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

Posted: Sat Apr 15, 2006 12:53 pm Post subject: DB Saturday Puzzle  April 15, 2006 


Here is today's David Bodycombe puzzle. A good one to do with pencil and paper.
Code: 
Puzzle: DB041506 ******
++++
 . 2 .  5 . .  . 7 4 
 . 7 3  . . .  . . . 
 4 . .  . . .  5 . 3 
++++
 9 . .  2 . .  3 . . 
 . 4 .  . 9 .  . 6 . 
 . . 8  . . 4  . . 1 
++++
 8 . 5  . . .  . . 6 
 . . .  . . .  2 3 . 
 7 1 .  . . 3  . 9 . 
++++

See if you can find the Unique Xwing, and figure out what to do with it! (Also known as a Type6 Unique Rectangle, it is an Xwing which is also a Unique Rectangle.)
Best wishes,
Keith 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Sat Apr 15, 2006 6:03 pm Post subject: 


Quote:  See if you can find the Unique Xwing, and figure out what to do with it! (Also known as a Type6 Unique Rectangle, it is an Xwing which is also a Unique Rectangle.) 
After the usual stuff, I used coloring to eliminate an "8." In thinking back, I don't think it contributed to the solution, but that's OK, I don't get to use coloring that often.
Then I found a rectangle. I never heard of a "Unique XWing" or a "Type6 Unique Rectangle." I don't think I could have done anything with it as an XWing, but as a Unique Rectangle I was able to solve a cell outside of the rectangle, and that opened things up.
Fun stuff. 

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

Posted: Sat Apr 15, 2006 8:11 pm Post subject: The details 


Code: 
Puzzle: DB041506 ******
++++
 . 2 .  5 . .  . 7 4 
 . 7 3  . . .  . . . 
 4 . .  . . .  5 . 3 
++++
 9 . .  2 . .  3 . . 
 . 4 .  . 9 .  . 6 . 
 . . 8  . . 4  . . 1 
++++
 8 . 5  . . .  . . 6 
 . . .  . . .  2 3 . 
 7 1 .  . . 3  . 9 . 
++++

After the usual standard moves (actually, only forced and pinned squares, or naked and hidden singles), you get to:
Code: 
++++
 1 2 9  5 3 8  6 7 4 
 5 7 3  46 246 26  1 8 9 
 4 8 6  79e 1 79f  5 2 3 
++++
 9 5 17  2 68 16  3 4 78 
 3 4 17  18 9 5  78 6 2 
 2 6 8  3 7 4  9 5 1 
++++
 8 3 5  479g 24i 279h 47j 1 6 
 6 9 4  178 58a 17  2 3 578c 
 7 1 2  468k 4568b 3  48 9 58d 
++++

Unique Rectangles:
You cannot have a solution where the corners of a rectangle contained within two blocks are
X Y
Y X
because this is not unique. You can interchange X and Y and still have a valid solution.
It follows that you cannot have the possibilities
XY XY
XY XY
as the corners of a rectangle contained within two blocks. Either of these (the solution or the possibilities) is called the "Deadly Pattern", to be avoided.
There are two unique rectangles in the position above, and both can be solved with reasoning involving "strong links" or "conjugate squares".
Rectangle 1:
Look at the cells labeled abcd. The Deadly Pattern is <58>. However, abcd is also an Xwing on <5>. Either the diagonal squares ad are both <5>, or bc are both <5>. If bc are <5>, ad must be <8>, and there is a Deadly Pattern. So, bc must both be <5>.
Rectangle 2:
Look at the cells labeled efgh. The Deadly Pattern is <79>. Looking only at the rectangle, at least one of gh cannot be <7> or <9>.
Now, look at the other squares in R7 and B7. One of gh must be <9>, there are no other possible squares to hold <9>. So, neither of gh can be <7>. Then, there are two paths:
1. j must be <7> (in R8), etc.
2. gih are now a triple in <249> (in B8), and we can eliminate <4> from b and k.
I had fun with this one!
Best wishes,
Keith 

