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Empty rectangles--another question

 
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Tue Sep 25, 2007 4:43 am    Post subject: Empty rectangles--another question Reply with quote

Can you have an ER in a box where the candidate occurs only twice. Consider the following grid fragment:

Code:

+-------------+-------+-------+
| 456 9   345 | . . . | . . . |
| 8   35  2   | . . . | . . . |
| 16  147 47  | . . . | . . . |
+-------------+-------+-------+
| .   .   .   | . . . | . . . |
| .   .   .   | . . . | . . . |
| .   .   .   | . . . | . . . |
+-------------+-------+-------+
| .   .   .   | . . . | . . . |
| .   .   .   | . . . | . . . |
| .   .   .   | . . . | . . . |
+-------------+-------+-------+

Play this puzzle online at the Daily Sudoku site

Can you start an ER on 3? If yes, there could be two ERIs depending on how the lines are drawn, one in r2c3 and the other in r1c2. Could either or both be used?
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Asellus



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

PostPosted: Tue Sep 25, 2007 5:14 am    Post subject: Reply with quote

The answer is definitely YES. It is a flexible ER!
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Tue Sep 25, 2007 3:42 pm    Post subject: Reply with quote

Asellus wrote:
The answer is definitely YES. It is a flexible ER!


Thank you Asellus!! I'm just getting started on this technique, even though I've known of it for quite some time, but I recently read a much clearer explanation that I had previously. I have the same rectangle that I showed here in a puzzle I'm working on and I'm anxious to see how much it helps.
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keith



Joined: 19 Sep 2005
Posts: 3174
Location: near Detroit, Michigan, USA

PostPosted: Tue Sep 25, 2007 10:20 pm    Post subject: Reply with quote

Marty,

The ER you describe is also a skyscraper, or possibly a box-line interaction.

Keith
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keith



Joined: 19 Sep 2005
Posts: 3174
Location: near Detroit, Michigan, USA

PostPosted: Tue Sep 25, 2007 10:58 pm    Post subject: An explanation Reply with quote

Here is the explanation of my previous comment:

The general Empty Rectangle logic:
Code:
Diagram 1
+----------+----------+----------+
| .  .  .  | .  #e .  | .  .  .  |
| .  .  .  | #d #d #d | .  #c .  |
| .  .  .  | .  #e .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  #a .  | .  #b .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+


For the strong link bc:

1. b is true. Then, a is false.

OR

2. b is false. Then, c is true, d is false, one of e is true, a is false.

Either way, a is false.

If the hinge box contains only two candidates, they are either

Code:
Diagram 2
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  #d .  | .  #c .  |
| .  .  .  | .  #e .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  #a .  | .  #b .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+

You do not need the strong link bc. The only occurrence of # in the hinge box is in the column; a is false (box-line interaction).

OR
Code:
Diagram 3
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  #d | .  #c .  |
| .  .  .  | .  #e .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  #a .  | .  #b .  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+

de is a strong link in the box. With cb it is a skyscraper (turbot fish, whatever).

At least one of d and e is false; at least one of e and b is true; a is false.

Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Wed Sep 26, 2007 5:00 am    Post subject: Reply with quote

Keith, thank you. I'd be lying if I said I follow this, but I will try and gradually digest it.
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Asellus



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

PostPosted: Wed Sep 26, 2007 5:39 am    Post subject: Reply with quote

Keith,

While it's true that in the usual simple direct ER elimination the 2-cell ER can be viewed alternately, there are other instances where the alternatives are not so readily available. For instance:
Code:
+----------+----------+----------+
| .  .  .  | .  .  .  | .  .  .  |
| .  .  .  | .  .  #d | .  #c #  |
| .  .  .  | .  #e .  | .  .  #  |
+----------+----------+----------+
| .  .  #  | .  #  .  | #  .  .  |
| #  .  #  | .  .  #  | .  .  .  |
| .  .  .  | .  #  .  | #  .  .  |
+----------+----------+----------+
| #  .  .  | .  .  .  | .  .  #  |
| #  .  #a | .  .  .  | .  #b #  |
| .  .  .  | .  .  .  | .  .  .  |
+----------+----------+----------+

