Unique Rectangles: The Essentials

Advanced methods and approaches for solving Sudoku puzzles

Diagonal Type 3

Postby keith » Sun Jun 24, 2007 7:15 pm

Danny,

Dusting the cobwebs from my mind ...

Editing your diagram, I think a diagonal Type 3 is:
Code: Select all
[cell #]<>34
 +---------------+
 |  12   . 123   |
 |   #   .   .   |
 |  34   .   .   |
 |---------------+
 |   .   .   .   |
 |   .   .   .   |
 | 124   .  12   |
 +---------------+

My exposition is entirely theoretical. I think you will agree with the above. Whether you ever find it in practice (and then find it useful) is another question.

I recall Mike Barker was on an unsuccessful hunt for a useful Type 5, which is a diagonal Type 2. Not yet ever found in the wild, so far as I know.

http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?p=2894#p2894

Best wishes,

Keith
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Re: My First Example

Postby keith » Mon Dec 31, 2007 6:43 am

re'born wrote:
keith wrote:
If you make these reductions, R1C9 is pinned to be <2>, and the puzzle falls apart.



Of course all of your reductions are correct and make nice examples, but only the first (removing 2 from r1c1) is needed to pin r1c9 to 2.


re'incarnated? re'surrected? Posted May 30, 2006, joined May 31, 2007.

A year and a day. Seems almost Biblical.

Keith
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re: rep'nA / re'born

Postby Pat » Mon Dec 31, 2007 11:58 am

keith wrote:re'incarnated? re'surrected? Posted May 30, 2006, joined May 31, 2007.

A year and a day. Seems almost Biblical.


rep'nA was mysteriously deleted

re-registered as re'born
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Re: The Guide

Postby Jean-Christophe » Wed Apr 16, 2008 12:55 pm

keith wrote:Unique Rectangles: The Essentials

...

Section 4: Suppose we have the following:
Code: Select all
+-------------+
|  #   -   -  |
| 12   -  123 |
|  #   -   -  |
+-------------+
| 123  -  12  |
|  -   -   #  |
|  -   -   #  |
+--------------+


...

There is an interesting diagonal variation, sometimes called a Type 6 UR. It was first noticed as the overlay of an X-wing on a UR. Consider the pattern of Section 4, above, and suppose the rectangle is also an X-wing on <1>. The result must be:
Code: Select all
+--------------+
|  -   -    -  |
|  1   -   23  |
|  -   -    -  |
+--------------+
| 23   -    1  |
|  -   -    -  |
|  -   -    -  |
+--------------+


The bottom right cell cannot be 2, because that would force the deadly pattern. (In fact, you do not need the full X-wing to make reductions. This will be explored in the examples.)


Hi,

I have problems understanding Type 6. For me, the reasoning is flawed.
It basically says something like "this cell (in the UR) cannot be 2 because it would give a dual solution". But if the cell is 2, then there is no dual solution anymore, because it breaks the UR. In other words, I do not see any other solution with the cell = 2
Because of the X-Wing on 1, the solution could be
Code: Select all
+-------------+
|  -   -   -  |
|  2   -   1  |
|  -   -   -  |
+-------------+
|  1   -   2  |
|  -   -   -  |
|  -   -   -  |
+-------------+

I see nothing wrong here.

What did I miss ? Can someone show me the two solutions with the cell = 2 ?

In your real example, this pattern cannot work because of some other reason (6 at c7 locked at r79c7), but not because of the UR and X-Wing alone

TIA
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Re: The Guide

Postby eleven » Wed Apr 16, 2008 2:28 pm

Jean-Christophe wrote:But if the cell is 2, then there is no dual solution anymore, because it breaks the UR.
The point is, that if there is a solution with this pattern for the given puzzle (and none of the numbers was given), then switching the numbers in the solution would always give another solution. Of course you would not reach it after setting the cell to 2, but it would exist, if this pattern would be valid.
IOW if setting it to 2 is valid for the puzzle, then also setting it to 1.
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Re: The Guide

Postby daj95376 » Wed Apr 16, 2008 2:38 pm

Jean-Christophe wrote:I have problems understanding Type 6. For me, the reasoning is flawed.
It basically says something like "this cell (in the UR) cannot be 2 because it would give a dual solution". But if the cell is 2, then there is no dual solution anymore, because it breaks the UR. In other words, I do not see any other solution with the cell = 2
Because of the X-Wing on 1, the solution could be <snip>

I see nothing wrong here.

What did I miss ? Can someone show me the two solutions with the cell = 2 ?

I have to think ahead to the final solution in order to follow UR contradictions. If the following pattern ends up in your solution, and none of the cells were a given, then I could come along and say that swapping the 1s and 2s is also a solution. Not good!

Code: Select all
+-------------+
|  -   -   -  |
|  2   -   1  |
|  -   -   -  |
+-------------+
|  1   -   2  |
|  -   -   -  |
|  -   -   -  |
+-------------+

The only way to keep this from happening is to reduce the UR Type-6 cells to:

Code: Select all
+--------------+
|  -   -    -  |
|  1   -   23  |
|  -   -    -  |
+--------------+
| 23   -    1  |
|  -   -    -  |
|  -   -    -  |
+--------------+

The resulting 23 cells can't have the same value. It's possible to capitalize on this constraint if you ever generate chains using either of these cells!

Note: eleven made his reply while I was composing mine. I'm still going to leave my message posted.
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Postby Jean-Christophe » Wed Apr 16, 2008 3:36 pm

Thanks for the reply.

I think I got the point, but my poor little brain need rest now ;)

PS: While investigating URs, I found a "UR hidden pair" which of course as its complementary naked "heptuplet" as a UR Type 3.
Although it's not really new, I believe it could be easier to spot a UR hidden pair than two ALS with seven candidates. Not sure it's very common, thought.
An example r1c12 = {12}, {12} at r4 locked in r4c125. Since r4c12 cannot have both 1 and 2 -> r4c5 = {12}, r4c12 must include one of {12}
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Postby eleven » Wed Apr 16, 2008 4:02 pm

My way to see the same is, that in one of the two UR rows (or columns,blocks) there must be a 1 or 2 outside the UR cells. In your sample the only possibility is r4c5 for both numbers, so 1 or 2 must be there.
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