I think what Wapati calls a vertex and you call a hinge, I call an ERI. But I don't understand what you're saying. Despite the talk of kites, grouped coloring, etc., in each of the two grids you show, I see an ordinary ER. What am I missing?

Wapati,

Thank you. I'll wait for your response to Keith before commenting further.
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Sun Sep 21, 2008 5:47 am    Post subject: Reply with quote

keith wrote:
wapati wrote:
You can use them to look for 2-string kites at the same time, if the vertex of the ER does NOT have the candidate.

X is the candidate, cells marked "-" are not X, @ is the vertex, which has not X as a candidate, # cannot be X.

Code:
+-----------+-----------+-----------+
|  -  X  -  |  .  .  .  |  .  .  .  |
|  X  @  X  |  -  -  -  |  -  X   - |
|  -  X  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  X  .  |  .  .  .  |  .  #  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
Wapati,

I do not understand. (First, what you are calling the "vertex" cell is usually called the "hinge" cell.)




Keith


Wow, you missed it all.

I call the pivot cell a vertex when it is not a hinge. I would accept pivot in both cases. Hinge is an early concept that does not cover the example I posted.

This is a similar position that is a hinge. "~" cannot be "X".
Code:

+-----------+-----------+-----------+
|  -  -  -  |  .  .  .  |  .  .  .  |
|  -  X  X  |  -  -  -  |  -  X   - |
|  -  X  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  ~  .  |  .  .  .  |  .  X  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+



Quote:
Suppose, in your diagram, R1C2 is X. I do not see how # must then not be X.


OK, set R1C2 to X. R2C8 must be X. Conclusion?

Code:
+-----------+-----------+-----------+
|  -  X  -  |  .  .  .  |  .  .  .  |
|  -  -  -  |  -  -  -  |  -  X   - |
|  -  -  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  -  .  |  .  .  .  |  .  #  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+


Code:
I do not see anything special about whether the hinge cell does, or does not, contain the candidate.


Well, set the vertex to X and what do we know?


Code:
+-----------+-----------+-----------+
|  -  -  -  |  .  .  .  |  .  .  .  |
|  -  X  -  |  -  -  -  |  -  -   - |
|  -  -  -  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
|  .  -  .  |  .  .  .  |  .  ?  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
|  .  -  .  |  .  .  .  |  .  .  .  |
+-----------------------------------+
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sun Sep 21, 2008 7:39 am    Post subject: Reply with quote

Are we talking apples and oranges here?

Apples: wapati's diagram can be viewed as a grouped 2-String Kite. He uses a subset of the Empty Rectangle pattern to make it work -- vertex cell void of the candidate. (Personally, I wouldn't call this a 2-String Kite.)

Oranges: keith's Diagram 2 uses the ER pattern as the Empty Rectangle technique. In it, the vertex cell does not have to be void of the candidate.
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Marty R.



Joined: 12 Feb 2006
Posts: 5063
Location: Rochester, NY, USA

PostPosted: Sun Sep 21, 2008 4:11 pm    Post subject: Reply with quote

Quote:
I call the pivot cell a vertex when it is not a hinge.

What are the differences between pivot, vertex, hinge and ERI?
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Sun Sep 21, 2008 5:27 pm    Post subject: Reply with quote

Marty,

You asked "Is there more that can be done with these ..."?


I replied, "I do not recall any explanation of ER's that talks about eliminating any candidate other than in the cell I have called @."

In other words, my answer to your question is no, not so far as I know.

It seems to me you could replace the strong link with a different inference. For example:

Code:
Diagram 3
+-------+-------+-------+
| . * . | . . . | . . . |
| * * * | . . . | . @ . |
| . * . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . # . | . . . | . . . |
| . . . | . . . | . # . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+


If the two cells # are the pincers of an XY-wing, you can make the ER elimination. (One, or both cells # are true.)



Wapati,

Clearly, I am missing something. I'll have to study your posts in more detail.

Best wishes,

Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5063
Location: Rochester, NY, USA

PostPosted: Sun Sep 21, 2008 5:39 pm    Post subject: Reply with quote

Quote:
I replied, "I do not recall any explanation of ER's that talks about eliminating any candidate other than in the cell I have called @."

