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A new "Unique Rectangle" Variant??

 
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keith
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Joined: 07 Feb 2006
Posts: 35
Location: near Detroit, Michigan, USA

PostPosted: Thu Mar 30, 2006 11:57 pm    Post subject: A new "Unique Rectangle" Variant?? Reply with quote

Ruud,

What do you think of this? It is a message I posted on Daily Sudoku. Although I do not use your SudoCue solver, and so do not know how you would solve it, no one has claimed that they can solve this puzzle with the logic given below

Best wishes,

Keith

===========================.

There is a method to solve this puzzle without chains:

Code:


2 5 1 |7 4 8|9 6 3
3 9 4 |6 5 2|7 18 18
6 7 8 |1 3 9|4 5 2
--------+-------+--------
5 26 267|9 8 3|12 4 17
1 8 3 |4 2 7|5 9 6
9 4 27 |5 1 6|28 3 78
--------+-------+--------
8 3 69 |2 679 4|16 17 5
4 26 5 |8 67 1|3 27 9
7 1 269|3 69 5|68 28 4




First, there is an X-wing on <9>. The corners are R7C3, R7C5, R9C3, and R9C5. This does not lead to any eliminations, but note that either R7C3 and R9C5 are <9>, or R7C5 and R9C3 are <9>.

Second, there is a possible non-unique rectangle on <69> lurking in the same squares as the X-wing. So, R7C5 and R9C3 CANNOT be <9>, for then R7C3 and R9C5 must be <6>, and there will be a non-unique rectangle.

So, R7C3 and R9C5 must be <9>.

Finally, to solve the puzzle, there is a BUG pattern which forces R4C3 = <2>.

Pretty cool, I think! I have not seen this variation of a unique rectangle described before.

Best wishes,

Keith
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Ruud
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Joined: 30 Dec 2005
Posts: 601

PostPosted: Fri Mar 31, 2006 12:38 pm    Post subject: Reply with quote

Very smart use of the Unique Rectangle.

I will introduce this method on the sudoku player's forum, to be added as uniqueness type 6, if you don't mind.

Ruud.
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keith
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Joined: 07 Feb 2006
Posts: 35
Location: near Detroit, Michigan, USA

PostPosted: Fri Mar 31, 2006 10:05 pm    Post subject: Please post Reply with quote

Ruud,

I would be delighted if you posted this on other forums. Please let me know the links. (I could not find a forum I thought was still active and current.) If this is indeed new, I will leave the classification up to the experts.

I think there are two observations:

1. There are useful variations of Unique Rectangles in which there is a "Diagonal Pair". For example, consider the following two blocks:

Code:


+--------------+
|  x   -    -  |
| 46   -   246 |
|  x   -    -  |
+--------------+
| 246  -   46  |
|  -   -    x  |
|  -   -    x  |
+--------------+



To avoid a non-unique rectangle, one of the squares with possibilities <246> must be <2>. Therefore, the squares labelled "x" cannot be <2>. This is comparable to a "Type-2 Unique Rectangle" (at least as I understand the explanation in the Susser manual of 12/31/2005).

2. If the "Diagonal Pair" defines a possible non-unique rectangle, and if that rectangle is also an X-wing on one of the components of the pair, then the diagonal pair must have the value of that component. This is the point of my original example. In the above example, if the rectangle is also an X-wing on <4>, the top left and bottom right squares must be <4>, i.e.,

Code:


+------------+
| -   -   -  |
| 4   -   26 |
| -   -   -  |
+------------+
| 26  -   4  |
| -   -   -  |
| -   -   -  |
+------------+



Best wishes,

Keith Meintjes
Waterford, MI, USA
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Ruud
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Joined: 30 Dec 2005
Posts: 601

PostPosted: Fri Mar 31, 2006 10:43 pm    Post subject: Reply with quote

I posted it here, where most new techniques are discussed: http://www.sudoku.com/forums/viewtopic.php?t=3709

Your last observation is already known as type 5 (or as a variety of type 2). It is not yet available in Sudoku Susser.

Lots of interesting threads on this forum.

Ruud.
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