Here, the ER effect is extended via three ERs in Boxes 2, 5 and 4. The "a" elimination is still valid. If Box 2 were not recognized as an ER, an opportunity would be missed.
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Asellus



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

PostPosted: Wed Sep 26, 2007 5:45 am    Post subject: Reply with quote

PS: In the grid I just posted, it is also possible to see the candidate at R5C1 eliminated via the strong link in C6 and the ERs in Boxes 3, 9 and 7. This isn't too surprising since either elimination implies the other.
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Asellus



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

PostPosted: Wed Sep 26, 2007 5:49 am    Post subject: Reply with quote

PPS: ERs are, admittedly, a bit confusing when first encountered. It takes a while to absorb the concept. But, once you get your head around them, you'll never be the same!!
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Asellus



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

PostPosted: Thu Sep 27, 2007 3:56 am    Post subject: Reply with quote

Here is another nice example of the use of an ER taken from today's (26-Sep-2007) Sudocue.net "Daily Nightmare (DM)":
Code:
+------------+-----------+-----------+
|  .   .   . |  .  .  .  |  .  .  .  |
|  .   .   . |  .  .  .  |  . x# x#  |
|  .   .   . |  .  .  .  |  .  .  .  |
+------------+-----------+-----------+
|  #   .   . |  .  .  .  |  #  .  #  |
|  .   .   . |  .  .  .  |  .  .  .  |
|  #   .   # |  .  .  .  |  . T# T#  |
+------------+-----------+-----------+
|  #   .   # |  .  .  #  |  #  .  #  |
| F#   .   . |  .  .  .  |  . x# x#  |
|  #   .   # |  .  .  #  |  #  #  #  |
+------------+-----------+-----------+

As above, # indicates cells that have the candidate digit of interest. (It is actually <9> in the "DM.") Notice the "Finned X-Wing"-like structure: the potential X-Wing is marked "x" and the Fin "F". This is useless as a regular Finned X-Wing because the Fin cannot "see" any of the cells that the X-Wing would eliminate. Or, can it?

As it happens, it can "see" the two cells marked T (for Target!) via the ER in Box 4. So, the candidate can be removed from these two cells!

For those who need a little more help seeing this: Either "F" is true or the X-Wing is true. If the X-Wing is true, then the two "T" cells are false (along with several other cells). On the other hand, if "F" is true, then R46C1 are false and R6C3 is true, which means that the two "T" cells are false. In either case, the "T" cells are false. Since there are only the two alternatives, the "T" candidates are eliminated.
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Asellus



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

PostPosted: Thu Sep 27, 2007 6:22 am    Post subject: Reply with quote

I had no idea I'd be posting yet again about ERs so soon. However, Sudocue.net is very obliging today (26-Sep-2007): The "One-Trick Pony" puzzle has another interesting application of ERs. Before this "Techniques" forum existed, I posted elsewhere about an "ER mirror," when a cell can see itself via a "loop" of ERs. But, it's nice to add an example to this forum. And, today's "Pony" provides a nice example:
Code:
+------------+-----------+-----------+
|  .   .   . |  .  #  .  |  .  #  .  |
|  .   .   . |  #  .  #  |  . a#  #  |
|  .   .   . |  .  .  .  |  .  .  .  |
+------------+-----------+-----------+
|  .   #   . |  .  #  #  |  .  .  .  |
|  #   .   . |  # b#  #  |  .  .  #  |
|  .   #   . |  #  #  #  |  .  #  .  |
+------------+-----------+-----------+
|  .   .   . |  .  .  .  |  .  .  .  |
|  #  c#   . |  #  .  #  |  .  .  .  |
|  .   #   . |  .  #  .  |  .  .  .  |
+------------+-----------+-----------+

The grid above shows the pattern for the digit <1>. Now, there is a Finned X-Wing and maybe other things that I'm going to ignore for the purpose of this post. Instead, I'm going to deal with the three cells marked "a", "b" and "c".