Sorry, I missed that sentence. Embarassed
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sun Sep 21, 2008 6:13 pm    Post subject: Reply with quote

In the Players' Forums, Havard introduced the Empty Rectangle pattern in this thread.

http://www.sudoku.com/boards/viewtopic.php?p=21366#21366

Havard wrote:
definition: An Empty Rectangle (ER) is when four cells in a box does not contain a specific candidate, and these four cells form a rectangle.

He then goes on to show how it can be combined with a single strong link to perform an elimination. This is the classic Empty Rectangle technique.

Multiple ER patterns can be combined to produce more complicated elimination grids. For example:

Code:
*-----------------------------------*
|  .  *  .  |  .  .  .  |  .  *  .  |
|  /  X  /  |  /  /  /  |  /  X  /  |
|  .  *  .  |  .  .  .  |  .  *  .  |
|-----------+-----------+-----------|
|  .  *  .  |  .  .  .  |  .  *  .  |
|  .  *  .  |  .  .  .  |  .  *  .  |
|  .  *  .  |  .  .  .  |  .  *  .  |
|-----------+-----------+-----------|
|  /  X  /  |  .  .  .  |  /  X  /  |
|  X *X  X  |  *  *  *  |  X *X  X  |
|  /  X  /  |  .  .  .  |  /  X  /  |
*-----------------------------------*

notation:
X cells containing candidate for optimal pattern
/ cells not containing candidate
* cells where candidate can be eliminated

re'born wrote:
Now an empty rectangle (or "hinge" as it was originally called by Rod Hagglund ...
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Sun Sep 21, 2008 7:02 pm    Post subject: Reply with quote

daj95376 wrote:

Havard wrote:
definition: An Empty Rectangle (ER) is when four cells in a box does not contain a specific candidate, and these four cells form a rectangle.


I find the "empty rectangle" hard to spot. I find it easier to look for candidates in a box that lie only in one row and one column.

daj95376 wrote:

re'born wrote:
Now an empty rectangle (or "hinge" as it was originally called by Rod Hagglund ...

Hagglund called the entire pattern a "Hinge". Hinge = Empty Rectangle.

What I called the "hinge cell" is called the "Empty Rectangle Intersection" (ERI) by Havard.

Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5063
Location: Rochester, NY, USA

PostPosted: Sun Sep 21, 2008 9:24 pm    Post subject: Reply with quote

keith wrote:
daj95376 wrote:

Havard wrote:
definition: An Empty Rectangle (ER) is when four cells in a box does not contain a specific candidate, and these four cells form a rectangle.


I find the "empty rectangle" hard to spot. I find it easier to look for candidates in a box that lie only in one row and one column.


After I read about that original definition I looked for a few, never found one, and gave up on the technique for a year or more. Then I read another description about how an ER is present when a candidate is in only two lines in a box. Then they became very easy to spot and I play a lot of them.

Unfortunately, with me being me, I couldn't figure that out on my own, but that's another story.
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Asellus



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

PostPosted: Sun Sep 21, 2008 9:40 pm    Post subject: Reply with quote

To resolve (I hope) the confusion above, we need to look a little more closely at the grouped link involved in a hinge/ER. If the ERI cell contains candidate X, then the grouped link is a strong (only) inference link. It is possible for both sides of the link to be true... IF the ERI cell is true. (It is impossible for both sides to be false, hence the strong inference.)

However, if the ERI cell does NOT contain candidate X, it is now a conjugate link, which means that it can serve as either a strong link OR a weak link, according to the need.

When used in the standard way as an ER, it is being used as a strong link. So, it doesn't matter whether or not X occurs in the ERI.

In wapati's example, the ER-like grouped link is being used as the weak link pivot of a Kite (or Turbot Fish). Thus, candidate X must not occur in the ERI. The strong links occur in the row and column to the lone Xs (and thus pincers) that occur outside the "pivot box". These are also grouped links, of course.

By the way, I would (and do) call this a Kite. I can see no advantage in denying the name to to something that has exactly the same logic and requires only a modicum of additional sodoku knowledge (a comfort with grouped links) to recognize and use.

I hope that helps clarify things.
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