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

Posted: Sat Apr 15, 2006 8:22 pm Post subject: Coloring on 8 


Marty,
Your coloring removes the <8> from square "b" in my previous post, thus destroying the first (Type6) Unique Rectangle.
The Rectangle logic is cool, but placing the last two <5>'s in this puzzle is not immensely helpful.
Keith 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Sat Apr 15, 2006 9:34 pm Post subject: 


Keith,
Correct, the coloring removed the "8" from cell b, thus destroying the rectangle. It's questionable whether I would have otherwise noticed it, and it's almost certain I wouldn't have found that the deadly pattern remained if one of the choices was used. My mind is not yet trained in reasoning out that configuration.
As to the 79 rectangle, it was immediately apparent that one of the corners had to be a "2" or a "4", but I thought it wasn't enough to do anything. But then when I noticed the "9s" were locked, I realized the "7s" had to go, after which I noticed the single "7" before I saw the newly formed 249 triple.
As an aside, did I read recently that the principles of Unique Rectangles don't apply if the corners occupy four boxes, i.e., the corners must be in only two boxes?
P.S. After posting the above, I noticed in your explanation you made reference to two boxes, so I guess that means the answer to my question must be "yes." 

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alanr555
Joined: 01 Aug 2005 Posts: 193 Location: Bideford Devon EX39

Posted: Sun Apr 16, 2006 9:39 am Post subject: 


Code: 
> As an aside, did I read recently that the principles of Unique
> Rectangles don't apply if the corners occupy four boxes, i.e.,
> the corners must be in only two boxes?
> P.S. After posting the above, I noticed in your explanation
> you made reference to two boxes, so I guess that means
> the answer to my question must be "yes."
CARE is needed here.
If the rectangle is a TRUE rectangle
(ie involving only two columns and two rows)
then it does not matter in how many boxes it is contained.
However, if there are other relationships involved (as may
be the case with only two boxes involved) then greater
care is needed and, indeed, the principles of the unique
rectangle may apply even to trapezoids or other
quadrilaterals  but it is likely that "Strong Links" will
be evident in such cases so that rectangle principles
are highly unlikely to be first or second port of call.
Essentially, if there is a constraint of a unique solution,
an "interloper" must be part of the solution if exclusion
of such interloper would leave a "deadly" rectangle
with the same pair of possibilities at each vertex.
If there is more than one interloper, then at least one
of them must be retained but further inspection will
be needed to determine which one.
Alan Rayner BS23 2QT



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

Posted: Sun Apr 16, 2006 12:02 pm Post subject: The vertices must lie in just two boxes 


Alan R wrote:  If the rectangle is a TRUE rectangle (ie involving only two columns and two rows) then it does not matter in how many boxes it is contained. 
I don't think this is right, Alan.
Suppose the "vertices" lie in four separate boxes. Just to make it concrete let's suppose that the "pair" appearing at each vertex is {1, 2}.
What happens if we reverse the values around the "rectangle"? Well, in box 1 we remove a "1" and insert a "2". In another box we remove a "2" and insert a "1".
After this transformation box 1 contains two "2"s, and the second box contains two "1"s. That's not a valid solution, so the vertices of the "nonuniqe rectangle" cannot lie in four different boxes. dcb 

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Marty R.
Joined: 12 Feb 2006 Posts: 5160 Location: Rochester, NY, USA

Posted: Tue Apr 18, 2006 3:55 pm Post subject: 


Coincidentally, just yesterday I finished a puzzle that had a fourbox rectangle. Note the "deadly pattern" at the four corners. This puzzle brought into focus for me the whole rectangle issue and illustrates what David is saying. Sorry about the nongrid.
Code:  291645873
587329146
463817295
615983724
832754619
749162358
928576431
154238967
376491582 


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