Starting at "a", you can trace the ER effects via Boxes 6, 4, 7, 8 and then 2 and then back to "a". [In detail: C7 hinges to R5 in Box 6, then R5 hinges to C2 in Box 4, then C2 hinges to R8 in Box 7, then R8 hinges to C5 in Box 8, then C5 hinges to R2 in Box 2, bringing us back to R2C7.] So, cell "a" can "see itself" via the "ER mirror" and can thus be eliminated.

A similar "ER mirror" cycle eliminates the candidate at "c". For "b", you have two choices: Boxes 2, 3, 6 or Boxes 4, 7, 8. Either way, "b" is gone.
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Johan



Joined: 25 Jun 2007
Posts: 206
Location: Bornem Belgium

PostPosted: Sun Dec 02, 2007 12:40 am    Post subject: Reply with quote

This puzzle can be solved with an UR Type 1 or a remote [34] pair.

Exercising ER's for several weeks, I always try to spot an ER after basics,

improving the technique to locate those ER's.

According to ER-logic i found a kind of bivalue-ER, for digits <3> and <4> in Box 7, they both share the same ERI in R9C3 in Box 7, and the strong link* on <3> and <4> in R6, eliminating <3> and <4> in R9C9.

The question is : ER's only just eliminate one single digit, in this case two, so can this elimination be defined as ER-logic or is this something else?
Code:

The original puzzle :

+-------+-------+-------+
| . 3 . | . . . | . 8 . |
| . . 1 | 8 . 2 | . . 9 |
| 2 . . | . . . | . . 6 |
+-------+-------+-------+
| . 9 . | 1 . 8 | . 6 . |
| . 6 . | . 3 . | . 1 . |
| . 8 . | 5 . 7 | . 2 . |
+-------+-------+-------+
| 9 . . | . . . | . . 2 |
| 8 . . | 4 . 3 | 1 . . |
| . 2 . | . . . | . 5 . |
+-------+-------+-------+


The puzzle after basics :


+--------------------------+--------------------------+-------------------------+
| 5         3         9    | 6           7       4    | 2         8        1    |
| 6         47        1    | 8           5       2    | 34        347      9    |
| 2         47        8    | 3           9       1    | 5         47       6    |
+--------------------------+--------------------------+-------------------------+
| 37        9         2    | 1           4       8    | 37        6        5    |
| 47        6         5    | 2           3       9    | 478       1        48   |
| 1         8        *34   | 5           6       7    | 9         2       *34   |
+--------------------------+--------------------------+-------------------------+
| 9 er      1 er      34   | 7           8       5    | 6         34       2    |
| 8 er      5 er      6    | 4           2       3    | 1         9        7    |
| 34        2     eri 7    | 9           1       6    | 348       5      -[34]8 |
+--------------------------+--------------------------+-------------------------+
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nataraj



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

PostPosted: Sun Dec 02, 2007 9:47 am    Post subject: Reply with quote

For me, the "empty rectangle" is just a rather graphic way of describing the concept of grouped links or grouped AICs. The reasoning is very similar to skyscraper/kite/turbot fish but at one point (the "empty rectangle") the weak link/strong link combination normally provided by single candidates is achieved by groups of candidates that follow the same logic:
if cell a = x (one end of the strong link in an ER) then
cell group b<>x (the cell(s) "seen" by a, lets say the column inside the box with the ER) then
cell croup c must contain x (the other cells in the box, i.e. the row) then (because cells c are aligned)
cell d (the target) cannot contain x

There is a thread with a more in-depth discussion about this here: http://www.sudoku.com/boards/viewtopic.php?t=3438 and a thread with nice graphics here: http://www.sudoku.com/boards/viewtopic.php?p=17612


___

edit 1024 GMT
P.S. I just realized Asellus already posted about the connection between ER and grouped AICsthe other day. (http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=2164&start=20)
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